An assessment method to ensure applicability of concrete capacity method for design of anchorages: Linear force distribution approach

The concrete capacity design (CCD) method for the design of anchorages comes with a pre‐requisite that the base plate connecting the anchors of a group should be “sufficiently stiff.” This article addresses a frequently and vastly discussed topic in fastening technology, namely “What a sufficiently stiff base plate actually means?” The qualitative definition used by the codes requires that the deformation of the base plate should be small compared to the anchor displacements, in addition to the requirement that the base plate should remain linear elastic. This requirement indirectly ensures that the forces among the anchors of the group are distributed in a linear manner, which is the primary requirement to apply the CCD method for design of anchorages. Such a qualitative definition has raised more questions than answers and haunts the fastening technology community. Perhaps the possible solution to this problem requires asking the question differently, that addresses the problem directly: How can it be ensured that the tension forces among the anchors of a group are distributed in a linear manner? This paper first discusses the problem in detail highlighting several issues in the current approach and later provides an overview of an assessment method in form of a set of criteria that can be used to evaluate whether the tension force distribution is linear for an anchorage with a given anchor pattern, baseplate, attachment, and load case. The principle of the approach is based on a combination of simple mathematical formulations and the evaluation of finite element calculations performed on anchorages modeled with base plate, attachment, and anchors. The proposed formulations are described, and it is explained why they are adequate to find practical and safe general solutions to the problems. The application of the set of criteria allows the calculation of an optimum base plate thickness, sufficient to distribute the forces linearly among the anchors, without being overly conservative.


| INTRODUCTION AND PROBLEM STATEMENT
The design of anchorages in fastening technology is carried out currently according to EN1992-4, 1 ACI 318, 2 fib Bulletin 58, 3 or other national standards, which are mainly the adaptation of one of the mentioned codes.The design of anchor groups is performed in two main steps: In the first step, the forces acting on individual anchors of the group are determined, assuming a linear distribution of forces in the anchors, or in other words, assuming a sufficiently stiff (rigid) base plate.In the second step, the resistance of the anchorage is calculated against all possible failure modes of the anchorage in both tension and shear, whereby the failure mode with the lowest resistance governs the design.In this paper, anchorages loaded in tension (also in combination of moment loading) are discussed.
Tension loaded anchor groups may fail by steel failure, pullout failure, concrete cone failure, combined pullout and concrete failure (for bonded anchors only), splitting of the concrete member, or side face blow-out failure.The "linear force distribution" assumption is central to the concrete capacity design (CCD) method 4,5 for the design of anchor groups against concrete cone breakout or combined pullout and concrete failure.Assuming a linear force distribution, ensures that the assumptions that are implicitly built in the CCD method are fulfilled.For example, if an anchor group is subjected to a concentric tension force only, all the anchors are considered to displace equally and assuming equal stiffness of the anchors, they will receive an equal amount of tension force.Also, for an anchor group subjected to an eccentric tension force or a tension force with a bending moment, it is assumed that the anchor tension forces will have a linear distribution (Figure 1).This requirement of linear force distribution is deemed satisfied provided the base plate stiffness is sufficiently high for it to be considered as almost rigid.Note the use of subjective terms such as "sufficiently high" and "almost rigid" in the previous sentence highlighting the non-scientific and non-objective nature of the requirement of sufficiently stiff base plate as currently included in the codes and standards.
Due to the finite stiffness of the base plate, however, the assumption of a rigid base plate can only be approximately fulfilled in reality, and therefore, the linear force distribution assumption can also only be approximately satisfied.The important question therefore is: How much approximation is sufficient to ensure that the CCD method is practically reasonable and acceptable to apply for the design of anchorages under tension loads without jeopardizing their safety?Unfortunately, this seemingly simple question is not so straight forward to answer in quantitative terms and has been haunting the fastening technology community for almost a decade.
In the current codes, the problem of the requirement of a rigid base plate is bypassed without giving clear guidelines for the designer.It is mentioned that the requirement of a rigid base plate is deemed satisfied provided (i) the base plate remains linear elastic under design loads and (ii) the base plate deformations are small compared to the anchor displacements.However, both of these statements are incomplete and partly ambiguous.First, the base plate remaining in the elastic range is a necessary but not sufficient condition of rigidity of the base plate.Second, no quantitative requirement is given as to how small the base plate deformations should be.Third, base plate deformation varies continuously at every point, so at which point of the base plate should the deformation be compared to the anchor displacement?
][11][12] In simplest terms, if the base plate is not rigid, the force distribution among the anchors of the group can be F I G U R E 1 Example anchor force distribution assumed in the current design codes and required for the application of the CCD method (assumption of rigid base plate): Eccentric tension or tension + bending.CCD, concrete capacity design.
significantly different from the force distribution calculated assuming a rigid base plate (Figure 2).Since some anchors will experience higher forces in reality than the forces calculated assuming a rigid base plate, the anchor group will offer a lower resistance to the applied loads than calculated assuming a rigid base plate.This can lead to a significant loss of safety of the anchorage.Such a case is clearly demonstrated by tests in Bokor, 9 where for a 1 Â 3 group, the resistance against concrete cone breakout failure under tension loads is found to be about 25% lower than the resistance calculated assuming a rigid base plate (CCD method).The base plate thickness was sufficient to ensure that the base plate did not yield and yet due to the nonlinear distribution of the anchor forces, the resistance of anchorage was significantly penalized.Furthermore, an anchorage with a 1 Â 4 anchor pattern was experimentally tested and numerically analyzed by Bokor et al. 13 The influence of base plate stiffness on the group behavior was investigated under concentric tension loading with base plates of thicknesses 5, 10, 40, and 80 mm (80 mm only numerically).A 40 mm thick base plate intuitively seems sufficiently stiff to ensure an equal force distribution within this particular anchor group.However, the study showed that the relatively small anchorage (base plate dimensions 350 Â 60 Â t mm 3 with an embedment depth of 60 mm) can only be considered to distribute the forces equally, if the base plate is at least 80 mm thick.The plate with 5 mm thickness yielded before the concrete breakout failure occurred.In the case of the anchor group with 10 mm base plate thickness, it was found that first the inner two anchors of the group got activated and only after their failure the outer anchors were activated and contributed to the resistance.For the studied case, the concrete cone breakout resistance of the group with rigid plate was obtained as 85 kN, whereas it was measured only as 73.5, 48.8, and 46.1 kN for base plates of thickness 40, 10, and 5 mm, respectively.Further details and extensive evaluation of the results for this case can be found in Bokor et al. 13 and Bokor. 9he rigid base plate assumption is relevant not only for concrete breakout failure modes but also for other failure modes.As per the design philosophy, for the failure modes steel failure and pullout failure, the highest loaded anchor is checked.Assuming a linear force distribution (considering a rigid base plate), while the actual base plate is not stiff enough, invariably leads to the under-estimation of the forces in some of the anchors of a group (Figure 2).This problem can be viewed from two different standpoints: 1. What is the required base plate stiffness to consider it as rigid? 2. Under which conditions can we ensure that the distribution of the anchor forces is linear and close to the distribution obtained assuming rigid base plate?
The qualitative guidelines given in the current codes view the problem from the first viewpoint and ask for the base plate deformations to be small compared to anchor displacements.Consider a case of two anchors in a row subjected to concentric tension force (Figure 3).In this case, if the base plate is just thick enough to ensure that there is no prying of the plate overhang, the anchor forces are independent of the actual base plate thickness.However, satisfying the requirement that base plate deformations are negligible to anchor displacements (typically, 1%-2% of anchor displacements) would require base plate thickness of unrealistic proportions.This indicates that the requirement of a rigid base plate is a sufficient but not really necessary condition for all the cases.
Using an alternate viewpoint, the necessary and sufficient condition for the application of the current design models given in codes for design of anchorages can be stated as: "The values and the distribution of anchor forces with the given base plate under applied loads on the anchorage should be close to the values and the distribution of anchor forces that would be calculated assuming a rigid base plate."This would ensure that not only the forces acting on individual anchors but also the internal eccentricity of the anchor forces, if any, are close to those calculated by assuming a rigid base plate.Note that for a simple group of two anchors subjected to a concentric tension force, this condition is satisfied with a base plate just thick enough to avoid any prying action of the plate overhang.However, the plate thickness should also be sufficient to prevent the yielding of base plate, which is a general requirement of the base plate design in steel design theory.
Therefore, the anchor design according to current codes and standards stands valid provided: 1. Base plate remains linear elastic under design loads.

The values and the distribution of anchor forces with
the given base plate under applied loads on the anchorage considering the actual stiffness of base plate are close to the values and the distribution of anchor forces that would be calculated assuming a rigid base plate.
The first condition is relatively straightforward to satisfy.However, the second condition has three main aspects to consider: i.The anchor force distribution calculated using the actual (finite) stiffness of base plate is strongly dependent on the anchor stiffness (Figure 4), ii.How "close" the distribution of the anchor forces should be to that calculated assuming rigid base plate (or how much deviation is acceptable), and iii.Is the same tolerance for the deviation of anchor forces applicable for all anchors of the group?
The requirement of a stiff base plate is contradictory to the approach followed in the design of the base plates for steel structures, 14 where the plasticization of the base plate is one of the desirable failure mechanisms.In particular, in the case of column bases subjected to bending moments, the anchor bolts are considered to take up the tension forces, while the bearing of the base plate on the concrete surface provides the internal equilibrium. 15enerally, two cases are considered for the column bases.In the first case, the bolts are flexible, the base plate is stiff, and the plate is separated from the concrete.In the second case, the base plate is in contact with the concrete resulting in prying action, and the bolts are loaded by additional prying force.Alternatively, the component method can be used for the design of steel column bases. 16he higher-order models, such as the component model, 1,17 can explicitly consider the different components of a connection including the attachment, the base plate, and the anchors.In INFASO model, the headed stud anchors are considered through a network of springs in series, where each component (steel, pullout, and concrete) is modeled as a spring.However, clear rules for considering the group effect in case of concrete-related failure modes are not given for anchorages with more than one row of anchors in tension.
Certain guidelines and standards such as fib Bulletin 58, 3 previously CEB Bulletin 206, 18 and CEN/TR 17081 19 allow for the plastic design of anchorages; however, it is required that steel failure of the anchors is the dominant failure mode and that the anchors should possess sufficient elongation capacity through requirements of stretch length.The method is not suitable for anchors where these requirements cannot be met such as concrete screws.The requirement of steel failure of anchors is satisfied by following a capacity design concept, where it is ensured that the steel failure capacity of the anchors is lower than their concrete cone breakout or pullout failure capacities.When the steel failure of anchors governs, the requirement of the base plate to be rigid does not exist provided the model is able to consider the flexibility of the base plate and can account for the prying forces while calculating the forces among the anchors of the group.Therefore, the discussion given in this paper is valid primarily for anchorages undergoing concrete-related failure modes such as concrete cone failure, combined pullout and concrete failure and indirectly for concrete pryout failure in shear.Note that, although plastic design requires the concrete breakout capacity to be higher than the steel failure capacity of the anchors, the concrete breakout capacity can be reliably calculated using the CCD method only if the base plate is rigid.
This paper discusses the aspects related to the required base plate stiffness in detail and in quantitative terms.A new set of criteria is proposed, which is based on sound mathematical principles and is applicable for anchorages under generalized loading (tension-loaded anchorages, anchorages under moment loading with tension, anchorages under combined tension and shear loads).Note that the base plate stiffness in case of anchorages loaded in pure shear or under torsional moment (in plane of the base plate) is deemed sufficient due to the high in-plane stiffness of the plate and is beyond the scope of this current study.

| Methodology
The aim of this study is to develop an assessment method to ensure that for a given anchorage, the anchor tension forces have a linear distribution so that the CCD Method within its current scope can be safely applied for the calculation of anchor group resistance governed by concrete failure modes.Once a linear force distribution is ensured-using the wording of the corresponding codes and guidelines-the base plate can be considered as sufficiently stiff (equivalent to rigid).Note, however, that the base plate stiffness is not the only parameter influencing the force distribution, but every further attachment to the base plate as well as the stiffness of the anchors have an effect and must be therefore considered.This also calls for using a different definition for the requirements of using the CCD method to calculate the anchorage resistance: Instead of asking for a sufficiently stiff base plate and linear strain assumption of the plate, a better requirement would be to ensure a linear force distribution within the anchors of the group.
Within the scope of this paper, theoretical considerations are discussed, and practice-relevant examples are numerically analyzed with the help of the spring model for anchorages within the framework of the finite element method (FEM).The numerical analyses are performed on different anchorage configurations under different load cases.Every configuration and load case combination are calculated by stepwise increasing the base plate thickness.Furthermore, the calculation of the corresponding anchorages with a rigid base plate and attachment is performed to determine the ideal, linear distribution of anchor tension forces within the group for reference.The evaluation of force distribution within a group in function of the base plate thickness facilitates finding the criteria relevant for judging an anchor force distribution, which can be considered as linear for all practical purposes.Note that the criteria themselves are not new inventions; they are mathematical functions.What is new is that the appropriate criteria that are important and applicable for group anchorages had to be developed and with the combination of these criteria, a practical solution to the problem of the base plate stiffness was found.
In the first step, a calculation method was selected with which the anchor force distribution within arbitrary anchor groups can be calculated realistically.Arbitrary anchor groups refer to anchorages with different arbitrary base plate geometry, thickness, anchor pattern, attached profile, stiffener, and load case.Arbitrary base plate thickness refers to any given base plate thickness that might yield, might not yield and might be sufficiently stiff.
The sufficient stiffness of a base plate is not an absolute value, but a question of the relative stiffness of the base plate (considering further attached elements and fixity to the profile or steel column) in relation to the anchor stiffness.This signalizes that the anchor stiffness plays an important role in determining the loaddisplacement behavior of anchorages and in the assessment of base plate stiffness.Therefore, it is necessary to calculate and evaluate the anchor force distribution within an anchor group using a method that is able to account for the influence of anchor stiffness as well as the influences of the geometry of the base plate, anchor arrangement, connected section, and stiffeners.

| Calculation of the axial anchor forces
Displacement-based approaches, such as spring models 9,15,16,[20][21][22] are suitable to calculate the anchor force distribution within the group through direct consideration of the stiffness and deformations of the base plate and the anchor stiffness.In a spring model for anchorages, calculations are carried out typically using a structural analysis software based on FEM.The stiffness and deformations of the base plate can be considered by modeling the base plate and attachments with finite shells or solid elements with an appropriate bedding on the concrete surface modeled by compression springs (see, e.g., in Bokor 9 ).The anchor stiffness can be modeled using tension-only springs.Such a spring model is conceptually shown in Figure 5.The concept of modeling the anchors by tension-only springs and the contact between the concrete and the base plate by compression-only springs is based on well-accepted assumption in fastening technology that when anchorages are subjected to tension loads (also in combination with bending moments), the anchors take up the tension force while the compression is transferred directly from the base plate to the concrete.It should be noted that due to the redundant system in the case of an anchor group with multiple finite elements and springs that may be in the nonlinear range, an iterative procedure is required to solve the equilibrium equations and compatibility conditions.Therefore, the spring models are best used with the help of a software (Finite Element Analysis Software).
To calculate the realistic force distribution among the anchors of a group, a basic spring model assuming linear tension-only springs for the anchors, the so-called linear spring model, is well-suitable.However, the more advanced methods, such as the nonlinear spring model Bokor 9 or 3D finite element models, might also be appropriate for the calculation.A linear spring model for anchorages considers the anchors as tension springs with a constant value for their stiffness, independent of the force acting on the anchors, anchor spacing, and close edges.The stiffness of the individual anchors within a group is assumed the same, since current design guidelines require that only anchors of the same type and size be used in a group.A linear spring model allows the calculation of the anchor force distribution of arbitrary anchor groups within the linear range in one calculation step, in which the load combination is applied on the anchorage and the forces and deformations are calculated in function of the geometry, material and stiffness conditions. 21,22That means that for any given base plate thickness, the realistic force distribution can be calculated within the linear range of the load-displacement behavior of the anchorage.An example is shown in Figure 6.
For the calculation model, the anchorage geometry, including the base plate and attachments, the applied load or/and bending moment, and the initial anchor axial stiffness, are known parameters.The load leads to deformations, which are determined at every node of the FE model.The individual anchor forces can be calculated by multiplying the anchor stiffness and the vertical displacement read at the anchor location.The results of the calculations are the anchor forces, the deformations, and the steel stresses in the base plate.If the same anchor group is analyzed with both, a given thickness of base plate and a rigid base plate, a comparison of the anchor forces (actual/rigid) is possible, and conclusions can be drawn about whether the base plate is stiff enough to distribute the forces linearly.
In this paper, the following nomenclature is used to describe the anchor forces calculated with a given thickness and rigid base plate: Anchor forces from the calculation with rigid base plate:

represents the index of the anchor under consideration)
Tension force in the highest loaded anchor with rigid base plate: N h,rigid: Tension force in the least loaded anchor with rigid base plate: N l,rigid Anchor forces from calculation with a given base plate thickness: Tension force in the anchor that was highest loaded assuming a rigid base plate, calculated with a given base plate thickness: N h Tension force in the anchor that was least loaded assuming a rigid base plate, calculated with a given base plate thickness: N l If, for example, with rigid base plate assumption, the highest loaded anchor is anchor 1 (N h,rigid = N 1,rigid Þ, then the force in the corresponding anchor calculated with the given base plate thickness is Similarly, if, for example, with rigid base plate assumption, the least loaded anchor is anchor 3 (N l,rigid = N 3,rigid Þ, then the force in the corresponding anchor calculated with the given base plate thickness is N l = N 3

| Approach to the solution
As explained above, the comparison of the anchor forces obtained from the calculation of anchorages with base plates of a given thickness (stiffness) and corresponding rigid base plates (incl.attachments) is helpful to judge whether the force distribution is linear among the anchors of an anchorage.However, the actual force distribution among the anchors of a group with a real value of base plate thickness will always deviate from the distribution calculated assuming a perfect rigid base plate, since in reality the base plate cannot be infinitely stiff.Strictly assessing, linearity is an exact term, and a force distribution of the arbitrary thick base plate can be considered linear only, if there is no deviation in the anchor forces as compared to the forces calculated assuming a rigid base plate.However, achieving this is unrealistic and practically impossible.Therefore, there is a need to give guidance, which conditions must be fulfilled and how much deviation in anchor forces may be allowed such that the distribution of anchor forces can be considered linear for all practical purposes.
There are different ways of defining a line (linear equation).If two different points (x 1 , y 1 ) and (x 2 , y 2 ) are given, there is exactly one line that passes through the points (Figure 7a).In case of three points, the equation is only linear if one line passes through all three points.The linear equation can also be defined by one point (x 1 , y 1 ) or (x 2 , y 2 ) and the slope of the line.The slope of line, m, is defined by the ratio Δy/Δx = (y 2 Ày 1 )/(x 2 Àx 1 ).Thereby, how close two lines are to each other can be adjudged by comparing either the individual coordinates or by comparing the coordinates of one point and the slope.
If this is applied to an anchorage with two anchor rows loaded by a combined bending moment and concentric tension, the given two different points can be considered as the highest and lowest loaded anchors with x for anchor locations and y for anchor forces as shown in Figure 7b.Based on the available information, the equivalent slope can be expressed as the change between the highest and lowest loaded anchor over the spacing s.Thereby, the similarity of the anchor force distribution for a group with a given base plate and a stiff base plate can be adjudged by comparing either the individual anchor forces or by comparing the force in one of the anchors (say, highest loaded) and the slope (difference) of anchor forces.
Note that the requirement is to ensure the linear distribution of anchor tension forces.One of the ways to do this is to ensure that the base plate remains linear (rigid), for which the deformation of several points on the base plate can be checked.However, here, a more straightforward and simpler approach is proposed that directly checks the distribution of anchor forces through a set of criteria.
In the following, several aspects are critically scrutinized and the found solution is explained by means of practical examples.

| Necessary condition: Base plate does not yield
The most basic and necessary condition for a base plate to be considered as rigid is that it shall remain linear elastic under the applied loads.However, this condition alone is not sufficient for a base plate to be considered as rigid or sufficiently stiff (Equation 1).This criterion is the state of the art and is used in practice for the evaluation/design of base plates and fixtures.The determination of the stresses in the base plate and attachments can be carried out, for example, according to the von Mises yield criterion.To do this, the equivalent stress on the positive surface side and on the negative surface side of the base plate are calculated and the maximum absolute value of the surface equivalent stresses σ Steel base plate are determined.Other evaluations may also be found appropriate.The base plate thickness at which the plate remains elastic is t stress .
Note that in practice, a stress averaging in the area, where the profile is attached to the baseplate is common.The distance, over which the stresses are usually averaged, is two times the plate thickness plus the wall thickness of the attachment. 8

| Assessing individual anchor forces-Method I
The easiest way to give a criterion ensuring linear force distribution among the anchors of a group could be to require that the difference in the individual anchor forces calculated assuming a rigid base plate and using an arbitrary thick base plate do not differ by more than a certain threshold value X 0 (Equation 2).
In principle, this is a rather straightforward criterion to ensure that the calculated values of anchor forces for a given base plate thickness are within reasonable accuracy with respect to the anchor forces calculated assuming a rigid base plate.
For certain cases, such as in case of a symmetric anchorage, with maximum of two anchors in a row (1 Â 2 or 2 Â 2 group) under an applied concentric tension force all anchors receive equal forces irrespective of the thickness of base plate.If the base plate would not pry on the concrete surface, the anchor forces calculated with a given base plate thickness would always be equal to the anchor forces calculated assuming a rigid base plate.However, if the base plate is flexible enough to pry on the concrete surface, in the overhang region, the anchor forces will be larger than the forces that would be calculated assuming a rigid base plate (Table 1).Therefore, for such cases, to suitably apply the CCD method, the requirement of small base plate deformations (refer to EN 1992-4 1 ) is redundant and strictly speaking only the requirement that the base plate does not pry on the concrete surface should be sufficient.The prying, if applicable, should be limited to the extent that would not significantly reduce the safety margin.In mathematical terms, if the deviation in the anchor forces is not greater than an acceptable margin (say 10%), the application of the CCD method might be considered acceptable.One such case is highlighted in Table 1, where for a base plate thickness of 17 mm, the anchor forces deviate by 9.2% from those calculated assuming a rigid base plate.From engineering viewpoint, this error may be considered acceptable, and the CCD method may be used to calculate the resistance of the anchorage.Note that the requirement of stresses remaining within the yield limit is satisfied with 15 mm thickness, but the deviation in forces calculated with a 15 mm thick base plate is more than the acceptable limit of 10%.
Although Equation ( 2) gives a rather straight forward criterion to determine the adequacy of the base plate thickness to apply the CCD method, the criterion can be quite sensitive for certain cases where some of the anchors of the group are loaded by a rather small tension force in case of a rigid base plate (N i,rigid is small).In such cases, a small value of N i,rigid can result in numerically high value of the % deviations and meeting the criterion defined in Equation ( 2) would become rather difficult or impractical.For example, in case of eccentrically loaded anchorages or anchorages subjected to bending moments, certain anchors of the group may receive rather low level of tension loads or even lie in the compression zone and carry no tension force.In extreme cases, even the nature of forces in an anchor (small tension or compression) may change as a function of the base plate thickness.
Such an example is depicted in Table 2 that shows a 2 Â 2 anchor group subjected to a uniaxial bending moment.In this case, anchors 1 and 3 are the highest loaded anchors in tension, whereas anchors 2 and 4 receive either a small tension force (for a stiff/thick base plate) or are in the compression zone (for relatively thin base plate of 27 mm thickness).Note that for this case, a base plate of 20 mm thickness would be enough to prevent yielding.Assessing the deviation of the anchors that either carry very small forces or zero forces; or changing from very small tension force to zero (compression) in case of any given plate thickness compared to the rigid plate, might result in very high value of deviation or dividing through zero making the evaluation numerically impossible.However, the question is that since these anchors are loaded with a very small force, they have a rather small influence on the resistance of the anchorage and on the group behavior.Therefore, making a decision on the required base plate stiffness by assessing the forces in these anchors that might result in unrealistically thick plates, is probably irrelevant.
To ensure that the calculated values of anchor forces for a given baseplate thickness are within reasonable accuracy with respect to anchor forces calculated assuming a rigid baseplate, appropriate application of this criterion should exclude anchors loaded with a rather small force from evaluation based on Equation (2).Table 2 shows that if all anchors are considered in the evaluation, the required base plate thickness would be 85 mm, while if anchors 2 and 4 are excluded from the evaluation, a base plate thickness of 27 mm is sufficient to satisfy Equation (2) with 10% as the critical value.
However, the question arises, which anchors should be excluded from the evaluation?To decide which anchors might be safely excluded from the evaluation, engineering judgment should be exercised.Intuitively, anchors receiving a tension force lower than a certain threshold (a critical % of the load taken by the highest loaded anchor) may be excluded.However, this critical value would depend on case-to-case basis and on the desired level of accuracy.This means, however, that again no clear guidance can be given for anchors that are not concentrically loaded.Therefore, further criteria or methods should be found to enable a generally applicable assessment method.This is attempted by the authors as explained in the next section.

| Linear force distribution approach, set of three criteria-Method II
The previous section showed that limiting the stresses in the base plate within elastic range is a necessary but not a sufficient criterion for the application of CCD method.On contrary, comparing the anchor forces in each anchor corresponding to a given base plate and assuming a rigid base plate is sufficient but not always necessary, and in certain cases results in unrealistically high base plate thickness values.To offer a solution, in the following, a set of three different criteria is presented which is necessary, sufficient and results in optimum base plate thickness values for which the CCD method can be safely applied.The criteria set includes: (i) assessing the forces in the highest loaded anchors, (ii) assessing the sum of anchor forces, and (iii) assessing the difference in anchor forces (slope).Each criterion and its purpose are explained below.

Assessing the highest loaded anchor
In this criterion, the anchor forces are calculated using a given base plate thickness and the corresponding rigid base plate, and the deviation between the corresponding force values is assessed only for the highest loaded anchor determined in the calculation with a rigid base plate (Equation 3).Note that this is again a necessary but insufficient criterion, since the highest loaded anchor defines only one point in a coordinate system where the anchor forces are plotted in function of anchor location and so is not sufficient to describe a linear equation (compare (x 2 , y 2 ) Figure 7b).
It is an important criterion to ensure that the highest loaded anchor is not overloaded by more than a certain threshold value X 1 , for example, 5%.Also, within the CCD method, in case of failure modes such as steel failure or pullout failure, the verification (E d ≤ R d ) requires checking the highest loaded anchor only.However, to define or recommend an appropriate value for X 1 , further considerations are necessary such as the influence of the overloaded highest loaded anchor on the group behavior, influence on the assumed probability of failure of the anchorage, and so on.To satisfy force equilibrium, if certain anchors are overloaded, other anchors must receive lower forces as compared to the reference case (rigid plate), provided there is no prying action in the force flow.
However, a difference in the calculated tension forces among the anchors of a group would also result in a difference in the internal eccentricity of the anchor forces.Furthermore, in the case, when the base plate pries on concrete surface, the sum total of the tension forces in all the anchors may be greater than the applied external action.Such questions and discussions on the definition of an appropriate threshold value already signalize that although this is an essential criterion, this criterion on its own is not sufficient because it does not account for the group behavior and the anchor force distribution.
Furthermore, in certain anchorage configurations and load cases, the highest loaded anchor with a rigid plate and the highest loaded anchor with a given base plate may not be the same anchor.In such cases the question arises: Which anchor should be considered as highest-loaded?The one, which receives the highest loads, when the force distribution is linear (rigid plate) or the one, which is highest loaded in the investigated case (given plate thickness)?For an objective application of Equation ( 3), the same anchor should be used to verify Equation (3).Since different anchors may be highest loaded for different base plate thickness, the authors define the highest loaded anchor as the anchor that receives the largest tension forces when analyzed assuming a rigid base plate.
The criterion "highest loaded anchor" ensures that the highest loaded anchor is not overloaded by more than a certain threshold value but is not a sufficient condition for linear force distribution among the anchors of a group.Considering only the deviation of the highest loaded anchor would result in most of the cases in much thinner base plates as compared to assessing the deviation at all anchors (if provided X 1 = X 2 ), which might not be safe in every case.Therefore, further criteria should be investigated.
Assessing the resulting anchor forces and their sum Depending on the anchorage configuration, base plate thickness and other dimensions as well as applied loading, it is possible that the base plate of the anchorage pries on the concrete surface resulting in additional tension forces being developed in the anchors of the group for a given applied load.The effect of prying in an anchorage with a given base plate thickness can be compared with the effect of prying in the corresponding anchorage with rigid base plate through the deviation of the sum of all anchor forces in both cases.Fulfilling Equation (4) ensures that the anchor group with a given base plate is not overloaded or underloaded by more than X 2 % compared to the same anchor group subjected to the same loading assuming a rigid baseplate.
This criterion accounts for the influence of prying forces, which is not possible if only the highest loaded anchor is used for the assessment.In case the base plate pries on the concrete surface, the sum of the anchor forces can be higher than the total applied load (refer to Table 1 with 17 mm base plate thickness).Since the number of anchors in a group is the same irrespective of the base plate thickness, the ratio of sum of anchor forces ( ) is equal to the ratio of the average force in each anchor (average force = total force divided by number of anchors).In this way, this criterion provides a smeared approach to verify the group forces because it takes into account the deviation of sum of all anchor forces.Note that the calculation of the group resistance in case of concrete cone failure according to the CCD method can also be visualized as a smeared approach, where the influence of adjacent anchors connected with a common base plate is addressed as a group effect and not on an individual anchor basis.
Satisfying Equation (3) ensures that the force in the highest loaded anchor for the group with a given base plate is close to force that would be calculated assuming a rigid base plate.Additionally, satisfying Equation (4) ensures that the prying action of the base plate, if any, and its influence on anchor forces is accounted for.However, even satisfying both Equations ( 3) and ( 4) do not ensure that the distribution of anchor tension forces is close to linear as in case of a rigid base plate.Therefore, an additional criterion that focuses on the linearity of the anchor force distribution is required.This is given in the next section.

Assessing the linearity of anchor force distribution
As discussed in Section 2.3, defining a line using a reference point and slope can be analogously seen as defining linearity of base plate using a reference point (force in the highest loaded anchor) and slope (variation in anchor forces over anchor spacing; refer to Figure 7).Earlier, it was discussed that for a correct and safe application of the CCD method, it must be ensured that the values and the distribution of anchor forces with the given base plate under applied loads on the anchorage should be close to the values and the distribution of anchor forces that would be calculated assuming a rigid base plate.Using the analogy with the line defined using a point and slope: two lines can be considered same if the reference point and the slope for both lines are the same; similarly, a base plate can be considered sufficiently stiff if the reference point (highest loaded anchor) and the slope (variation in anchor forces) is close to the corresponding values obtained assuming a rigid base plate.Verification of the reference point is already done using the Assessing the highest-loaded anchor section.Here, the verification of the variation of anchor forces, hereafter referred to as slope due to the analogy with the straight line, is presented and discussed.
The anchor force distribution within a group can be described with the simple linear function f(x) = ax + b, where "a" corresponds to the difference between the highest and lowest anchor force over the distance between them (s), and "b" is the highest anchor force (refer to Figure 7).The parameter "a" can also be expressed as the slope of the line, which is the constant rate of change of f(x) per unit change in x.
The linear force distribution among the anchors of a certain group can be ensured if Equation ( 5) is satisfied.First, the force difference between the highest loaded anchor (N h,rigid ) and the lowest loaded anchor (N l,rigid ) is calculated assuming a rigid base plate (denominator).Then, a second force difference is calculated between the same anchors (N h and N l ) with a given base plate (numerator).Subsequently, the quotient between the second force difference and the first force difference is formed.The first force difference and the second force difference can be understood as the first and second slopes of the connecting line.The anchor spacing between the highest loaded and lowest loaded anchors "s" (decide based on the rigid case) cancels out in the equation because always the corresponding anchor pairs are taken.
Note that in case of anchorages with bi-directional eccentricity and/or biaxial moment loading, the equation should be satisfied additionally in both loading directions: First, the force distribution is calculated, and the highest loaded anchor (rigid base plate) is determined.
Then, the anchors in two perpendicular rows that intersect at highest loaded anchor are identified.And finally, the slope is checked along those two lines.
To explain how to assess whether the force distribution is linear among the anchors of a group with a slope criterion, calculations on two anchorages are analyzed in the following.The geometry of the steel components such as base plate and attached profile of the anchorages are the same (Figure 8).The assumed steel strength corresponds to S235.However, in the first example, a 3 Â 2 anchorage, and in the second case a corresponding 2 Â 2 anchorage (leaving out the two middle anchors of the 3 Â 2 configuration) are analyzed.The dimensions of the rectangular base plate are 455 mm Â 300 mm by varying the thickness t.The load case is the combination of a concentric tension force of 80 kN for 3 Â 2 anchorage and 50 kN for 2 Â 2 anchorage along with a bending moment of 9 kN m, applied about the strong axis of the concentrically placed HEAA200 steel profile.For the anchors, an embedment depth of 100 mm and an initial axial stiffness of 100 kN/mm were assumed.
Several numerical analyses were performed by stepwise increasing the base plate thickness starting from 10 mm up to 82 mm.In addition, an analysis assuming a rigid base plate of the corresponding configurations was performed for reference as described in Section 2.2.In Figure 9a, the deviation in tension force obtained for each (ith) anchor for a given base plate thickness compared to the force in the same anchor obtained for reference case (rigid base plate) calculated using Equation ( 6) are plotted in function of base plate thickness for the 3 Â 2 anchorage (Figure 8a).Figures taken from FiXperience Online. 23 As expected for relatively thin base plates, the anchor forces calculated for the given base plate thickness deviate significantly from those calculated assuming a rigid base plate.It can be seen that with increasing the base plate thickness, the deviation converges to zero, meaning that the force distribution is getting close to the reference linear distribution.Note that the highest deviation is obtained for anchors 2 and 5, which correspond to the anchors in the middle anchor row.This is attributed to the fact that thin base plates tend to deform in double curvature due to applied bending moment resulting in large displacements and therefore high anchor forces for both outer anchor rows (see Figure 10a).On contrary, the rigid base plate rotates linearly and therefore the middle anchor row receives lower forces than the outer anchor row (see Figure 10b).Consequently, the forces in the highest loaded anchors (outermost anchors) converge at lower base plate thickness to the target value, whereas the forces in the middle anchor require thicker base plates to reach the required level of convergence.Note that the 0 and 450 mm correspond to the left and right ends of the base plate along the x-axis and 0 and 300 mm on the y-axis.The anchor locations are 50, 227.5, and 405 mm on the x-axis, and 50, 250 mm on the y-axis for the 3 Â 2 configuration and 50, and 405 mm and 50, 250 mm for the 2 Â 2 configuration, respectively.
This can be visualized clearly in Figure 9b, where the anchor forces obtained for different base plate thicknesses are plotted for each anchor.In case of a rigid base plate, a linear anchor force distribution is represented by the three symbols for the individual anchors on the solid black line.It can be noticed that for rigid base plate, all three anchors are in tension with individual anchor forces as approx.25 kN for anchors 1 and 4, approx.13 kN for middle anchor row (anchors 2 and 5) and approx. 2 kN for anchors 3 and 6.For a 10 mm thick base plate (shown by red dotted line and red empty circles), both outer anchor rows (anchors 1, 4 and 2, 5) receive almost equal force of approx.30 kN while anchors 3 and 6 lie in the compression zone and receive no tension force.As the thickness of base plate is increased, the force distribution as well as the individual anchor force values tend to get closer to the results obtained assuming a rigid base plate.
The closeness of the anchor force distribution can be judged by comparing the force in the highest loaded anchors (anchors 1 and 4) and by the slope of the lines (difference of forces between anchors 1 or 4 and anchors 3 or 6).Thereby, simultaneously satisfying Equations ( 3) and ( 5) will result in assessing the base plate thickness that would result in anchor force distribution close to that of a rigid base plate.Only for such base plate thickness values, the CCD method can be applied safely.For the example shown in Figure 9, if Equations ( 3)-( 5) are satisfied and a threshold value of 10% and 5% is assumed, a base plate thickness of 33 mm or 52 mm, respectively, can be applied.Similarly, if Equation ( 2) is satisfied and a threshold value of 10% and 5% is assumed, a base plate thickness of 61 mm or 82 mm, respectively, might be used.
Similarly, in Figure 11a, the individual anchor force deviation compared to the reference case in function of base plate thickness is shown for the 2 Â 2 anchorage.In this case, with increasing the base plate thickness, anchors 1 and 3 that are the highest loaded converge fast.Once the base plate is not yielding, they already converge to the reference case.Anchors 2 and 4 converge rather slowly.Plotting the anchor force deviation compared to the reference case in function of anchor location as shown in Figure 11b is not helpful in judging linear force distribution because when connecting two points, there is always exactly one linear line going through them suggesting linear force distribution; however, not giving information about the relation to the reference rigid case.For that, the slope of the line should be evaluated to assess the linearity of the force distribution (refer to Section 2.3).
For further illustration and clarification of the criterion given by Equation ( 5), additional evaluations were performed using the FE Model of the connection following a linear spring model approach (Figure 12a).The base plate deformations were plotted for different base plate thicknesses (see Figure 12b-d) and the corresponding vertical displacements were evaluated, in particular at the section cut A-A over the anchorage in the longer side through two anchor rows as shown in Figure 12a.This allowed to evaluate displacements at the anchor location, which in turn allowed the direct calculation of anchor forces because of the linear anchor stiffness assumed in these models (N i ¼ δ i Á k 1 ).Note that in case of uniaxial bending it is rather straightforward to choose the locations that should be identified for the assessment.However, in case of complicated loading scenarios, this might be difficult.
It may be questioned whether the evaluation according to Equation ( 5) is accurate enough since once certain anchors lie in the compression zone, zero load is assumed as their contribution.If the displacements are evaluated, even "negative displacements" (considering the bottom of the base plate as zero and the tension direction as positive) can be taken into account, and so a more precise evaluation could be achieved.However, these negative displacements on the compression side of the base plate take up very small values (to achieve negative displacements the plate has to be pressed into the concrete) which may cause numerical problems.Furthermore, if the criterion according to Equation ( 5) is combined with the criterion according to Equation (4), the "missing" negative force values can be captured due to considering the sum of anchor forces, whereas fulfilling Equation (3) ensures that the highest loaded anchor is not too much overloaded.Therefore, a combination of Equations ( 3)-( 5) would ensure that the anchor forces, force distribution as well as the prying action, if any, obtained for a given base plate thickness are close to the corresponding values obtained assuming a rigid base plate.

| Criteria set for evaluation of required base plate stiffness
The investigations have shown that linear force distribution among the anchors can be ensured if a set of criteria is fulfilled at the same time.For anchorages under concentric tension load, both Equations ( 1) and ( 2) must be satisfied (Table 3).For anchorages under eccentric tension, bending moment, or combined tension load and bending moment, Equations ( 1) and ( 3)-( 5) must be satisfied as summarized in Table 3.
If the corresponding set of criteria are satisfied, the anchor force distribution can be considered close to linear, 24 Results of numerical analyses displaying the base plate displacements of an anchorage with a base plate thickness of: (b) 11.5 mm, (c) 44 mm, and (d) 98 mm. and the base plate can be considered sufficiently stiff to distribute the forces linearly and apply the CCD Method for the calculation of resistance against concrete cone failure safely.Note that satisfying Equations ( 1) and (2) (=Method I) delivers a safe solution for every case, independent of loading.However, as explained in Section 2.3.2, it gets sensitive to the small values of anchor forces and results in unrealistically thick base plates for certain anchorages with unequal force distribution.For the majority of the cases with bending moment or eccentric loading, satisfying Equations ( 1) and ( 3)-( 5) (Method II) would result in optimum and safe values of base plate thicknesses.For a generally applicable solution that is safe but also practical, a force distribution might be considered as linear, if either the equations of Method I or Method II are satisfied.
In the next step, appropriate threshold values should be defined for X 0 -X 3 to ensure the linear force distribution among the anchors of a group.For this, the sensitivity of the criteria, such as the influence of the threshold values on the sufficient thickness of the base plate, and the convergence of the criteria are studied in Section 2. 4. Verification examples are given in Section 3.

| Proposed convergence criteria and threshold value
In the current design of anchorages according to EN1992-4, 1 a probabilistic approach using the partial factors is followed.The characteristic resistance, which is calculated corresponds to the 5% fractile of the resistance (value with a 95% probability of being exceeded, with a confidence level of 90%).A design using the partial factors given in EN1992-4 1 and the partial factors given in EN 1990 is considered with the reliability class RC2, that is, a β-value of 3.8 for 50-year reference period.In the verification for ultimate limit state, an appropriate degree of reliability is used so that the anchors can sustain all actions and influences, which are likely to occur during execution and use.

Method I Method II
Primarily for anchorages loaded under concentric tension Anchorages loaded in eccentric tension, bending moment, or combined axial loading and bending moment If all criteria are satisfied, the force distribution is close to linear among the anchors of a group, and the base plate can be considered sufficiently stiff to distribute the forces linearly.
In case of anchorages, which are loaded mainly in concentric tension via the rigid fixture or rigid base plate results in almost equal axial tension in all anchors.In such cases, the deviation of the anchor force compared to the reference linear case (rigid base plate) at every anchor should be checked and should be smaller than a threshold value, for example, X 0 according to Equation (2) described in Section 2.3.2.For simple groups subjected to a concentric tension force, this condition is satisfied with a base plate that is thick enough to avoid any prying action of the plate overhang.Note that since the resistance is calculated by the CCD method, an overloading of the anchorage with a given base plate by X 0 % would result in a straightforward reduction in the safety margin of X 0 %.Following certain simplifications, the probability of failure can be calculated as per theory for an exemplary anchorage that is subjected to concentric tension loads.For this, a normal distribution is assumed with 100% utilization (E d = R d ) in case of 0% overload (=rigid base plate), 5% overload, 10% overload, and 15% overload.In this context, "overload" refers to the maximum deviation between the anchor forces calculated using a rigid base plate and those calculated using a non-rigid (or arbitrary thick) base plate.The maximum deviation between the anchor forces calculated assuming a rigid (N i,rigid Þ and non-rigid base plate N i ð Þ required for the evaluation of the failure probability can be calculated according to Equation (2).The performance function, the standard deviation and finally the reliability index β and probability of failure are determined.The probability of failure of the exemplary anchorage results in ca. 1 out of 1.000.000if there is no overload (β = 4.73), 2 out of 1.000.000if the overload is 5%, 3 out of 1.000.000if the overload is 10% and 5 out of 1.000.000if the anchorage is overloaded by 15%.Note that these values obtained for the probability of failure are calculated for the utilization of 100% considering the highest loaded anchor.In case of certain failure modes, the utilization is checked for the anchor group and therefore, the influence due to over-loading might be smeared on the group and the utilization degree might be lower than 100%.Consequently, for the same anchorage, the failure probability might be different than that determined based on the highest loaded anchor.Taking into account the considerations and discussion above, a threshold value such as X 0 = 5% may be recommended, however, even up to X 0 = 10% may be acceptable.If required, a more detailed reliability analysis can be performed if values of deviation higher than 15% are used.
In case of anchorages, which are loaded in a way that the tension force in the anchors of the group is not equal, for example, due to eccentric tension, bending moment, or combined tension load and bending moment, a set of three criteria must be satisfied simultaneously to ensure that the anchor force distribution is close to linear.All three equations should result in a value smaller than the threshold values (deviation), for example, X 1 , X 2 , and X 3 .This threshold can be determined by evaluating the convergence of X 1 , X 2 , and X 3 in function of the base plate stiffness.Once the plotted curve becomes gradual and stable and shows no oscillation, it can be assumed that the base plate thickness convergences to a value, from which the anchor force distribution remains linear and further increasing the thickness does not result in changes in the anchor force distribution.
Several numerical Finite Element calculations were performed to analyze, which threshold values are appropriate and whether X 1 , X 2 , and X 3 should take up the same value or any of them should be stricter than the others.Three select examples are given in Table 4: Anchorages (a) and (b) correspond to those discussed in the Assessing the linearity of anchor force distribution section, Figures 9 and 11.The third example (c) is a 2 Â 2 anchorage with a square base plate of side length 300 mm, anchor spacing of 200 mm and with a square hollow section QSH 100 Â 10 loaded in combined concentric tension (50 kN) and biaxial bending (5 kN m about both axes).For all three cases, the embedment depth was h ef = 100 mm, and the axial anchor stiffness was taken as k 1 = 100 kN/mm.In Table 4, the anchorage configuration, the loading combination, and the base plate thickness corresponding to no yielding, Δ highest ≤ X 1 = 10%, Δ sum ≤ X 2 = 10%, and Δ slope ≤ X 3 = 10% are given.
The analyses have shown that in different anchor configurations and load combinations, different criteria might govern.For case (a), for base plate thickness smaller than 33 mm, highest loaded anchor always shows the largest deviation among the three criteria.Beyond 33 mm base plate thickness, the slope criterion results in slightly larger deviations compared to the highest loaded anchor criterion.Interestingly, if the critical value of convergence (deviation) is taken as 10%, the required base plate thickness is governed by the highest loaded anchor, while if the critical value is taken as 5%, the required thickness is governed by the slope criterion.For this case, the sum of anchor forces gives the lowest values of deviation for all base plate thicknesses investigated.For case (b) with two anchor rows, the convergence of highest loaded anchor and slope criterion is rather similar with a slower convergence for the slope criterion and therefore it remains the decisive criterion to determine the required base plate thickness.In case of case (c), the biaxial eccentricity requires the slope to be checked for the absolute highest and lowest loaded anchors, and for both orthogonal directions leading to thicker plates for satisfying linearity.
For a general applicability, it is recommended to require the same threshold for X 1 , X 2 , and X 3 to satisfy the set of criteria.Considering the investigated cases, 10% seems reasonable.However, for more strict evaluation, 5% or smaller value can be recommended.Furthermore, note that the necessary but not sufficient condition to ensure a linear force distribution is that the base plate does not yield.This means that if a baseplate yields at a certain thickness, then the behavior of the anchorage and the convergence of the criteria X 1 , X 2 , and X 3 is not relevant for the assessment.

| COMPARISON WITH EXPERIMENTAL TESTS
In this section, the suitability of the proposed criteria set to determine the required base plate thickness is assessed against experimental results from the literature.
3.1 | Anchor groups of 1 Â 3 configuration loaded in concentric and eccentric tension

| Experimental results
Experimental test series G51R and G52 on 1 Â 3 anchor groups (three anchors in a row), from Bokor et al. 10,11 were taken for the assessment.The tests were carried out in concrete slabs with measured mean concrete cube compressive strength of 66.6 N/mm 2 using bonded anchors installed away from the edge, under concentric and eccentric loading.The effective embedment depth of the M16 anchors was h ef = 70 mm, and the anchor spacing was s = 120 mm.The series G51R was tested under centric load with the actual load applied at two points equidistant from the middle anchor. 10,11In series G52, the load was applied at an eccentricity e = s/2 = 60 mm.Detailed evaluation of the test results is reported by Bokor et al. 10,11 and Bokor. 9Note that the purpose of these tests was not to verify the assessment criteria presented in this work; however, the tests are suitable for it.Table 5 summarizes the test configurations and measured mean ultimate loads along with the mean concrete cone resistance calculated according to the CCD method.For the tests, where the measured mean ultimate load is significantly smaller than the calculated value, it is assumed that the base plate thickness was not sufficient to distribute the forces linearly.In this paper, the tested configurations were numerically analyzed with different base plate thickness values, assuming a steel grade of S235, and applying a tension force of 28 kN, which corresponds approximately to the design tension resistance of a single anchor and approx.70% of the design resistance of the group G52.The applied force lies in the linear range of the load-displacement curves obtained in the experiments.It was necessary to remain in the linear range, because a spring model with linear spring characteristics was used for the analyses as described in Section 2.2.Note that the assessment of the linear force distribution is independent of the applied force level.However, the minimum base plate thickness requires a non-yielding base plate, which is force-dependent.For the corresponding tests, an initial stiffness of ca.k = 300 kN/mm is reported in Bokor et al. 10,11 and Bokor. 9he calculations were performed considering two different values of anchor stiffness, namely 100 and 300 kN/ mm to show the influence of anchor stiffness on the base plate thickness, which is sufficient to distribute the forces linearly.Each anchorage configuration and with both stiffness values 100 kN/mm (red symbols) and 300 kN/ mm (black symbols) was calculated with 25 and 50 mm base plate thickness and assuming rigid base plate.Furthermore, the thickness, at which the base plate is not yielding (symbol at the smallest base plate thickness) and the base plate thickness that satisfies the criteria according to Table 3 (vertical lines in the graphs), with a threshold value of 10% for all three criteria were determined.In Figure 13, the evaluation of the analysis results in terms of anchor force deviation compared to the reference case (rigid base plate) and convergence of criteria in function of base plate thickness are given.First, in Figure 13a,c, the deviations of individual anchor forces are plotted in function of base plate stiffness for different anchor groups tested assuming a lower and higher stiffness for the anchors.Correspondingly, in Figure 13b,d, the convergence values for different criteria defined in Table 3 are plotted for each case.
For the group tested under concentric tension load (series G51R), the criteria of Δ max , Δ highest , and Δ sum are evaluated.Due to concentric tension, in this case, the Δ slope criterion is not applicable.From Figure 13b, it can be noticed that assuming 300 kN/mm anchor stiffness, the required thickness for linear force distribution is 65 mm, which is controlled by Δ max or Δ highest criteria.In the tests by Bokor et al., 10,11 a base plate of 50 mm thickness was used, and the mean test failure load was found to be significantly smaller compared to the mean concrete cone resistance calculated as per the CCD method (compare Table 5).This suggests that a 50 mm thickness for base plate is not sufficient to ensure a linear force distribution among the anchors and thereby not enough to safely apply the CCD method.With a different anchor stiffness, however, assuming 100 kN/mm stiffness, a base plate thickness of 45 mm would be sufficient.This suggests that satisfying the proposed criteria set can safely and economically ensure a linear anchor force distribution.
In case of series G52, the load was applied with an eccentricity of 60 mm.For this case, the required thickness to satisfy the criteria set of Δ highest , Δ sum , and Δ slope is obtained as 49 mm for 300 kN/mm anchor stiffness (Figure 13d).The tests were performed with a 50 mm thick base plate, and the measured and calculated mean resistance showed only a 2% difference (Table 5).This again indicates that satisfying the criteria set proposed in this paper results in the correct assessment of sufficiently stiff base plate.Note that for the case with assumed anchor stiffness of 100 kN/mm, the required base plate thickness comes down to only 36 mm.

| Numerical analysis
In this paper, only the stiffness of the base plate is evaluated.To this end, the tested configurations were numerically analyzed using a linear spring model assuming an anchor stiffness of 430 kN/mm.The configuration described in the experimental paper was simulated, but without considering the shear lug and the steel plates placed on the top of the anchors, which served as large washers to cover the holes in the base plate during the experiments.The base plate thickness was set to 20 mm, consistent with the value reported in the experimental setup.In the tests, an axial compression of 400 kN was applied and the failure of the connection was obtained at a moment of 128 kNm.This work does not evaluate the resistance of the connection, which may be obtained using the component model or the nonlinear spring model.Nevertheless, it is intended to check whether the base plate of 20 mm would be considered stiff enough according to the proposed criteria or not.Therefore, a compression force and bending moment ratio of 100 kN and 32 kNm, respectively, were applied to the attached profile, which gave the force-to-moment ratio consistent with the test results (400:128).Figure 14a displays the von Mises stresses in the base plate with a thickness of 20 mm, as obtained from the analysis.The stress profile aligns well with the reported results.The experimental results reported by Bajer et al. 25 indicate that the base plate is not sufficiently stiff, resulting in a nonlinear deformation profile.The analysis results agree well with the experimental findings (Figure 14b).Due to the nonlinear deformation of the base plate and the corresponding prying action, the tension-loaded anchors are significantly overloaded compared to the values that would be calculated assuming a rigid base plate.The tension forces are only transmitted through a single anchor row (anchors 1 and 2).If a rigid base plate is assumed, the force in the tensionloaded anchors is calculated as 21.14 kN.For the case analyzed with actual base plate thickness (20 mm), anchors 1 and 2 each bear a load of 35.65 kN.This corresponds to 68.6% overload compared to the same anchorage with an assumed rigid base plate.Considering the same resistance from concrete, this means that the anchorage will fail at 68.6% lower load compared to that obtained using the CCD method assuming a rigid base plate.This again highlights that assuming a rigid base plate when it is not can be very unconservative for the application in the design according to the CCD method.
Further analyses were carried out with increasing base plate thickness to find, which thickness value will be considered sufficiently stiff as per the proposed criteria.It was found that if a threshold value of 10% was chosen for Δ highest , Δ sum , and Δ slope , a base plate thickness of 108 mm met the criteria.
Figure 14c shows the linear displacement profile of the investigated base plate connection with 108 mm base plate thickness.In this case, anchors 1 and 2 receive 23.24 kN tension, which is 9.9% overload compared to the same anchorage with rigid base plate.If a more stringent threshold value is chosen, the overload on the anchors can be reduced.However, this comes at a cost of an even thicker base plate.

| SUMMARY
The CCD method is currently the only recognized method for the design of anchorages against concrete cone breakout failure under tension loads.An inherent assumption of this method is that the forces among the anchors of a group are distributed in a linear manner (the so-called rigid base plate assumption).This means that for anchorages under centric tension loads, all anchors of a group are assumed to be subjected to equal tension forces, while for anchorages under eccentric tension load, the forces among the anchors of a group are assumed to be distributed linearly.Even though this is an inherent assumption of the CCD method, the current standards do not provide any quantitative guidance on how to ensure that the anchor forces obtained for an anchorage with a given base plate thickness agree with this assumption.Using the base plate thickness that will not lead to the satisfactory fulfillment of this basic assumption can lead to a significant reduction in the safety of the anchorages.
In this paper, an assessment method is proposed that ensures that the tension forces among the anchors of a group are distributed in a linear manner and thereby the CCD method formulations can be safely applied to the design of anchorages.First, to provide an overview, the problems are discussed in detail highlighting the limitations and several issues in the current approach for the design of anchorages in concrete.As a solution to the problem, the paper provides an assessment method in form of a set of criteria that can be used to evaluate whether the tension force distribution is linear for an anchorage with a given anchor pattern, baseplate, attachment, and load case.The principle of the approach is based on a combination of straightforward and logical mathematical formulations and the evaluation of finite element calculations performed on anchorages modeled with base plate, attachment, and anchors, modeling the anchors as linear elastic springs.
The proposed formulations are described, and it is explained why they are adequate to find practical and safe general solutions to the problems supported by example calculations.Furthermore, the application of the set of criteria allows the calculation of an optimum base plate thickness that is sufficient to distribute the forces linearly among the anchors, without being overly conservative.As the calculation of anchor forces requires the use of finite elements, all attachments and stiffeners can be taken into account, so that the stiffness of the connection and not only that of the base plate itself is evaluated.Results of numerical analyses are compared with experimental test results to verify the applicability of the proposed assessment method.
The proposed method serves as a simple tool to assess whether a linear tension force distribution among the anchors of a group can be ensured and to calculate the base plate thickness that is sufficient for that.This way, it ensures the applicability of the CCD method and helps to comply with the design rules.In general, the method should be applicable for any given base plate geometry and anchor pattern.However, further experimental verification is necessary, especially on anchorages with more than one row of anchors under tension and under different load combinations to verify the suitability of the proposed criteria as a method to determine the required base plate thickness to satisfy the rigid base plate requirement safely.
The proposed assessment method serves the "action side" only and does not recommend to calculate the resistance for arbitrary anchor configurations or anchorages with "not sufficiently stiff" or "flexible" base plates.There is a need to develop design methods for concrete cone failure that are able to capture the behavior of "flexible" base plates in a rational and safe manner.However, calculations based on assuming a linear behavior of the anchorage do not allow to consider force re-distributions and are therefore limited for use in the linear range of the anchorage load-displacement behavior.For anchorages with arbitrary base plate and anchor pattern, and when considering the realistic load-displacement behavior of the anchorage is essential, it is recommended to use more advanced methods such as the nonlinear spring model proposed by Bokor.9

F
I G U R E 2 Example of actual force distribution among the anchors of a group with the base plate of finite stiffness: Eccentric tension or tension + bending.

F I G U R E 3
Example of a case where actual stiffness of base plate does not influence the anchor force distribution (assuming no prying of plate overhang).F I G U R E 4 Example: Influence of anchor stiffness on the force distribution among the anchors of a group with a given base plate stiffness (thickness): In this example, t = 35 mm; k = 20, 100, and 200 kN/mm.

F I G U R E 5
Spring model of an anchorage in concrete-Example.

F
I G U R E 7 Linear equation and its adaptation for the evaluation of anchor force distribution.(a) Linear equation.(b) Anchor force distribution of rigid (red) and given base plates (blue).

F
I G U R E 8 Analyzed anchorages: (a) 3 Â 2 configuration, (b) 2 Â 2 configuration.F I G U R E 9 Results of the 3 Â 2 anchorage: (a) anchor force deviation compared to the reference case in function of base plate thickness, (b) anchor force in function of anchor location (0 and 450 mm correspond to the left and right ends of the base plate along the x-axis).

F I G U R E 1 0
Deformations of the base plate and profile (a) with base plate thickness of 20 mm, (b) with rigid base plate-The deformations are scaled up by a factor of 100.F I G U R E 1 1 Results of the 2 Â 2 anchorage: (a) anchor force deviation compared to the reference case in function of base plate thickness, (b) anchor force in function of anchor location.

T A B L E 3
Sufficient condition to ensure linear force distribution.

3. 2 |
Steel-concrete joint with compressive force and bending moment-2 Â 2 anchor pattern3.2.1 | Experimental testBajer et al.25 investigated a steel-concrete joint subjected to a compressive force and bending moment.The connection consisted of a base plate of dimensions

440 Â 330 Â
20 mm 3 , an attached HEB 240 profile, a shear lug IPE 100 with length of 100 mm welded to the bottom of the base plate and four M20 cast-in anchors made of threaded rods of steel grade 8.8.The steel grade of the base plate, profile, and shear lug was S235.The anchor pattern corresponded to a 2 Â 2 configuration.The concrete strength class was reported as C20/25.

F I G U R E 1 3
Evaluation of results in terms of anchor force deviation and convergence of criteria of test series G51R (a, b); G52 and (c, d).

F
I G U R E 1 4 (a) von Mises stresses in the base plate with 20 mm.Deformations of the base plate and profile (a) base plate thickness of 20 mm and (b) 108 mm.
T A B L E 1 Example 2 Â 2 anchorage under uniaxial bending.
T A B L E 2 y = 8 kNm Embedment depth: h ef = 100 mm Anchor stiffness k 1 = 100 kN/mm 10,11 L E 5 Test results of anchor group series G51R and G52 from Bokor et al.10,11