This multimodal stewardship intervention was associated with a decrease in antibiotic exposure for patients with COVID-19 (44.4% vs 61.8%,  Understanding that viral RTIs are more common than bacterial, empirically treating all patients with antibiotics leads to unnecessary antibiotic exposure, increasing risk of antibiotic resistance, cost, and adverse events. This was common practice with the emergence of COVID-19, however, studies conducted early in the pandemic found antibiotics were prescribed in 56%–72% of patients despite low bacterial co-infections (3%–8%). With this information, judicious antibiotic prescribing became a stewardship imperative.Respiratory tract infections (RTI) are a leading cause of hospitalization and death in the United States. Modeled after this success, we replicated the process for patients with COVID-19. We concurrently implemented targeted prospective audit for patients with COVID-19 to identify opportunities to de-escalate antibiotics, representing a combined active and passive intervention.Previous internal research demonstrated that implementation of an electronic medical record (EMR)-based provider best practice alert (BPA) significantly decreased antibiotic days of therapy.This was a pre-post, multi-site study at 7 hospitals within Saint Luke’s Health System (SLHS) and received investigational review board waiver approval. All data points were collected via chart review from the EMR, EPIC .In November 2020, an automated BPA was implemented for all system hospitals. This alert fired upon any provider opening a patient chart when alert criteria were met encouraging antibiotic de-escalation with documented viral illness Figure . The alePatients were included in the study if they were 18 years or older and had a PCR-confirmed COVID-19 infection on admission. Patients were excluded if they transferred from outside of SLHS, the primary admission diagnosis was not COVID-19, or required antibiotics for a non-respiratory indication. The preintervention group included patients admitted in October 2020 and the postimplementation group included those admitted in January 2021.Clostridioides difficile infection, re-initiation of antibiotics after discontinuation for any cause, percent of discharge antibiotic prescriptions, and inpatient mortality. Bacterial co-infections were confirmed based on positive blood or respiratory culture results. Blood and respiratory cultures were considered relevant if they were a possible respiratory pathogen and treatment was pursued based on provider documentation. A preplanned subgroup analysis assessed patients who had a BPA fire in the postimplementation group compared to patients in the preimplementation group who would have also met criteria.The primary endpoint was percent of patients prescribed antibiotics within the first 7 days of hospitalization. Secondary endpoints include duration of antibiotic therapy (days), antibiotic days of therapy, length of hospital stay, bacterial co-infections on admission, n (%) for categorical data and evaluated using chi-square tests. Data for the primary outcome was non-normally distributed data so we used the non-parametric alternative tests. All analysis was done with SAS 9.4 .Data are shown as mean +/− SD or median (IQR) for continuous data and compared between pre- and postintervention cohorts using either the Student’s T-test or Wilcoxon Test. Data are shown as p = 0.039). There was no difference in confirmed bacterial co-infections . For the primary outcome, significantly fewer antibiotics were prescribed in the postintervention group in the first 7 days . For secondary endpoints, the duration of antibiotic therapy within the first 7 days of hospital admission  and total antibiotic days of therapy  were significantly lower in the postintervention group. There was no difference between groups among other secondary endpoints (Table n = 33) vs those in the preintervention group that would have met criteria for the BPA (n = 64) had a lower duration of antibiotic therapy  and antibiotic days of therapy .This study included 298 patients . Baseline characteristics were similar between groups except for a higher postintervention Charlson Comorbidity Index score  of patients receiving antibiotics within 7 days of admission, duration of antibiotic therapy, and antibiotic days of therapy. Similar to previous studies, we found low rates of bacterial co-infections.Limitations of our study included 2 comparator groups during 2 separate waves of the COVID-19 pandemic, which may have led to unforeseen confounding factors, including clinical management standards, which continuously evolved throughout the entirety of the pandemic. Given the rapidity of practice change, there was a concern that even greater bias may have been introduced if we had not compared populations close in time. Second, it is possible that antibiotic prescribing may have been impacted by factors outside of formal stewardship recommendations and/or BPAs leading to decreased prescribing. Third, the BPA was fired based on initial PCT results on admission, possibly at a time when PCT may not have peaked. However, we did assess antibiotic re-initiation and there was no difference between groups. Additionally, the study was limited to the first 7 days of inpatient admission and does not characterize subsequent effects on antibiotic use. Finally, we did not differentiate, nor stratify outcomes based on severity of illness.Our study also had strengths. First, we allowed for an adaptation period between initiation of the BPA and more active ASP efforts in November 2020 to post-implementation data collection. This may have improved the clinical discernment related to BPA agreement when it triggered and/or ASP recommendations over time. Second, we limited the study population to only those with potential respiratory co-infections to provide a more accurate representation of attributable antibiotic use in the COVID-19 population, without skewing our results with other unrelated infections. Third, we replicated this practice after previous research demonstrating success using similar BPA strategies. These strategies could be replicated by other health systems to improve antimicrobial stewardship.This study suggests that a multimodal antimicrobial stewardship intervention effectively decreased antibiotic use in patients with COVID-19. We found success leveraging a passive approach using BPAs and active ASP pharmacist efforts to further impact prescribing. However, there are multiple mechanisms by which decreasing antibiotic prescribing can be targeted. One study found success by creating an ASP-developed institutional guideline. Compared to our study, they found similar success in reducing antibiotic prescribing (32.5%) and antibiotic duration (1.3 d), highlighting the importance of institution-specific process design.For newly admitted patients with COVID-19, our study found leveraging ASP pharmacists and EMR alerts is associated with decreased antibiotic exposure leading to enhanced patient care.


To investigate the influence of polypropylene–basalt hybrid fibers (PBHFCC) on the durability of ceramsite concrete, this study determined the appearance change, mass loss rate, relative dynamic elastic modulus, compressive strength and splitting tensile strength of ceramsite concrete with four kinds of hybrid fibers volume admixture under chloride erosion and dry–wet cycles. The results reveal that under this effect, the apparent damage of each group of specimens increased with the growth of the erosion time. The quality, compressive strength and splitting tensile strength of the specimens all increased gradually during the erosion age period of the first 72 d and gradually decreased after 72 d. The relative dynamic elastic modulus was similarly mutated in 48 d. When the hybrid fiber content of the specimens is 0.15 vol %, the enhancement effect of ceramsite concrete is better than that of the other three amounts. The relative dynamic elastic modulus value is used as a damage variable to establish the damage equation, and the damage evolution equation of PBHFCC considering the volume of hybrid fiber under chloride erosion and dry–wet cycle is derived. The conclusions can be used as a reference for the durability design and construction of PBHFCC. Ceramsite concrete is a kind of light aggregate concrete made of ceramsite granule, ceramsite sand and cementitious materials, which has the advantages of high strength, lightweight, low elastic modulus and good sound absorption and noise reduction performance . In ordeWith the development of fiber-reinforced concrete, hybrid fiber-reinforced concrete has gradually become a research focus. Hybrid fiber-reinforced concrete is concrete in which two or more fiber-reinforcing materials are added to the cement matrix in a reasonable proportion, which can play the advantages of a single fiber, but also reflects the synergistic effect between the fibers. Therefore, hybrid fiber-reinforced concrete is a new type of composite material with synergistic effect ,7,8, whiIn recent years, the research and use of basalt fiber and polypropylene fiber has gradually increased. Jiang et al.  state thMost of the existing studies mainly focus on the enhancement law of hybrid fibers on the mechanical properties of concrete under normal conditions. When located in the range of frequent water-level fluctuations such as tidal and wave splash zones in marine or salt lake areas, concrete is subjected to both dry–wet cycles and chloride erosion ,19,20. ATherefore, it is of great significance to study the durability of PBHFCC. Relevant scholars have studied the durability of different types of fiber-reinforced concrete under chloride erosion and dry–wet cycle. Chen et al.  studied In this paper, polypropylene fiber and basalt fiber were composed of hybrid fibers. After several trials, the hybrid ratio of polypropylene fiber:basalt fiber = 1:1 was selected. Four kinds of ceramsite concrete with hybrid fiber volume content were prepared. The changes in appearance, mass damage, relative dynamic elastic modulus, compressive strength and splitting tensile strength of each group of PBHFCC under a chloride dry–wet cycle were measured, and the damage model of PBHFCC under a chloride dry–wet cycle was established with the change in relative dynamic elastic modulus as the index. The research results can provide a reference for the engineering application of PBHFCC in coastal areas.(1)Cement: Huaxin brand P.O.42.5 ordinary Portland cement. The technical indicators are shown in (2)Coarse aggregate: Yichang Guangda brand 900-grade high-strength gravel–shale ceramsite, performance parameters are shown in (3)3, fineness modulus is 2.91.Fine aggregate: ordinary river sand (medium sand), apparent density is 2.54 g/cm(4)Fiber: polypropylene fiber and basalt fiber. The performance parameters are shown in (5)Water-reducing agent: HSC polycarboxylate high-performance water-reducing agent, the solid content of the water-reducing agent was 20%.(6)Water: tap water.The volume addition rate of polypropylene fiber is 0.1 vol %, and the volume addition rate of basalt fiber is 0 vol %, 0.05 vol %, 0.1 vol % and 0.15 vol %, respectively. Four kinds of mix proportion specimens were made for pre-experiment, and the cube compressive strength, splitting tensile strength and prism axial compressive strength of the four kinds of mix proportion specimens were tested. The results showed that when the volume addition rate of polypropylene fiber and basalt fiber is 0.1 vol %, the above three kinds of mechanical strength reach the maximum. Therefore, polypropylene fiber and basalt fiber were mixed at a ratio of 1:1. Three kinds of PBHFCC of 0.10 vol %, 0.15 vol % and 0.20 vol % were used for test-mixing by the absolute volume method. According to the 28 d compressive strength and 28 d splitting tensile strength measured by each test group, the optimal mix ratio was selected for the test. The specific mix ratio parameters are shown in The chloride corrosion test of PBHFCC was carried out under the condition of a dry–wet cycle. A 3.5 vol % NaCl solution was used as an erosion solution. Referring to the dry–wet cycle test in the References ,29,31, tThe grouping of specimens is shown in The relative dynamic elastic modulus is one of the important indexes of the mechanical properties of materials, which can reflect the response ability and deformation degree of concrete under external loading. The model established by using the dimensionless index of the relative dynamic elastic modulus can predict and evaluate the overall mechanical properties of PBHFCC, so as to better understand the behavior and performance of PBHFCC.The appearance of the test piece at 0 d, 120 d (15 cycles) and 240 d (30 cycles) under the interaction of dry–wet cycles and 3.5% NaCl solution corrosion are, respectively, shown in The mass loss rate of PBHFCC specimens under the combined action of dry–wet cycle and 3.5 vol % NaCl solution is shown in The relative dynamic elastic modulus of PBHFCC specimens under the combined action of dry–wet cycle and 3.5 vol % NaCl solution is shown in The compressive strength of PBHFCC specimens under the combined action of dry–wet cycle and 3.5 vol % NaCl solution is shown in It can be seen from the above analysis that the addition of hybrid fibers of 0.15 vol % had the most obvious effect on improving the compressive strength of ceramsite concrete, followed by the addition of hybrid fibers of 0.10 vol %. The negative effect of hybrid fiber appears in the hybrid fiber specimens with 0.20 vol %, which makes the compressive strength of concrete lose more, and the compressive strength is lower than that of the specimens without fiber.The splitting tensile strength of PBHFCC specimens under the combined action of dry–wet cycle and 3.5 vol % NaCl solution is shown in − reacts with C3A in concrete to form expansive crystalline products such as Friedel’s salt , f(t) ≥ 0;1 = t2 = t12, f(t1) = f(t2) = f(t12);When t12, f′(t) > 0; When t12 ≤ t2 ≤ tmax, f″(t2) < 0.When 0 ≤ t ≤ tEquation (1) satisfies the following boundary conditions:Among them, f′(t) represents the speed of concrete damage deterioration, and f″(t) represents the acceleration of concrete damage deterioration.rd:Derivation of EThe equation is derived twice:Among them, 2C represents the damage acceleration of concrete after entering the second stage, that is, the concrete accelerates the damage uniformly with the initial speed of D and the acceleration of 2C. The greater the 2C, the faster the damage rate, and the more serious the damage and deterioration of concrete.The data of relative dynamic elastic modulus are substituted into Equation (1) for fitting, and the fitting results are shown in 12 of the GC-2 group is the smallest, followed by the GC-1 group, and t12 of the GC-0 group is the largest. This is because the addition of hybrid fibers can reduce the pores of concrete. The expansion crystallization products generated by the specimens during the dry–wet cycle are faster than the specimens without hybrid fibers at filling the pores of concrete, which makes the damage and deterioration of PBHFCC enter the second stage faster.Among the four groups of specimens, tIn order to analyze the influence of mixed fiber content on the chloride corrosion resistance of concrete, the influence coefficient K of mixed fiber is introduced. It can be seen from the table that the coefficients of the equation increase first, then decrease and then increase with the increase in fibers. Therefore, the cubic polynomial is used to characterize the effect of the volume of mixed fibers in chloride environment on the damage evolution equation of concrete.SetIn the equation, K is the influence coefficient of the volume content of hybrid fiber on the damage evolution equation of concrete, X is the content of hybrid fiber, a, b, c, d are the test-fitting constants. The influence factors are as follows.2 − 105.2403X3Coefficient A: k = 0.17927 + 0.29976X + 21.56864X2 − 0.20436X3Coefficient C: k = 3.55004E − 4 + 0.01487X − 0.20436X2 − 221.0406X3Coefficient D: k = −0.31113 − 4.79672X + 66.71511X2 + 7379.3544X3Coefficient E: k = 125.44314 + 210.72335X − 2415.70692XTherefore, the damage evolution equation of concrete considering the volume content of hybrid fiber is as follows:(1)The durability of PBHFCC is affected by the number of erosion d and fiber content. The measured mechanical indexes increased first and then decreased with the increase in action time. An appropriate amount of fiber incorporation  can effectively improve the durability of ceramsite concrete, while too-high fiber incorporation (0.20 vol %) will produce negative effects of hybrid fibers and aggravate erosion damage.(2)The apparent damage degree of each group of specimens increased gradually with the increase in action time. The durability damage of the specimens with hybrid fiber of 0.20 vol % was the most serious, and the damage degree was even slightly larger than that of the specimens without hybrid fiber. The damage degree of the specimen with hybrid fiber of 0.15 vol % was lighter.(3)The mass of PBHFCC in each group increased gradually during the first 72 d of erosion, and decreased rapidly after 72 d. The mass loss rate of the specimens with hybrid fiber of 0.15 vol % was the smallest in each stage of the test.(4)The relative dynamic elastic modulus increased gradually during the first 48 d of erosion and decreased gradually after 48 d. The relative dynamic elastic modulus of the specimen with hybrid fiber of 0.15 vol % was larger than that of the other three groups of specimens with the same erosion age.(5)The compressive strength of PBHFCC in each group increased gradually in the first 72 d and then decreased gradually. When the erosion age is 72 d, the compressive strength and splitting tensile strength of the hybrid fiber specimen with 0.15 vol % volume content reach the peak, which are 66.2 MPa and 5.16 MPa, respectively. Compared with the specimens without hybrid fibers, they were increased by 12.2% and 14.2%, respectively.(6)Taking the relative dynamic elastic modulus value as the damage variable, the linear function is used to represent the rising stage in the early stage, and the quadratic polynomial is used to represent the falling stage in the later stage. The damage evolution equation of PBHFCC considering the volume content of hybrid fiber under the action of chloride erosion and dry–wet cycle is well fitted, which can provide reference for the engineering design of PBHFCC in coastal areas.In this paper, through the dry–wet cycle–chlorine salt erosion experiment of PBHFCC with different volume fiber content, the apparent damage degree, mass loss rate, relative dynamic elastic modulus, compressive strength and splitting tensile strength of PBHFCC are analyzed. The main conclusions are as follows:


This paper is devoted to the fastening system that consists of a number of anchors of approximately equal effective embedment depth, called “anchor group”, embedded into hardened concrete, used to transmit forces transverse to the anchors from an attachment to the concrete. The anchor group is far from the edges and is subjected to no more than marginal axial forces. Being post-installed, rather than cast-in, the embedded end of each anchor is not hooked, and no nuts, washers, or plates are attached to the embedded shaft. The paper focuses on the transverse forces that can be transmitted across an anchor group from an attachment to the concrete. The paper provides an analytical model for predicting the maximum (ultimate) shear force that an anchor group can bear, thus called “shear strength”. The model hence allows the structural designer to predict the shear strength of an anchor group post-installed into concrete. The model is based on five mechanical assumptions, which were established from a wide-ranging numerical analysis. Model predictions turned out to be, on average, 20% lower than the results of experiments performed on cast-in anchor groups borrowed from literature. The comparison verifies model accuracy, considering that the tested anchor groups benefitted from the extra-strength furnished by nuts and washers attached to the embedded shaft. Model predictions were also compared to code provisions; the former resulted to be up to one third of the latter. The paper presents and comments those comparisons, as well as all mathematical development. Applications of the model to wide-ranging case studies is presented and discussed as well. There has been much valuable work conducted in the field of adhesive and mechanical anchors intended for use in concrete, and concrete anchors have been the subject matter of many valuable papers. However, a detailed insight into those documents reveals that most of the papers are devoted to the anchor’s extraction strength, i.e., pull-out resistance ,5,6,7,8,Moreover, the phenomena that have been investigated for cast-in and post-installed shear anchors are premature failure modes, i.e., concrete edge failure, concrete splitting, anchor failure, concrete breakout, concrete pry-out failure, cyclic load failure, collapse due to prying actions ,54,55,56Eventually, shear anchors post-installed into hardened concrete have received less attention by research communities.Another lacuna in the literature is that almost all the studies have investigated the post-installed anchors from the experimental and numerical perspective ,64,65,66Ultimately, not one of the papers that are present in the published literature furnishes an analytical model to calculate the shear strength of anchors post-installed into hardened concrete, with adequate clearance to the edges.Anchors intended for use in concrete are dealt with by many codes, requirements, specifications, standards, provisions, and test methods, edited by organizations for technical assessment ,83,84,85Some of the above documents are entirely devoted to anchors, such as , which fMore specifically, shear strength of anchors post-installed into concrete that fail by pure shear (and not by a premature failure mode) is touched upon in Chapter 17 of American Concrete Institute (ACI) 318-14 . In factAnother important document that mentions shear strength of anchors post-installed into concrete that fail by pure shear is . AlthougIn conclusion, no analytical model can be borrowed, not only from scientific literature, but also from structural codes to reliably predict the shear strength of post-installed anchor groups .There are instead reports (instructions) edited by producers and manufactures devoted to giving technical provisions, which include formulas for predicting the shear strength of anchors post-installed into concrete structures. Some of those formulas are implemented into software for practitioners. Unfortunately, that software and the predictions of those formulas are unrealistic, as shown in As a result, the designer who wants to use shear anchors is obligated to perform on-site tests to directly measure the shear strength of the designed anchors. Unfortunately, measuring the shear strength of an anchor is labor intensive, and even more so for an anchor group. The reason is that the reaction of the shear force applied to the anchor is not directly resisted by the concrete structure (as it occurs in the pull-out test) but must be resisted by the concrete structure by means of a specific test set-up. In addition, on-site tests must be numerous in order to be statistically significant.The testing procedure performed by the designer has to comply with codes or instructions. Namely, the performance of post-installed anchors in concrete must be evaluated by a testing procedure that is recognized and accepted. A popular testing procedure is that presented in . That doHowever, the reality can be even more demanding. If the test results are lower than the expected strength, the design must be changed, which entails serious problems for the designer, the client (or the owner of the building), and construction activity, since design changes must be carried out at the construction site stage.It was due to exactly such a problem, found in his professional activity, that the author of this paper set up this research activity. This activity includes both single anchors and anchor groups, post-installed into either concrete or masonry. This paper is exclusively devoted to anchor groups post-installed into hardened concrete, while the other research topics are or will be published in other papers.Ultimately, this paper provides an analytical (closed-form) mathematical model for predicting the shear strength of an anchor group post-installed into a hardened concrete whose compressive strength is known, far from the edges. The shear strength predicted by this model must be compared to the shear strength of the anchor group dictated by the metal of the anchors and by the distance from the edges. The actual shear strength of the anchor group is the lowest of those strengths.The present paper deals with post-installed anchors intended for use in concrete. The anchors form an anchor group, which transmits loads to or receives loads from a structural assembly, external to the surface of the concrete, called “attachment”.The anchor group that is dealt with is sufficiently far from the edges of the concrete structure, so that the strength does not depend on the distance from the boundaries.The author specifies that, since he is not enthusiastic about invention and deployment of new acronyms, this paper does not use any acronym.aV. The sum of the shear forces aV that are applied to the anchors of the group is denoted by gV. Research activity is devoted to analytically predicting the maximum (ultimate) value that the shear forces gV can reach, which is called “shear strength” of the anchor group and is denoted by the symbol guV.Each anchor of the group is subjected to a transverse force applied to the end that protrudes from the concrete surface, i.e., a force orthogonal to the anchor. That force is called “shear force” and is denoted by the symbol aV is ever eccentric to the concrete surface, which implies some bending moment in the embedded anchor sections. Second, a hinge is never a pure pin, which implies some bending moment acting on the anchor group. Third, there is always some prying action, which implies further bending moment acting on the anchor group.The attachment that is here considered does not transmit bending moment to the anchor group, which implies that the connection between the anchor group and the attachment must be a hinge. That in theory. In reality, an attachment always transmits some bending moment together with the shear force. In fact, first, a forces aV with respect to the concrete surface . Namely, the model reproduces the bending moment induced by that eccentricity.The bending moment acting on the anchor group is totally different than the bending moment acting on a single anchor. Actually, the latter has the same effect as the shear force—namely, the bending moment acting on a single anchor adds contact pressures to those due to the shear force. So, the bending moment acting on a single anchor does not modify the resisting mechanism of the anchor, which continues to be provided by contact pressures exchanged between the concrete and the embedded system. The model presented here accounts for the eccentricity of the shear forces On the contrary, the bending moment acting on the anchor group  gives rise to a couple composed of a compressive force, which induces compressive contact pressures on the concrete surface, and a tensile force, which induces tensile axial forces in some anchors of the group. While the compressive contact pressures play no role in the shear strength, the tensile force impacts on the shear strength of the anchor.More specifically, tensile force in an anchor induces longitudinal shear stresses at the interface between concrete and anchoring material (or anchor), whose ultimate resultant is the extraction force (pull-out resistance).Eventually, the bending moment acting on the anchor group due to the fact that the hinge is not a pure pin and due to some prying action, implies a resisting mechanism of the anchor group that combines shear strength and axial strength.Non only does the prediction of the extraction force (pull-out resistance) require a specific model, but also there is a substantial interaction between the shear strength and the pull-out strength. The effects induced by the shear force cannot hence be superimposed to the effects induced by the axial force. So, that anchor must be modeled under combined shear and axial loading.However, the point is not the prediction of the strength when the anchor group is simultaneously subjected to shear force and substantial axial force, but the optimal design of an anchor group. The interaction between shear force and axial force drastically reduces the strength of an anchor. Thus, design should minimize the bending moment acting on the anchor group. Namely, the connection should be designed to behave as close as possible to a pin and to reduce prying action as much as possible. Ergo, an optimally designed connection transmits no more than moderate bending moment to the anchor group, so that the axial force introduced into every anchor is marginal.This paper refers to optimally designed connections. Accordingly, the anchors that this paper is devoted to, are subjected to substantial shear force and marginal axial force. Ultimately, the axial force in each anchor of the group that is dealt with does not interact with the shear force, so it can be neglected.Va  applied to the end that protrudes from the concrete surface, but no parallel force . It follows that the anchors subjected to substantial axial force together with the shear force is not represented by this model.For the sake of clarity, this paper assumes that each anchor is subjected to a transverse force For the same reason, the attachment that is considered here does not transmit axial force, which however is quite a normal condition.It is noteworthy to mention that the collocation “shear force”, used in this paper (including the title) refers to the external force acting on an anchor, while the internal forces in an anchor are shear force and bending moment.Anchors can be adhesive or mechanical. This paper refers to both types of anchors.The insertion of an adhesive anchor into hardened concrete first requires drilling a hole into the concrete. Drilled holes shall be perpendicular to the surface of the concrete member . All holes must be cleaned using a stiff brush (a stiff bottle brush to loosen as much dust as possible) and then vacuumed using an industrial vacuum cleaner (with adequate nozzle) to remove dust from the sides and bottom of hole. This procedure must be continued until a gloved finger rubbed on the walls came out dust-free.Then, post-installation goes on filling the hole with an anchoring material , and inserting the anchor into the hole, which implies that the anchoring material in excess is expelled outside (excess adhesive must be removed from the concrete surface).Post-installation is complete when the anchor touches the bottom of the hole. Anchors must be cured at adequate temperature and number of days before being loaded.This type of anchor is called “adhesive anchor” after the anchoring material. For both structural and non-structural anchors, it is important to ascertain that the adhesive anchor system that is selected reliably performs in rigorous or adverse jobsite conditions.If the anchoring material is a resin, the drilled hole must be tight to the anchor. In fact, resins are relatively soft and, if the gap between anchor and drilled hole were not small, the layer of resin would result rather thick, which would imply that the anchor will undergo relative movements to the drilled hole. If conversely the anchoring material is made of mortar, the drilled hole can be appreciably greater than the anchor.Alternatively, the anchor can be fixed to the concrete without anchoring material. This type of anchor is called “mechanical anchor” after the way it is bonded.In this case, the anchor achieves resistance either by taping its own hole as the screw is driven into the concrete or by means of expansion against the concrete or against a sleeve (this procedure includes drilling a hole slightly larger than the sleeve).The way a mechanical anchor is post-installed implies that the drilled hole is as wide as the anchor.This paper refers hence to anchors that achieve resistance by bonding with adhesives and anchors that are bonded to the concrete without any adhesive. Actually, the pull-out mechanism is not involved, as the anchor resists only transverse forces. So, the way the anchor is bonded to the concrete plays no role in the strength of the anchor.The whole system—anchors, anchoring material, drilled holes, and concrete that surrounds the drilled holes—is here called “anchorage”.Each anchor that composes the anchor group this paper refers to is a metal element post-installed into a hardened concrete member and used to transmit applied loads. Typically, anchors are made of steel .aV is parallel to the surface of the concrete. If the anchor were not perpendicular to the concrete surface or the external force were not parallel to the concrete surface, such anchor would be subjected to axial force together with the shear force.Every anchor is perpendicular to the surface of the concrete member, so the external force L. One end of the anchor is embedded into the concrete, while the other end protrudes from the concrete, in order to allow the anchor to be connected to the attachment. Each shear force aV is applied at the protruding end, whose distance from the concrete surface is denoted by e. The embedded depth of the anchor is hence L − e. The length e depends on the anchor and the attachment.The anchors of the group have the same length and embedded depth. The total length of the anchor is denoted by An anchor installed into a hardened concrete member  is different than an anchor installed before placing concrete (cast-in anchor). Typically, the latter has a hook at the embedded end, and the embedded part includes nuts, washers, and/or plates. Moreover, the member the anchor is cast into includes specific bars and stirrups that embrace the anchor.A post-installed anchor requires drilling a hole into the concrete, which prevents the embedded end from being hooked and any attachment from being placed along the shaft of the anchor. Consequently, the post-installed anchor is a straight shaft (shank) with no nut, washer, or plate attached to the shaft. Moreover, the concrete member that the anchor is post-installed into does not include any reinforcement devoted to transferring the shear from the anchor to the member, as the attachment is a load not included in the original design.Ultimately, the post-installed anchor this paper refers to, cannot rely on any strengthening component in addition to the straight shaft embedded into the concrete.Each anchor should be installed into uncracked concrete. If the concrete is cracked, the anchors should have adequate clearance from the cracks, so that the failure mode is not dictated by cracking existing prior the installation. Where this criterion cannot be met, the shear strength of the anchor is lower than that predicted by this model, but however only moderately ,12,16,65gV reaches high values are considered by the model, as these cracks belong to the main events that lead to shear failure .Needless to mention, the cracks that are referred to just above are those which have occurred before installing the anchors, while the cracks that occur in the concrete that surrounds the anchors when aV, which sum up together giving rise to the resultant force gV transmitted between the anchor member and the attachment. However, the reason why the anchors are dealt with as a group instead of as individual elements independent of one another is that relatively close anchor spacings can generate some structural effects due to interaction between anchors.Literally, a shear anchor group, which is the subject that this paper focuses on, is a number of anchors that bear shear forces Appendix B of  . The resultant force gV is the sum of N force aV transmitted across the N anchors of the group.Typically, the anchors of a group are collected by a steel plate, which is connected to the attachment  that the anchor group connects to the concrete member. Accordingly, the anchor group transmits the resultant force D and its center by C.Design of an anchor group shall start defining the largest circumference within which the anchors can be placed satisfying all the design geometric restraints, the design constraints, and the design prescriptions. That circumference drawn onto the concrete surface is called “circumscribing circumference”. The diameter of that circumference is denoted by C. Then, if necessary, the pattern shall be integrated by anchors inside the circumference, placing these anchors approximately axially symmetric around the center. The anchors shall be placed equally spaced on that circumference, so as to be axially symmetric around the center N and δ, respectively  and the latter is called internal circle .guV acts through the axis of that cylinder, at a distance e from the center C of the external circle .The force x-y-z system, with origin on the surface of the concrete, which is mainly used to identify the depths .Two coordinate reference systems are used . The firx-y plane at a given depth z. The coordinate α of a point A at a given z is the angle between two radii from A that pass through the edges of a segment that is tangent to the circumscribing cylinder at the origin of the Cartesian coordinate system, orthogonal to the shear force gV, and having length D. Of course, a given α does not identify a single point (apart from α = π), but a curve  of the structure, so that failure is not dictated by the distance of the anchors from the edges. Ergo, the shear strength is not reduced by edge effects.Where clearance from the edges is not large, the shear strength obtained from this model must be compared to that obtained from formulations that allow for edge effects ,83,84,85Ultimately, this paper models anchors that form an anchor group, post-installed into uncracked concrete, however installed and bonded, subjected to pure shear forces applied at the external ends, which do not suffer from interaction with the edges of the concrete member, whose failure is dictated by the concrete that surrounds the anchors.Namely, this paper studies anchor groups composed of either adhesive or mechanical anchors, post-installed in hardened concrete, with no global bending moment acting on the anchor group , which fail by shear load mechanism, whose strength does not depend on the distance from the edges (no effects of proximity to edges), in which cracks trigger only approaching the ultimate shear force .This paper uses the term “concrete” because the contribution that the reinforcement provides shear strength with is usually marginal. Moreover, that contribution cannot be generalized for post-installed anchors. Nevertheless, the members that the anchors are post-installed into are typically made of reinforced concrete.The author performed numerical non-linear analyses of an exhaustive variety of anchor groups embedded into concrete and subjected to shear forces, in order to understand the mechanical behavior and failure modes.Anchors and anchoring material were modeled as linear elastic, using the relevant elasticity moduli.In the case of adhesive anchors, the interface between an anchor and the surrounding anchoring material was modeled as a bilateral contact. The interface between anchoring material and surrounding uncracked concrete was modeled as a bilateral contact, while the interface between anchoring material and surrounding cracked concrete was modeled as a unilateral contact.In the case of mechanical anchors, the interface between anchor and surrounding uncracked concrete was modeled as a bilateral contact, while the interface between anchor and surrounding cracked concrete was modeled as a unilateral contact.c and the compressive strain εc, in which compressive stresses and strains are positivecmf is the concrete compressive strength, expressed in N/mm2.The stress–strain relationships of concrete was modeled using the following function between the compressive stress σcmf and the strain εc1 of Equation (1) account for the confinement action that the anchor pattern provides the concrete of the circumscribing cylinder with, and for the confinement action that the biaxial stress state provides the concrete surrounding the circumscribing cylinder with. It is to note that the former confinement is drastically greater than the latter, although the latter is not negligible.The strength cmf of Equation (1) was estimated by the following equations, in which compressive stresses are positive:cf is the compressive strength of concrete under a uniaxial state of stress.The increased strength cf has to be defined performing in-place tests, considering the necessary factors in planning in-place tests, as well as considering the specified design strength, if available.The member that the anchors are embedded into is existing, which entails that A variety of techniques are available for estimating in-place strength of concrete. The methods to estimate in-place strength of concrete include rebound number, penetration resistance, pullout, break-off, ultrasonic pulse velocity, maturity, and cast-in-place cylinders. Adequate procedures are then needed to derive compressive strength from in-place results, which include using appropriate statistical techniques to interpret test results.2 of Equations (2) and (3) is the lowest compressive principal stress. Namely, σ2 is the smallest confining stress of the triaxial stress state.The stress σc1 of Equation (1) was estimated by the following equations, in which compressive strains are positive:The increased strain εcE of Equation (1) is the tangent modulus given by Equation (5), with both cmf and cE in N/mm2.The elasticity modulus cmf is the increased strength given by either (2) or (3).In (5), cE1 of Equation (1) is cE1 = cmf/εc1, with both cmf and cE1 expressed in N/mm2, in which cmf is given by either (2) or (3), and εc1 by (4).The elasticity modulus c/cmf ≤ 0.5. The strain εcL at σcL = 0.5·cmf is given by Equation (6).For the descending part of the stress-strain function, Equation (1) is valid only for values of σc > εcL the descending branch of the σc − εc function was described using Equation (7):For strains εCompressive failure of concrete is often a discrete phenomenon, i.e., there is a fracture region of limited width, in which compression strains are concentrated .For practical reasons and due to lack of sufficient experimental data, these strain concentrations generally are smeared, as has been done in Equations (1)–(7). As a consequence, the descending branch of the stress-strain relation in compression is influenced by the length of the member subjected to compression.c or without imposing any limit on εc. That limit is the crushing strain of concrete, which is herein denoted by εcu. In the former case, the contribution to the anchor’s shear strength provided by the concrete depends on εcu, while in the latter case does not depend on εcu.Due to such uncertainty and highly variable performance, the descending portion of the stress-strain relation is considered as an envelope to all possible stress-strain relations of concrete which tends to soften as a consequence of concrete micro-cracking. In view of that, the concrete constitutive law of Equation (7) can be used either imposing a limit on εcu  was estimated by the following equation:The increased strain εcu. On that account, Equation (7) was used without any limit on εc, i.e., without applying εcu.The results of the non-linear analysis demonstrated that the shear strength of an anchor group depends slightly on εTensile failure of concrete is always a discrete phenomenon, as compressive failure. Differently than compressive failure, however, tensile failure has been extensively studied by fracture mechanics and the results that were obtained allow cracks to be accurately modeled. Furthermore, smearing the strain concentrations would yield lower accuracy than for compressive failure .To describe the tensile behavior, thus, a stress-strain diagram was used for the uncracked concrete, and a stress-crack opening diagram for the cracked section. More specifically, for uncracked concrete subjected to tension, a bilinear stress-strain relation was used, composed of two ascending branches with decreasing slope. For a cracked section, a bilinear stress-crack opening relation was used, composed of two descending branches with decreasing negative slope.A general non-linear finite element model was then prepared, which incorporated the above-described constitutive laws and reproduced each component using the relevant geometry and mechanical properties. In so doing, accurate simulations of the responses of anchor groups to shear forces were obtained.If on one hand the results of that model are accurate and comprehensive , on the other hand, the numerical model is not easy to use in daily practice. So, the model cannot be proposed for practitioners’ activity. In fact, to run the numerical model, and to understand and visualize the data outputs require spending amounts of time and effort that are excessive. Research was directed towards providing a tool for daily practice of the structural designer that retains the accuracy of the non-linear finite element model. To that end, the non-linear finite element model was used to simulate a sufficiently comprehensive series of cases, in order to identify general behaviors shared by all the anchor groups. Those general behaviors allowed an analytical model to be developed.The analyses performed using the above-described non-linear finite element model showed that all the anchor groups have the same type of ultimate behavior and the same failure mode, which allowed mechanical assumptions to be established.D  and the circle where the embedded ends of the anchors lie on .The key to understanding the ultimate behavior is to focus on the circumscribing circumference , whose dD , and on D , whose hThe circumscribing cylinder includes concrete, anchors, and anchoring material .L − e. Those ranges are defined by the limits reported immediately below, expressed in mm.Numerical modeling results show that the anchors can substantially confine the concrete. The level of confinement depends on how much the anchors exploit the area of the circumscribing circumference. The results pointed out three ranges of behavior depending on the spacing δ of the anchors and embedded length Close or intermediate spacings of the anchors provides the circumscribing cylinder  with a substantial level of confinement, which drastically increases its stiffness and strength, with respect to the stiffness and strength of the concrete that surrounds the circumscribing cylinder. That level of confinement makes the deformation of the circumscribing cylinder be negligible with respect to that of the surrounding concrete. Moreover, that level of confinement makes the concrete of the circumscribing cylinder be substantially stronger than that of the surrounding concrete.Eventually, by virtue of the confinement provided by the anchors for close or intermediate spacings , the cirBy virtue of that confinement, hence, the surrounding concrete reaches strains much greater than the circumscribing cylinder, so that the ultimate shear force is dictated by the concrete that surrounds the circumscribing cylinder, while the concrete of the circumscribing cylinder reaches stresses far from the concrete crushing strength.Eventually, the “group effect” consists of those drastic increases in stiffness and strength of the concrete circumscribed by the cylinder.On the contrary, wide spacings of the anchors provide the circumscribing cylinder with marginal confinement, which does not appreciably increase its stiffness and its strength, with respect to the surrounding concrete. Thus, wide spaced anchors do not give rise to any group effect.D and embedded length L − e.For close and intermediate spacings, hence, modeling can replace the anchors and the enclosed concrete with the circumscribing cylinder, whose strength and stiffness can be assumed to be infinite. That is, the circumscribing cylinder behaves as if it were a sole anchor of diameter Ultimately, the concrete of the circumscribing cylinder  does not fail but dictates the strain profile that causes the failure of the surrounding concrete.For an anchor group with wide spacing, each anchor behaves as an individual member, due to a marginal confinement action induced by the anchor group. Such confinement increases neither the stiffness nor the strength of the circumscribing cylinder. It follows that no anchor group effect occurs for wide spacing, and the shear strength depends on the concrete that surrounds each individual anchor. The shear strength of wide spaced anchors has therefore to be obtained for the model of the single  anchor, whereas the model delivered here does not apply.The analysis of the numerical results focused on the ratio between the shear strength of an anchor group and the sum of the shear strengths of each anchor of the group behaving as an individual anchor . The three ranges that have been introduced above derive from those ratios.For close spacing of anchors, that ratio is lower than unity. Accordingly, the impact of the anchor group effect on the ultimate behavior consists not only in a confinement but also in a reduction in the potential strength of the anchors. That is, the shear capacity of the anchor group is lower than the sum of the individual shear capacity of the anchors.For intermediate spacing of the anchors, that ratio is regularly greater than unity, which means that the impact of the anchor group effect on the ultimate behavior consists of an increase not only in the stiffness but also in the potential strength of the anchors. That is, the shear capacity of the anchor group is often greater than the sum of the individual shear capacity of the anchors.While the result obtained for close spaced anchors is intuitive, the result obtained for intermediate spaced anchors is counterintuitive. The reason of a ratio greater than unity lies in the fact that the anchors provide the concrete of the circumscribing cylinder with substantial confinement, at which concrete compressive strength and stiffness are enhanced. As a result, the strength is dictated by the concrete that surrounds the circumscribing cylinder, and not by the concrete that surrounds each individual anchor. At the same time, intermediate spacing implies that the number of anchors is not high, so that the sum of the individual strength of the anchors is not high too, in particular may be lower than the shear strength of the anchor group. Namely, the anchors do not completely exploit the area of the circumscribing circumference, which in turn implies that the anchor group effect boosts the shear capacity of the group.For wide spacing of the anchors, that ratio is equal to unity, which means that the anchor group effect does not impact on the strength of the group. This is the logical result of the fact that such spacing does not provide the concrete with any appreciable confinement. So, wide spaced anchors cannot be called anchor group since no anchor group effect exists.Since research activity was directed at investigating anchor group effect, hereinafter, the paper considers only close and intermediate spacings, whereas wide spacing enters the topic of the single anchor.y-z plane. The uniform translation is slight and can be neglected compared to the linearly varying translations due to the rotation.The rigid body motion of the circumscribing cylinder that results for close or intermediate anchor’s spacings is composed of a uniform translation parallel to the shear force and a rotation in the The rotation makes the cylinder translate parallel to the shear force. Translations of the points of the circumscribing cylinder are in proportion to the distance from the rotation center. The direction of the translation of a point depends on the side of the point with respect to the rotation center. In particular, the rotation causes the external circle of the cylinder to exhibit a translation in the direction of the shear force (positive direction) and the internal circle to exhibit a translation in the opposite direction (negative direction), as shown in L = λ + β + e  is denoted by λ and that from the internal circle (embedded ends) by β, where  + β + e . The len + β + e .The rotation generates both tensile and compressive strains in the concrete. The tensile strains induce tensile stresses, which cause concrete to crack. Cracking causes most of the tensile stresses to disappear. Eventually, the tensile strain and stresses, as well as the cracks, can be neglected, and the sole effect of the rotation that is here considered is the compressive stress state.Made that assumption, the elaboration of the results from the numerical model aimed to define the compressive stresses in the concrete at the ultimate—i.e., when the anchor group reaches the shear strength .Vg applied to the anchor group is balanced by the stresses acting on the lateral surface of the circumscribing cylinder. Thus, elaborations were directed at defining the stresses acting on the lateral surface of the circumscribing cylinder at ultimate.In other words, the shear force Elaborations of the numerical results allowed the stresses acting on the lateral surface of the cylinder to be replaced by two resultant forces acting on the lateral surface of the cylinder , while tgV, which is denoted by Fλ. The second force is the stress resultant acting on the segment β of the circumscribing cylinder at ultimate, whose direction is that of gV, which is denoted by Fβ  and (15) accurately estimate Fλ and dλ.Upon inspection of the numerical results, gV along the cylindrical lateral surface  is a straight-line, and along the width  is uniform  in the concrete that surrounds the external circle of the circumscribing cylinder. The rotation that induces εc1 in the concrete surrounding the external circle provides hence an accurate estimation of the ultimate rotation.In all the cases that were analyzed, the rotations at ultimate were found to differ marginally from the rotation that induces the compressive strain εβ. The stress produced by εβ, which is elastic too, is denoted by σβ.Compression induced by the ultimate rotation in the concrete that surrounds the internal circle is elastic, since β is always much shorter than λ. That elastic compressive strain is denoted by ε2 of Equations (2) and (3) is zero in the concrete that surrounds the external circle. Plugging that value into Equation (4), the compressive strain εc1 turns out to be 2.2‰.The compressive stress σc1 = 2.2‰ at the external surface to zero at the rotation center (segment λ), and from zero to a substantially elastic strain εβ at the embedded ends of the anchors (segment β).At ultimate, hence, the compressive strains on the lateral surface of the cylinder  range frFβ, is equal to the volume of the triangular prism whose sides are β, σβ, and D. The application point of Fβ is the centroid of that prism.The resultant force of the compressive stresses acting on the segment β at ultimate, which is denoted by Elaborations of the numerical results demonstrated that the stress state that exists in the concrete prior the installation of the anchor does not influence the shear strength of the anchorage.z, τzx, and τzy are marginal in the concrete that surrounds the circumscribing cylinder  of the members surpasses L − e does not influence the shear capacity of a given anchor group.That conclusion agrees with another numerical results, which is that the shear strength of an anchor group with given embedment length z, the strain do not vary along x for—ϕ/2 ≤ x ≤ ϕ/2. Equations (14) and (15) are based on that result.Insight into the compressive stresses induced by the circumscribing cylinder in the concrete that it pushes against allowed another general result to be achieved. Let us consider a circle of the circumscribing cylinder . The stresses in the concrete included into the square that circumscribes a circle are constant. That result agrees with the fact that, at a given The circumscribing cylinder that presses against the surrounding concrete can hence be decomposed in the segment λ that pushes against the surrounding concrete in one direction and the segment β that pushes against the surrounding concrete in the other direction .z can be modeled by a semi-space  made of concrete. The surface of that half-space is a plane that passes through the origin of the Cartesian coordinate system and lies on the x-axis .Considering that the stresses in the concrete included into the square that circumscribes a circle are constant and that the stresses due to loads different than the shear force gV. The half space is loaded at that surface by a strip load of width D and magnitude cp(z). Although the half-space extends infinitely along the axes x and z , the strip load reproduces only the stresses at a given z, as the stress state is biaxial, so stresses do not depend on z.In the end, the mechanical model ,90,91,92z induced by the contact pressures cp(z) is equal to the two-dimensional stress-state induced by cp(z) smeared along a strip of infinite length acting on a semi-space composed of the concrete that the cp(z) presses against. Consequently, the compressive stresses induced by either λ or β at a given abscissa z are simulated by a strip load whose value is that of the stresses at this z applied at the plane horizontal surface of the half-space .Equation (14) requires knowing the maximum compressive stress Elaborations of the numerical results demonstrated that, in the concrete that surrounds the cylinder, the major principal stress σ1 and minor principal stress σ3 of the stress biaxial state, together with the principal stress in the third direction σ2 can be satisfactory approximated by the following expressions (compressive stress are assumed to be positive):cmf tolerated by that concrete to be defined. In so doing, Equation (14) can be used.Equations (16)–(18) define the level of confinement of the concrete that surrounds the circumscribing cylinder, which in turn allows the maximum compressive stress 1 and σ3 on z is dictated by cp(z), i.e., given cp(z), σ1 and σ3 do not depend on z . Equation (17) shows that the principal stress σ2 is almost zero everywhere.Equations (16) and (18) show that the dependance of σ1 occurs for α = π, i.e., at the middle of the circumscribing cylinder, and Equation (18) shows that σ3 reaches the maximum for α = π too, i.e., at the same point.Equation (16) shows that the maximum σ1 causes the concrete to crush, the principal stress σ3 confines the concrete in the second direction, and the principal stress σ2 does not confine the concrete in the third direction. As a result, confinement is moderate. Thus, the effect of σ1 on crushing prevails over the effect of σ3 on confinement.The principal stress σcmf is dictated by the stress state σ1 = cmf (compression), σ2 = 0, and σ3 = cmf (compression), at the point at α = π.Ergo, cmf:cf is the strength of concrete under a uniaxial state of stress, which has been introduced about Equation (1). Eventually, Equation (19) is simultaneously an accurate and safe estimation of cmf.For the above biaxial stress state, all the formulations that agree well with test data provide values of the biaxial concrete strength that only slightly exceed the following value of cmf to plug into (14) is not the uniaxial compressive strength of concrete cf, but is that of Equation (19)—namely, it is higher, as it benefits from some confinement provided by the biaxial state of stress. It is to note that cmf from (19) was used in the numerical model too –(5)).In brief, the strength D concentric to the cylinder be uncracked. That result agrees with [Equations 16)–(18) were obtained from uncracked concrete. Nevertheless, in order to transmit those compressive stresses is sufficient that a circumference with diameter 1.5·ees with ,12,16,65–(18) werThe results of the numerical analysis presented in The assumptions refer to the circumscribing cylinder introduced in z of the Cartesian coordinate system  are negligibleStresses induced by loads different than the shear forces applied to the anchors’ ends that protrude from the concrete surface  to zero at the rotation center (z = λ), and from zero to an elastic value εβ at the internal circle (z = L − e)The compressive strains at the interface between the lateral surface of the circumscribing cylinder and the surrounding concrete have a straight profile, which ranges from ε.Assumption 3.g, so that the resultant force of the shear stresses is zeroNo tensile stresses act on the lateral surface of the circumscribing cylinder, and the shear stresses on the lateral surface are symmetric with respect to V.Assumption 4.λ of the compressive stresses acting on the lateral surface of the circumscribing cylinder is defined by Equation (14), in which fcm is defined by Equation (19)The magnitude of the resultant force F.Assumption 5.λ between rotation center and application point of Fλ is defined by Equation (15)The distance d.Assumption 1 allows the model to account only for the stresses induced by the anchorage. The position at which the anchor group is installed plays hence no role, as long as adequate clearance from the edges is guaranteed.c1 = 2.2‰ the shear force transmitted by the anchor group is lower than the ultimate value.Assumption 2 is tantamount to say that the circumscribing cylinder has a rigid body rotation. The profile of the compressive strains in the concrete that surrounds the circumscribing cylinder is a straight line that starts from a known value of the strain and passes through the rotation center . The knoz along the lateral surface can be calculated using direct proportionality.Once the position of the rotation center is known, hence, the strain at any β is elastic, which implies that the compressive stresses acting on the segment β are elastic too. Thus, the resultant force Fβ is a known save for σβ and β. Furthermore, the distance between the application point of Fβ and the anchor’s rotation center dβ is known save for β.Assumption 2 also establish that εAssumption 3 allows the model to neglect the tensile strains, and, in turn, the tensile stresses. This assumption is justified by the fact that the tensile strains cause the concrete to crack . It follows that the strength of the anchor group stems from the compressive stresses acting on the lateral surface of the circumscribing cylinder.Fλ, which depend only on λ.Assumption 4 allows the compressive stresses acting on the segment λ of the cylinder lateral surface to be replaced by the resultant force Fλ and the anchor’s rotation center to be set equal to dλ, which depends only on λ.Assumption 5 allows the distance between the application point of The five assumptions of This section presents the analytical model that predicts the shear strength of an anchor group, which is the contribution made by this paper.Fβ (resultant force of the compressive stresses acting on the segment β) can be obtained as a function of both β and the compressive stress at the internal circle σβ:Given that the stresses acting on the segment β of the cylinder lateral surface are elastic (Assumption 2), the force Fβ is transverse to the anchors and acts on the lateral surface of the circumscribing cylinder. The direction of Fβ is that of guV  and (21).Both β and σβ at the internal circle (at z = L − e) is in proportion to the concrete compressive strain at the external circle (z = 0). The concrete compressive strain εβ can thus be expressed as a function of λ and β:Assumption 2 implies that the concrete compressive strain εβ is elastic, which allows σβ to be obtained by using the concrete elastic modulus cE:cE is provided by Equation (5).Assumptions 2 also entails that σFβ can be expressed as:Eventually, the force e = L (which implies that the unknown is only one distance); 2- and the shear strength of the anchor group guV, which is the prediction the model aims at.The problem that has to be solved in order to determine the shear strength of an anchor group has two unknowns: 1- the position of the rotation center, whose distance is λ from the external circle and β from the internal circle of the circumscribing cylinder, where β + λ + guV can be obtained from the translational equilibrium.The position of the rotation center can be obtained from the rotational equilibrium equation. Then, the shear strength The rotational equilibrium equation is written with respect to the ends that protrude from concrete (free ends), so that the shear force, which now is unknown, is not included.cmf is the biaxial compressive strength of concrete defined by Equation (19).Replacing Equations (14), (15), (20) and (21) into Equation (25), the following equation turns out.Equation (26) allows the diameter ϕ to be eliminated.Equation (27) shows that the position of the rotation center does not depend on the cylinder diameter, while it depends on the concrete constitutive law.cE and cmf are known, as they are provided by Equations (5) and (19), respectively, in which cf is data. Both β and λ are unknown, but the geometric parameters satisfy the relationship: β + λ + e = L  includes only one unknown, which can be derived solving the equation. In so doing, β and λ are determined and, in turn, the position of the rotation center is determined.In Equation (24),  + e = L , in whicThe translational equilibrium equation that governs is that in the direction of the shear force:guV of the anchor group embedded into concrete is gained and the problem is eventually solved.The values of β and λ of Equation (28) are those obtained solving Equation (27). Once those values are put into Equation (28), the shear strength cf, the length L and diameter ϕ of the anchor, and the length e, while cmf and cE are provided by Equations (5) and (19).Ultimately, the model—Equations (27) and (28)—requires knowing the uniaxial concrete strength The substitution of β as a function of λ or vice versa in Equation (27), and the resulting explicit formula that gives λ or β is not presented here, since it would be not only cumbersome but above all useless. In fact, the Reader who will use this model will find easier to solve the two-equation system employing a mathematical program, which is what the author has performed in the case studies presented here.The predictive capacity of the analytical model stems from that of the non-linear numerical model, which was the tool used for performing a broad-spectrum analysis , whose rIn order to evaluate the predictive capacity of the analytical model not exclusively by theoretical means, experimental results were borrowed from literature and those tests were simulated by the analytical model of Unfortunately, most of the tests reported in the literature were performed on anchors cast into new structures, i.e., installed before the concrete is poured ,69,77,86On the contrary, an anchor installed into hardened concrete consists of a straight shaft (shank), without any strengthening element. Moreover, the reinforcement of the member that the anchors are post-installed into was not designed to bear those anchors, which entails that the shear strength of the anchor group cannot rely on the reinforcement.Thus, the tests included in this section are only those where the shear strength did not benefit of substantial extra-strength due to those devices.The strength of the vast majority of the tested specimens reported in the literature was dictated by premature failure modes ,54,55,56For the sake of completeness, it should also be mentioned that this research work included the comparison between the shear strength of some tests that collapsed by a premature failure mode, i.e., by edge effect ,16, pryiIn particular, the results of those comparisons agree with , whose mN = 4. The paper listed the characteristic dimensions for each anchor specimen and each block, so that sufficient information was provided to reproduce the experiments described.Reference  presenteL − e = 102 mm and 152 mm) and edge distances of the anchor specimens were selected by the Authors targeting either steel fracture or concrete failure (shear strength based on the design capacity equations in ACI 318 [The embedment depths ( ACI 318 ). As a r2 and an ultimate strength of 527 N/mm2. Ready-mixed concrete with a specified compressive strength of 27.6 N/mm2 was designed. The actual compressive strength of the concrete was 46.1 N/mm2, measured using tests of three 100 × 200 mm cylinders with a coefficient of variation of 5.9% .The test anchor consisted of a 19 mm (3/4”) diameter ASTM F1554 Grade 55 threaded rod and a heavy hex nut at the embedded end, which both provided extra strength and extended the embedded length by adding 57 mm. The measured stress–strain relationship of the anchor’s steel indicated a yield strength of 476 N/mmcf = 46.1 N/mm2; N = 4; L = 169 mm and 219 mm; ϕ = 19 mm; e = 10.0 mm. The average shear strengths measured by the monotonic tests were guV = 226.62 kN and 322.19 kN for the 169 mm and 219 mm length, respectively.The data of those 12 tests to plug into the analytical model are: guV = 187.81 kN and 241.66 kN, respectively. The theoretical predictions resulted to be lower than the experimental results, as expected. The maximum difference between theoretical and experimental results turned out to be 17.1% and 25.0%, which is a satisfactory result considering the extra-strength given by the heavy hex nut, with respect to the anchor group simulated by the analytical model.The shear strengths from the analytical model were Load-controlled cyclic push-out tests were conducted in ,93 as weTypically ,15,32, tReference  presenteThe comparisons were carried out on other experimental tests, which exhibited differences having the same order of magnitude of those presented above.As a final remark about the comparison between analytical and experimental results, the latter validated the former, i.e., the analytical model presented in this paper satisfactory agrees with experiments borrowed from literature.This model was applied to a wide-ranging series of practical cases, in order to achieve two kinds of objectives. First, the case studies aimed to test the capacity of the analytical model of being easily applied. Second, the case studies aimed to explore the ultimate behavior of anchor groups embedded into concrete structures. The analysis of those results allowed a discussion to be developed and conclusions to be drawn . Some ofcf of the concrete that the anchors are embedded into, the length L of the anchors, the external protruding length e, the diameter of the circumscribing circle D—which are the data of the model—and the shear strength of the anchor group guV—which is the result of the model. As explained in the caption, each column of The cases of N independent anchors). That analysis considered all the possible anchor patterns and diameters, and referred to the best combination of diameter and number of anchors that was found for each pattern. The results of that analysis are reported in The case studies of N of anchors and their diameter ϕ . Then, the strength of each anchor group of N anchors of the group behaving as individual elements (no group effect).As explained in the relevant caption, guV provided by this model . The strength uV was calculated using a specific model of the author.The results of that analysis allowed the effect of the group on the anchors’ shear strength to be measured. To that end, the second to last row of guV and ΣN·uV. That ratio is shown in the last row of In order to quantify the penalization due to the anchor group effect, an exploitation ratio was introduced, which is defined as the ratio between This section presents the critical discussion about the analytical model and about the major findings obtained from an analysis carried out using this model, which has included the case studies presented in This research has studied anchor groups post-installed into concrete members, with adequate clearance from the edges, and subjected to transverse force. This paper has delivered an analytical model that simulates the ultimate behavior of an anchor group subjected to pure shear force.The model presented in this paper consists of analytical exact mathematical formulation expressed by two closed-form equations, which allows the structural designer to predict the shear strength of an anchor group embedded into concrete. The model requires knowing one mechanical data, which is included in the basic information that must be available for sound analyzing an existing concrete building, and three geometric data related to anchor pattern and embedment.More specifically, the model requires 4 pieces of data—namely, the uniaxial compressive strength of the concrete the anchors are embedded into, the diameter of the circumference that circumscribes the anchors, the embedded length of the anchors, and the length of the anchors that protrudes from the concrete surface connected to the attachment.The analytical model presented here simulates anchors installed into hardened concrete, which entails that the anchor is a straight shaft (shank), with no hook at the embedded end, and with no nuts, washers, or plates attached to the embedded shaft. Differently from cast-in anchors, hence, post-installed anchors are not strengthened by those devices.Moreover, the members that the anchors are post-installed into are existing, which entails that no bars and stirrups had been designed to bear the anchors. Differently from cast-in anchors, hence, post-installed anchors are not even strengthened by any reinforcement.The analytical model derives from five mechanical assumptions. Consequently, the predictive capacity and accuracy of this model stems from the representativeness and accuracy of those assumptions. The mechanical assumptions were established based on a non-linear numerical analysis that simulated the behavior of a complete range of anchorages, which allowed the ultimate behavior of an anchor group to be described, comprehended, and interpreted. So, the assumptions are realistic and exhaustive. It follows that the model almost guarantees an equal degree of accuracy in predicting the shear strength of an anchor group.Eventually, the numerical analysis allowed complexity to be accurately simulated, then to be reduced into five statements. In so doing, complexity was concentrated in the assumptions, so that the analytical model is easy and fast to use. Nevertheless, the accuracy of the model is that which is guaranteed by the assumptions.That statement is confirmed by the comparisons to results of experimental tests found in literature . It is hD of the circumscribing circumference, while it depends neither on the diameter of the anchors nor on the diameters of the individual drilled holes of each anchor. That conclusion holds true as long as the anchor group provides the circumscribing cylinder with substantial confinement, which can be guaranteed by the patterns of The final equations—Equations (27) and (28)—show that the shear strength of an anchor group depends on the diameter The results of the analytical model demonstrate that anchor group effect is different than what is usually referred to. Namely, the anchor group effect is not the phenomenon that causes the behavior an anchor group to be worse than the behavior that the anchors could display as individual components . Actually, that behavior can be a consequence of anchor group effect, but is not the direct effect; hence, it is not its definition.The direct effect of anchor group is to provide the concrete of the circumscribing cylinder with substantial confinement, which enhances its stiffness and strength.Stiffness enhancement makes the circumscribing cylinder move as a rigid body, so that it rotates rigidly .Strength enhancement makes the concrete of the circumscribing cylinder become much stronger than the concrete that surrounds the circumscribing cylinder, so that the shear strength is dictated by the latter but not by the former.What reported above is the actual definition of anchor group effect that stems from this research work, while the consequence of this effect on the shear strength depends not only on anchor group effect (as defined here), but also on how much the anchors exploit the area of the circumscribing circumference.When the area of the circumscribing circumference is highly exploited, the shear strength of the anchor group is lower than the sum of the shear strength of the anchors of the group as individual elements, i.e., without any group interaction. However, when the area of the circumscribing circumference is no more than moderately exploited, the shear strength of the anchor group is greater than the sum of the shear strengths of the anchors of the group as individual elements.In In the case studies of The case studies of The former result was expected, since it agrees with the conventional definition of anchor group effect, whereas the latter result was unexpected and is one of the novelties provided by this research work.More specifically, this research work has proven that the anchor group effect consists of confining the concrete of the circumscribing cylinder, whereas its impact on the shear strength depends on the spacing of the anchors and can result either in a decrease or in an increase.Those results allow the optimal design of an anchor group to be defined. Actually, there are two design approaches, according to the level of conservativeness that the designer wants to guarantee, i.e., according to the margin of safety that must be achieved.guV, and the sum of the shear strengths that each anchor would reach if it did not interact with the other anchors, i.e., ΣN·uV = N·uV. Under this assumption, an exploitation ratio greater than unity is useless.For a drastically conservative safety assessment, the shear strength of the anchor group can be taken equal to the lowest value between the anchor group shear strength, i.e., guV, while for wide spacing the shear strength must be taken as ΣN·uV = N·uV. Under this assumption, an exploitation ratio greater than unity can be useful.For a realistic but safe safety assessment, the shear strength of the anchor group with either close spacing or intermediate spacings shall be taken equal to N and diameter ϕ are optimized. If, on the contrary, that ratio is lower than unity, the number and/or diameter of the anchors is/are excessive and can be reduced. While if the ratio is greater than unity, the area of the circumscribing circumference is not exploited, so its diameter D could be smaller.The former approach  implies a specific design strategy for anchor groups. The circumscribing circumference is the starting point of the design, since it stems from issues that are at a higher level of the anchor design process and that being so it cannot be modified. Given the circumscribing circumference, the pattern and spacing of the anchor group shall be chosen so that the ratio is as close as possible to unity. In so doing, the number The former approach implies hence finding the greatest exploitation ratio for the given circumscribing circumference.The latter approach should start placing a minimum number of anchors and increasing the number up to reaching the strength that the anchor group has to transmit.2 to 100.0 N/mm2. The ultimate shear stress of all the anchors that can be found on the market is drastically greater than those shear stresses. Therefore, the shear strength of the case studies of In the case studies of Model predictions were compared to the strength obtained from code provisions, requirements, specifications, standards. The comparisons revealed that the actual shear strength of an anchor group post-installed into concrete is substantially different than the shear strength furnished by documents edited by organizations for technical assessment of anchorages. The differences resulted to be even greater comparing model predictions to the strength from technical reports edited by producers and manufacturers.Those differences are due to two facts. The first is that the formulations included into those documents are empirical, while this model is analytical, so it is more accurate. The second is that the tests from which the formulations were derived are devoted to anchorages for new structures (cast-in anchors), while this model simulates anchors whose strength cannot take advantage of any of the strengthening devices that are customarily used for anchors cast in new structures .One of the most popular approaches to the design of anchors embedded into concrete is Appendix B—Anchoring to concrete included in . This apThe outputs of that analysis are presented in For large-sized anchors and strong concrete, the predictions of equations in ACI 349  resultedOn the contrary, for small size anchors and weak concrete, the predictions of equations in ACI 349 resulted to largely underestimate those of the analytical model. One of the reasons for those differences is that, when the concrete is weak, the edge effect and anchor group effect are substantial.Since the 1990s, numerical structural analyses have almost replaced analytical methods in the profession. In particular, practitioners usually carry out structural analyses employing commercial software codes that apply the finite element methods, which also perform the final safety verifications according to a selected code.This research work has proven that commercial software codes cannot correctly describe the ultimate behavior and reliably predict the shear strength of an anchor group embedded into concrete, and that they are not suited to use for assessing the safety conditions of an anchor group. Actually, to be accurate, a numerical model should be three-dimensional, and should use non-linear constitutive laws up to failure for concrete and interfaces, which goes beyond the capability of structural software systems directed at structural practice and used by practitioners.The sole alternative to such numerical modeling is an analytical model derived from assumptions that accurately and exhaustively represent the ultimate behavior of anchor groups obtained from the numerical modeling. That is exactly what has been performed in this paper.Activity included the verification that the model reproduces experiments present in the published literature with close agreement. The average difference between model simulations of experiments performed on cast-in anchors and the results of such experiments turned out to be approximatively 20%, which is a satisfactory result considering that the tested anchors benefitted from the extra-strength provided by nuts and washers attached to the embedded shaft.Model predictions were also compared to code provisions and the differences turned out to be huge. In particular for large diameters model predictions resulted to be up to one third of the shear strengths calculated using code provisions, which proves that codes edited by organizations for technical assessment, as well as reports and instructions edited by producers and manufactures, are not reliable.Accuracy of the model predictions depends on how much the analyzed anchor group is represented by the reference structure, which is an anchor groups that fails by pure shear strength of the surrounding concrete, whose anchors transmit no more than marginal axial force. Accordingly, strength assessment has also to ascertain that the bending moment transmitted by the anchor group from the attachment to the concrete is no more than moderate, so that the axial forces in the anchors be negligible. Moreover, the strength assessment has to ascertain that the edges do not imply any reduction in the strength (edge effects). Strength assessment has also to ascertain that the ultimate shear forces dictated by the steel of the anchors (rupture of the shafts) and by the anchoring material (adhesive anchors) are greater.The results have shown that for close-spaced anchors the shear strength of the anchor group is lower than the sum of the shear strength of each single anchor of the group as individual member, while for intermediate spaced anchors the shear strength of the anchor group may be even higher than the sum of the shear strength of each single anchor as individual members.The initial remark of this paper would also be suitable to close the paper. On one hand, post-installed anchors are marginally included in the papers present in the published scientific literature and in the documents edited by organizations for technical assessment, which are mainly devoted to cast-in anchors. On the other hand, the behavior of post-installed anchors is substantially different than the behavior of anchors cast in new concrete structures, i.e., before concrete has been poured. This paper has been written aiming to fill that gap for the anchor groups.