REVIEW PAPER

P. Mahrenholtz1 and R. Eligehausen2

Definition and Quantification of Anchor Ductilityand Implications on Seismic Design

ReferenceMahrenholtz, P. and Eligehausen, R., “Definition and Quantification of Anchor Ductility and Implications on

Seismic Design,” Journal of Testing and Evaluation, Vol. 46, No. 1, 2018, pp. 370–384, https://doi.org/10.1520/

JTE20160369. ISSN 0090-3973

ABSTRACT

Ductility plays an important role in structural and seismic engineering. Its beneficial effect on

load distribution and performance during earthquakes is assumed to be true also for anchors

connecting concrete and steel elements. However, a conclusive definition of ductility for post-

installed concrete anchors is not given yet in the relevant qualification guidelines and design

codes. The goal of this paper is to define anchor ductility based on the load-displacement

behavior of installed anchors, and to evaluate available displacement capacities and their impact

on seismic anchor design. The background of anchor ductility was investigated and meaningful

anchor ductility parameters were developed based on the database of several hundred tests on

diverse anchor types. Finally, the effects of anchor displacements on the behavior of structural

and nonstructural connections during earthquakes are discussed.

Keywords

post-installed anchors, ductility, strength and deformation, displacement capacity, seismic load, anchor design,

product qualification

Introduction

In structural design, ductile failure modes are generally considered as desirable because of their beneficial

effects on the behavior of structures. This also applies to post-installed concrete anchors [1–4], which are

often used to fix nonstructural components and systems to the structure or to connect structural elements

to each other for retrofitting or for new construction. In response to any load acting on an anchor in-

stalled in concrete, it displaces due to elastic and plastic deformation or slip. The load-displacement

behavior of anchors installed in concrete ranges from very brittle to moderate or pronounced ductile

(Fig. 1). It is easy to imagine what ductility means; however, to date neither the concept of anchor ductility

has been formally defined nor has the available anchor ductility been generally quantified. Moreover, the

effects of anchor ductility in the context of seismic design have not been discussed in detail.

Notwithstanding these deficits, current anchor design and qualification guidelines use anchor ductility

as a criterion.

Manuscript received July 16, 2016;

accepted for publication

November 2, 2016; published

online October 31, 2017.

1 R&D, Stanley Black & Decker,

Holbeinstr. 51, 60596 Frankfurt,

Germany (Corresponding author),

e-mail: philipp@mahrenholtz.net

https://orcid.org/0000-0003-

0906-7490

2 Institute of Construction

Materials, University of Stuttgart,

Stellaweg 28, 70563 Stuttgart,

Germany

Journal of Testing and Evaluation

Copyright © 2017 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959 370

doi:10.1520/JTE20160369 / Vol. 46 / No. 1 / January 2018 / available online at www.astm.org

The designation of whether an anchor is ductile or brittle af-

fects several aspects of anchor design. According to the European

Technical Specification for the design of fastenings for use in con-

crete CEN/TS 1992-4 [5], which implementation as Part 4 of the

structural concrete design code Eurocode 2 (EN 1992-4 [6]) is

expected for 2017, a plastic design approach of structural connec-

tions is acceptable only when the failure is governed by ductile

steel failure of the anchor. Furthermore, the strength reduction

factor Φ for steel failure under tension and shear load as given

in the US structural concrete design code ACI 318 Chapter 17

[7] depends on the ductility classification of the anchor.

For earthquake engineering, ductility capacity is particularly

important to overcome the extreme demands. This fundamental

idea was also transferred to seismic anchor design. EN 1992-4 [6]

and also the 2008 edition ofACI 318 [8] require seismic design of the

anchorage using an overstrength factor of 2.5 if it cannot be shown

that either the element fixed by the anchor yields before anchor fail-

ure, or the anchor is qualified as ductile. It is noted that the 2011 re-

vision of ACI 318 [9] dropped this factor, but stipulates the

application of amplification factorsΩ0 (IBC [10]) on the design load

in case of brittle anchor failures, which has basically the same effect.

Despite these design provisions, a practical procedure for

qualifying an anchor as ductile is neither given yet in design codes

nor in relevant guidelines used for the qualification of concrete

anchors. In Europe, EN 1992-4 [6] requires the rupture elonga-

tion to be at least 12 %, measured over a gage length equal to 5d,

with d = diameter of specimen, but does not give any details on

the testing method. However, it may be reasonably assumed that

the elongation is to be determined in an analogues way to ISO

6892-1 [11], which is a common standard for material tensile

tests. EN 1992-4 allows explicitly ductile failure modes other than

ductile steel failure, if the equivalency to ductile steel failure can

be shown in the relevant European Technical Specification; how-

ever, no further details are given. Also, the European anchor

qualification guideline ETAG 001 [12] and the German

Guideline for Anchorages in Nuclear Power Plants DIBt NPP

[13] do not specify any requirements for anchor ductility. In

the U.S. ACI 318 [9] defines an element with a rupture elongation

of at least 14 % and reduction in area of at least 30 % as

ductile. The percentage elongation should be measured over

the gage length specified in the appropriate ASTM standard.

Furthermore, ACI 318 states that a steel element meeting the re-

quirements of ASTM A307 [14] shall be considered as ductile.

ASTM A307 guides to ASTM F606 [15], a standard providing

test methods for determining mechanical properties of fasteners

and which defines a gage length of 4d. For ductile design of an-

chors to resist seismic forces, EN 1992-4 [6] as well as ACI 318 [9]

for seismic design categories C to F require the use of ductile steel

material and a free strain length of 8 times the rod diameter d.

The US standard for anchor testing ASTM E488/E488M-15

[16], as well as the anchor qualification guidelines for mechanical

concrete anchors ACI 355.2 [17] and adhesive concrete anchors

ACI 355.4 [18] do not address ductility. However, the acceptance

criteria for the qualification of mechanical anchors AC193 [19]

and adhesive anchors AC308 [20], which are published by the

ICC Evaluation Service and are based on ACI 355.2 and ACI

355.4, require material tension tests on machined coupon spec-

imens according to ASTM F606/F606M-16 [15]. Being in line

with ACI 318 and ASTM F606, AC193 and AC308 classify an-

chors as ductile if the reduction of area is at least 30 % and

the elongation is at least 14 % measured over a gage length equal

to 4d. However, it remains unclear where the criteria originate

from. Moreover, the requirements address per se only tension

ductility but no shear ductility.

In many cases the behavior of an installed anchor systemmay

differ significantly from what the material test result may suggest.

As demonstrated in Hoehler et al. [21], the threaded rod of an

adhesive anchor system may develop ductile load-deformation

FIG. 1 Load-displacement behavior of anchors installed in concrete; (a) displaced anchor loaded in tension; (b) brittle and ductile anchor load-displacement behavior and associated performance characteristic.

MAHRENHOLTZ AND ELIGEHAUSEN ON ANCHOR DUCTILITY 371

characteristics in a material tension test. However, when being

bonded to the concrete, the threaded rod cannot generate any

substantial deformation because the available free strain length

between concrete surface and nut is very small. On the other

hand, many mechanical anchor types experience considerable

overall displacement when the installed system is loaded, result-

ing in displacement capacities larger than indicated by the

material test (Mahrenholtz et al. [22]). This leads to the conclu-

sion that the requirements for ductile steel failure of anchors

based on percentage elongation alone do not necessarily achieve

the intended goal of large anchor displacement capacities.

Furthermore, the requirement for a minimum area reduction

of 30 % for anchors loaded in tension is not meaningful, because

this parameter does not allow any conclusion on the physical

behavior of the installed anchor.

In conclusion, ductility of concrete anchors should rather be

tested on installed systems than by material tests. The measured

anchor load-displacement curve can then be taken for the evalu-

ation of ductility characteristics. This approach was evaluated in

the study presented in this paper. The definition of anchor duc-

tility based on the load-displacement behavior of the installed an-

chor will be more general than a definition based on steel strain

alone and may also allow for failure modes other than steel failure

to be classified as ductile. To this end, a database of several hun-

dred tests on diverse anchor types was developed and evaluated to

characterize and quantify their load-displacement behavior. The

results allow a better understanding of anchor ductility and give

benchmarks for available anchor displacement capacities.

Background

DUCTILITY IN MATERIAL SCIENCE

In material science, ductility is a pure material property tested by

tensile tests, e.g., according to ISO 6892-1 [11] or ASTM F606 [15].

Size and shape of the used specimen is in principle not limited by

these standards, but the layout of test machines generally favors

specimens cylindrical in shape and 200 mm maximum in size.

The plastic behavior and thus material ductility is generated by dis-

locations of the atoms. Ductile materials, such as steel, exhibit a

linear stress-strain relationship up to the yield point at which

material strain changes from elastic to plastic, causing the specimen

to permanently deform. As deformation continues, some steel

types develop a stress plateau where increased strains occur at con-

stant stress, and then the stress increases due to strain hardening

until it reaches the ultimate strength. Until this point, the strain is

almost equal over the whole length of the specimen. Beyond this

point, necking of the specimen occurs at a random location and

eventually the specimen ruptures due to increased stress (Fig. 2a).

The reduction in area can be measured and given unambig-

uously as a percentage value. For the determination of percentage

elongation, i.e., plastic strain, the two halves of the ruptured speci-

men are joined (Fig. 2b) and the measured absolute elongation is

related to a reference length. The definition of the reference length

is critical since it directly affects the percentage result. The pro-

visions given in ISO 6892-1 [11] fall back on the so-called propor-

tional specimen with original reference length fixed in proportion

to the square root of the original cross-section area by Lo =5.65 · S0.5o . Compliance with this proportion ensures that the cal-

culated strains of different specimen dimensions and cross sec-

tions are comparable. For round specimens, the equation

yields Lo = 5do. In case of non-proportional specimen length,

the elongation value has to be converted, for which ISO

6892-1 refers to ISO 2566-1 [23]. With 2(do/Lo(π/4)0.5)0.4 given

therein as conversion factor, elongation values derived from a test

with Lo = 4do have to be divided by the factor 1.094. In conse-

quence, an elongation requirement of 14 % for a 4d specimen, as

given in ACI 318 [7], is equivalent to a requirement for a 5d speci-

men of 14/1.094 = 12.8 %, which almost agrees with the value of

12 % required in EN 1992-4 [6].

FIG. 2

Tensile testing according to ISO 6892-1[11]; (a) stress-strain diagram, Ag = strainbefore necking and A = strain afterrupture; (b) specimen before and aftertesting, Lo = original reference lengthbefore testing and Lu = reference lengthafter testing.

372 Journal of Testing and Evaluation

DUCTILITY IN EARTHQUAKE ENGINEERING

Ductility is a very important property in earthquake engineering.

It is generally used to describe the ability of a structure or its com-

ponents to offer resistance in the inelastic range of the response.

The goal of designing an earthquake-safe structure is to have suf-

ficient tolerance with respect to the inevitable inaccuracy in pre-

dicting earthquake imposed displacements. The ductility factor μ

is defined as the ratio of maximum to yield deformations

μ = Δm/Δy [24], where Δm and Δy are deformation response

parameters of the structure such as displacement, rotation or cur-

vature (Fig. 3a). The ductility factor is often illustrated by mon-

otonic load-displacement curves. However, and most important

for earthquake engineering, a ductile element is also able to

undergo cyclic deformations in the inelastic domain without a

substantial reduction in strength. During the hysteretic deforma-

tions (Fig. 3b), the element dissipates energy, potentially protect-

ing other parts of the structure against the destructive power of

earthquakes [25].

DUCTILITY IN ANCHOR TECHNOLOGY

Also in anchor technology, adequate ductility is generally consid-

ered as desirable. The encouragement of ductile steel failure in the

seismic anchor design provisions of EN 1992-4 and ACI 318 is

based on several assumptions [21]: In contrast to anchors with

load-displacement behavior and strength controlled by brittle

concrete failure modes, anchors failing in steel show a good re-

sistance to load cycling even after steel yielding (Fig. 4a). Steel

failure is also associated with robust load-displacement behavior

which is prolonged in the post-peak range and provides warning

prior to failure (Fig. 4b). Displacement capacity can be further

distinguished between absolute displacement capacity and rela-

tive displacement capacity. In anchor technology, a large absolute

displacement capacity within small loss of resistance (Fig. 4c) is

important for displacement controlled failure modes, and enables

load redistribution between anchors at different supports [26].

Large relative displacement capacity (Fig. 4d) describes an activa-

tion of full strength for a large displacement domain and is ex-

pressed by a large ratio of the absolute displacement to the

displacement at the onset of full strength. It is critical for anchor

groups because large absolute displacement alone does not guar-

antee a favorable load distribution [27]. The general load-

displacement characteristics of the different anchor types have

been considered in earlier studies [28–30] with an emphasis

on anchor load capacity. However, displacement capacities

needed for the definition of quantitative ductility parameters have

not been investigated yet.

Characterization and Quantificationof Anchor Ductility

IDENTIFICATION OF CRITICAL DUCTILITY PARAMETERS

Before quantifying anchor ductility, those ductility parameters

have to be identified which control the behavioral objectives re-

sistance to load cycling, robust load-displacement behavior, large

relative displacement capacity, and large absolute displacement

FIG. 3

Load-displacement behavior of reinforcedconcrete members; (a) idealization ofload-displacement curve [24]; (b) typicalmeasured hysteresis loop indicatingenergy dissipation [25].

FIG. 4

Schematic illustration of the objectives onthe load-displacement (L-d) behavior ofanchors for ductility; (a) resistance to loadcycling; (b) robust load-displacementbehavior; (c) large absolute displacementcapacity; (d) large relative displacementcapacity.

MAHRENHOLTZ AND ELIGEHAUSEN ON ANCHOR DUCTILITY 373

capacity. Fig. 5 shows examples of the load-displacement behavior

of anchors loaded monotonically and cyclically in tension or

shear. The diagram in Fig. 5a depicts a pulsating tension load

of a cycling test carried out on an expansion anchor according

to a test protocol developed by the Structural Engineers

Association of Southern California (SEAOSC [31]) as a standard

method for cyclic load tests on anchors. Failure was caused by

concrete cone breakout. Anchors loaded in tension can endure

a large number of load cycles at load levels below peak

(Fig. 5a), irrespectively of the failure mode [32]. In case of steel

failure, several load cycles can also be performed after passing the

yield load. Axially loaded anchors primarily transfer tension

loads, which are generally introduced via the fixture as shown

in Fig. 1a. When the anchor displaces and is then unloaded,

e.g., due to a reversed moment in the connection, it remains

in the displaced position and the fixture slides along the anchor.

The compression load is then directly transferred by the fixture to

the concrete, and on the reverse stroke, the anchor tension load

steeply increases once the fixture contacts the anchor washer and

nut again [33,34].

Fig. 5b depicts the load-displacement curve of an alternating

shear load cycling test on an expansion anchor according to the

FEMA 461 [35] test protocol for determining the seismic perfor-

mance characteristics of components. Failure was caused by steel

rupture due to low cycle fatigue. In this context, it is important to

note that anchors qualified according to ACI 355 or ETAG 001

for seismic loading are only approved for a small number of cycles

at high stress levels to prevent low cycle fatigue of the anchor. If

there is an annular gap between anchor and fixture, anchors take

up shear load only after closing this gap (Fig. 5b). This behavior

results in an extremely pinched load-displacement hysteresis. As

for anchors loaded cyclically in tension, the load-displacement

curves of anchors loaded cyclically in shear show stiff and closely

spaced loading and unloading branches. It can be clearly seen that

hysteretic energy dissipation is in both cases nearly non-existent.

In conclusion, resistance to load cycling can generally be assumed

for qualified post-installed anchors, although energy dissipation,

which is generally sought for applications in earthquake engineer-

ing and associated with load cycling, appears to be small or non-

existing.

Large absolute displacement capacity describes the ability of

the anchor to develop large displacements until it ultimately fails.

Fig. 5a demonstrates that even concrete failure is not necessarily

abrupt and the anchor may displace beyond the peak load, with

decreasing load resistance and negative stiffness. Large relative

displacement capacity, however, does not necessarily come along

with large absolute displacement. Relative displacement is known

in earthquake engineering as the ductility factor μ relating the

maximum to the yield displacement. The concept of yield and

maximum displacement implies that the load-displacement

curves flattens or softens after the yield point. The capability

to develop a robust displacement behavior is largely expressed

by an adequate relative displacement capacity.

For the above reasons, absolute and relative displacement

capacities are the key parameters characterizing anchor ductility.

They can be evaluated and quantified on the basis of load-

displacement curves of installed anchors loaded to failure. It is

noted that the envelope curve of cyclic load-displacement curves

resulting from stepwise increasing load or displacement ampli-

tudes follow the monotonic curve for increasing demands

[36,37], as demonstrated in Fig. 5 for SEAOSC [31] and

FEMA 461 [32] test protocols. However, cyclic load-displacement

curves derived from simulated seismic load tests with a stepwise

decreasing load regime according to ACI 355 or from simulated

seismic crack tests with a constant load level [38] are not suitable

for the evaluation of displacement capacities.

DETERMINATION OF DISPLACEMENT CAPACITIES

To determine the absolute displacement capacity (Δm) and rela-

tive displacement capacity (Δy/Δm), the load-displacement curve

FIG. 5 Examples of load cycling tests; (a) example of large number tension load cycling (Hoehler und Eligehausen [32]), test was performed afterSEAOSC [31]; (b) example of displacement controlled shear load cycling (Mahrenholtz at al [22].), test was performed after FEMA 461 [35].

374 Journal of Testing and Evaluation

has to be evaluated for the yield displacement Δy and the

maximum displacement Δm. In earthquake engineering, the

load-displacement curve is often idealized by a bilinear elasto-

perfectly-plastic system defining Δy and Δm. Various approaches

for curve idealizations can be found in the literature [24,25,39,40].

When the load-displacement curve does not have a well-defined

yield point, the curve may be idealized by balancing the areas

between actual and idealized curves. A common approach to find

the yield displacement is by means of the secant stiffness at 75 %

of the peak strength. Furthermore, the maximum available dis-

placement capacity is often estimated by limiting the allowable

loss in post-peak strength. In earthquake engineering, a reduction

in strength of 20 % is generally considered as acceptable.

For the evaluation of relative and absolute displacement

capacities for a large data base of anchors with typically five test

repeats per test series (Fig. 6b), a practical approach based on

characteristic points on the load-displacement curve was devel-

oped for the study presented in this paper. Fig. 6a schematically

depicts the load-displacement curve of an anchor loaded in ten-

sion. During loading, the anchor displaces with decreasing stiff-

ness. The peak strength Nu and the corresponding displacement s

(Nu) can be directly determined. However, a clear change in stiff-

ness after the elastic range is often not visible and a pseudo yield

point has to be defined for Δy. The authors of this paper tried to

establish reliable rules for the pseudo yield point based on more

sophisticated algorithms, e.g., the reduced tangent stiffness ap-

proach given in Part 5 of ETAG 001 [12], which was also adopted

in ACI 355.4 [18], for the determination of the load at loss of

adhesion in case of bonded anchors. However, these turned

out to be impracticable or less meaningful than a simple rule tak-

ing the yield strength as 75 % of the peak strength (Ny = 0.75 Nu

and s(Ny) = Δy). For the maximum displacement capacity Δm,

recent studies focusing on anchorage details have limited the al-

lowable loss in post-peak strength to 15 % [41,42]. In view of the

scatter in load-displacement data of anchor test series the post-

peak strength was in the presented study in general conservatively

neglected (s(Nu) = Δm), or in specific cases taken at 85 % of

the peak strength (Npp = 0.85 Nu and s(Npp) = Δm). Load-

displacement curves of anchors loaded in shear show, in

principle, the same characteristic points when neglecting the

shear displacement due to closing the annular gap. Therefore,

above considerations were also applied to anchors loaded in shear.

Data Base Evaluation

The database is based on the results of extensive qualification tests

performed at the Institut für Werkstoffe im Bauwesen (IWB),

Universität Stuttgart. The results of more than 1500 tests on an-

chors of various types and manufacturers were evaluated. Test

data were made anonymous and assessed with reference to an-

chor type, anchor size, concrete strength, crack width, and type

of loading. It is noted that the evaluated displacement capacities

should not be mistaken as available for a specific post-installed

anchor. Rather, the database evaluation is intended to indicate

potentially available displacement capacities of various anchor

types. For a better understanding of the load-displacement behav-

ior of post-installed anchors, a brief introduction in the character-

istic behavior of different anchor types and their corresponding

load-displacement curves is given.

ANCHOR TYPES AND LOAD-DISPLACEMENT CURVES

Fig. 7 presents typical load-displacement curves of different post-

installed anchor types derived from monotonic tension tests on

anchors installed in concrete members. The load-displacement

curves represent the average behavior of anchors with a diameter

of d = 12 mm or 1/2 in. and an effective embedment depth

hef of 75 to 90 mm, tested in concrete with a nominal strength

f0c = 25 MPa. It is noted that the load-displacement curves have

not been normalized to the nominal concrete strength, and the

actual concrete strength was in general substantially higher than

25 MPa. This has in general a small effect on the displacement

capacities, as will be shown later, but the peak loads of the plotted

load-displacement curves are higher than one may expect for a

concrete with this nominal strength. If not otherwise stated,

the anchors were located in predefined cracks (w = 0.3 mm),

which were initiated before and opened after installation. In

the following, the load-displacement characteristics of the anchor

FIG. 6

Characterization of anchor load-displacement curves (Mahrenholtz [37]);(a) schematic of anchor load-displacementcurve and key characteristic points;(b) example of measured load-displacement curves from a test serieswith five test repeats.

MAHRENHOLTZ AND ELIGEHAUSEN ON ANCHOR DUCTILITY 375

types and their typical failure modes are briefly discussed. A com-

prehensive description of anchor behavior and the background of

anchor testing in uncracked and cracked concrete can be found in

Eligehausen et al. [43]. It is important to note that the curves in

Fig. 7 are used for the following discussion on general behavioral

characteristics of different anchor types and no load or displace-

ment data for specific anchors can be drawn from these curves.

BONDED ANCHORS

Bonded anchors transfer tension loads by mechanical interlock

from the steel element into the adhesive mortar and by bond

and micro-interlock (due to the geometric imperfection of the

drilled hole) from the mortar to the concrete. When set deep

enough to exclude concrete failure, bonded anchors generally fail

after a steep elastic ascending branch in pullout failure mode due

to bond failure. Depending on concrete properties, actual embed-

ment depth and the bond strength of the mortar, also steel failure

might occur. For illustration, Fig. 7 shows the load-displacement

curve of a test on a bonded anchor installed with high strength

mortar in uncracked concrete. The anchor ruptured after a pro-

nounced inelastic load plateau which, however, is short because of

the small free strain length of the installed anchor. A similar

behavior is observed for other anchor types failing by steel rup-

ture if the available length along which inelastic strain occurs is

small, e.g., for bolt-type expansion anchors with necked bolts.

SCREW ANCHORS AND UNDERCUT ANCHORS

Screw anchors transfer tension loads over the entire embedment

depth into the concrete, similar to bonded anchors but by

mechanical interlock. For the tested embedment depth, the peak

load is reached when the concrete consoles between the thread

pitches are sheared off, typically resulting in a combined pull-

out/concrete failure mode with a shallow cone. The displacement

capacities are small.

Also undercut anchors function by mechanical interlock with

the concrete which is created by an undercut element. The con-

centrated load transferred at the anchor base allows a deep con-

crete cone to develop. Load and displacement capacities for

undercut anchors are larger than for screw anchors with the same

embedment depth. For larger embedment depths, steel failure

will occur.

EXPANSION ANCHORS

Expansion anchors transfer tension loads by friction between an-

chor and concrete. Torque-controlled expansion anchors are in-

stalled by using torque to expand the expansion elements against

the concrete. During loading, the anchor is pulled further into its

expansion elements (follow-up expansion), and therefore experi-

ences larger displacements than, e.g., an undercut anchor with the

same embedment depth. Sleeve-type torque-controlled expansion

anchors fail predominantly by concrete breakout. Bolt-type

torque-controlled expansion anchors may also fail by pulling

the expansion cone through the expansion element, in particular

in the case of thin expansion elements. This failure mode shows

the characteristic bell-shaped curve with large displacements at

peak load. Bolt-type expansion anchors may fail in various

modes within a test series. The load-displacement curve of the

bolt-type expansion anchor in Fig. 7 is based on tests with

pull-through failure.

Displacement-controlled expansion anchors typically consist

of a sleeve into which a plug is driven by a defined axial displace-

ment. In contrast to torque-controlled expansion anchors,

FIG. 7

Average tension load-displacement curvesmeasured for anchors d = 12 mm (crackedconcrete if not otherwise stated):BA = bonded anchor (hef = 70 mm);SA = screw anchor (hef = 75 mm);UA = undercut anchor (hef = 90 mm);EA-s = torque-controlled expansionanchor, sleeve-type (hef = 90 mm);EA-b = torque-controlled expansion,anchor bolt-type (hef = 80 mm);EA-d = displacement-controlledexpansion anchor (hef = 55 mm).

376 Journal of Testing and Evaluation

displacement-controlled expansion anchors cannot develop any

follow-up expansion and therefore generally do not perform well

when installed in a regular sized crack. They are relatively stiff and

develop only small displacements before they typically fail in a

concrete or pullout failure mode.

Fig. 7 demonstrates that most anchor types, including bonded

anchors with insufficient free strain length, have only limited dis-

placement capacities in tension. Only torque-controlled expan-

sion anchors exhibit substantial displacements before failure,

especially in the case of a pull-through failure mode. All load-

displacement curves have in common a very short elastic range,

and a lack of a distinctive yield point and plateau.

Anchors loaded in shear fail in steel, provided that the anchor

is located far from edges to exclude concrete edge failure, and its

embedment is deep enough to exclude pry-out failure. Since this

holds for all anchor types, the behavior of anchors loaded in shear

is not much influenced by the anchor type and is therefore not

discussed in detail herein.

TENSION DISPLACEMENT CAPACITIES

Depending on size, embedment depth, concrete strength, etc.,

every anchor develops different failure modes and corresponding

displacements, resulting in a pronounced database fragmentation.

Assessment of the database showed that the influence of concrete

strength on the displacement parameters is small. Therefore, the

displacement data derived from anchors tested in various

concrete strengths (low to high) was merged. In the following,

the results of tension test data are first presented for absolute dis-

placement capacities. The maximum displacement capacity Δm is

conservatively taken as the displacement at peak load s(Nu).

Table 1 depicts the range of displacements at peak load for bonded,

screw, and expansion anchors for common sizes. For undercut

anchors and displacement-controlled expansion anchors, there

are not enough test data available to give reliable results.

However, the values of s(Nu) for undercut anchors may be as-

sumed to be somewhere between the bounds given by sleeve-type

expansion anchors and screw anchors, and for displacement-

controlled expansion anchors approximately as small as for screw

anchors. As indicated before, torque-controlled expansion an-

chors provide much larger displacements at peak load than screw

and bonded anchors. Virtually all anchors incorporated in Table 1

failed in a mode other than steel as anchor products are in general

designed to activate the maximum concrete breakout capacity for

the given embedment depth. The embedment depth is therefore

relatively fixed for each anchor size; approximate values are in-

dicated in Table 1.

The database evaluation shows a wide range for the displace-

ment values s(Nu). Sensitivity studies revealed that the scatter of

the results cannot be reduced by differentiating for failure mode,

which is in line with other anchor test data analyses [44]. The

detailed evaluation further showed that the displacement at peak

load s(Nu) of a specific anchor product is generally larger in

TABLE 1 Data base evaluation: Range of absolute anchor displacement at ultimate load s(Nu) in (mm).

Anchor Type Crack Width

M10 (3/8 in.) hef:

60 ÷ 80 mm

M12 (1/2 in.) hef:

70 ÷ 90 mm

M16 (5/8 in.) hef:

80 ÷ 110 mm

M20 (3/4 in.) hef:

90 ÷ 120 mm

M24 (1 in.) hef:

100 ÷ 130 mm

Bonded Anchor 0.0 mm 0.7 ÷ 2.6 0.8 ÷ 3.0 0.8 ÷ 3.4 1.1 ÷ 1.2 1.3 ÷ 4.0

0.3 mm 1.1 ÷ 1.4 1.4 ÷ 3.8 1.1 ÷ 4.3 4.6 ÷ 4.8 1.4 ÷ 5.0

0.5 mm 1.0 ÷ 1.4 –a 1.3 ÷ 1.6 –a 1.4 ÷ 1.6

Combined 0.7 ÷ 2.6 0.8 ÷ 3.8 0.8 ÷ 4.3 1.1 ÷ 4.8 1.3 ÷ 5.0

Median 1.7 2.3 2.6 2.9 3.2

Screw Anchor 0.0 mm 1.0 ÷ 2.1 1.6 ÷ 1.8 1.4 ÷ 3.4 2.8 ÷ 3.3 –a

0.3 mm 1.0 ÷ 2.3 1.0 ÷ 3.1 2.4 ÷ 3.3 3.0 ÷ 3.2 –a

0.5 mm 1.4 ÷ 2.6 1.9 ÷ 3.1 2.3 ÷ 3.8 2.2 ÷ 4.5 –a

Combined 1.0 ÷ 2.6 1.0 ÷ 3.1 1.4 ÷ 3.8 2.2 ÷ 4.5 –

Median 1.8 2.1 2.6 3.4 –

Torque-controlled expansion

Anchor, sleeve-type

0.0 mm 3.7 ÷ 7.5 4.1 ÷ 9.5 5.4 ÷ 12.6 6.4 ÷ 10.9 9.0 ÷ 9.4

0.3 mm 3.9 ÷ 9.5 4.0 ÷ 10.7 3.7 ÷ 11.6 7.9 ÷ 9.8 10.1 ÷ 12.7

0.5 mm 4.3 ÷ 8.5 4.2 ÷ 9.0 4.1 ÷ 10.4 8.4 ÷ 12.9 –a

Combined 3.7 ÷ 9.5 4.0 ÷ 10.7 3.7 ÷ 12.6 6.4 ÷ 12.9 9.0 ÷ 12.7

Median 6.6 7.4 8.4 9.7 10.9

Torque-controlled expansion

Anchor, bolt-type

0.0 mm 4.3 ÷ 6.8 5.8 ÷ 12.0 5.6 ÷ 18.2 9.3 ÷ 11.6 –a

0.3 mm 7.5 ÷ 12.1 5.3 ÷ 14.4 7.0 ÷ 18.8 10.0 ÷ 15.9 –a

0.5 mm 4.2 ÷ 12.9 6.2 ÷ 15-3 10.1 ÷ 18.9 9.1 ÷ 14.8 –a

Combined 4.2 ÷ 12.9 5.3 ÷ 15.3 5.6 ÷ 18.9 9.1 ÷ 15.9 –

Median 8.6 10.3 12.3 12.5 –

a No or too few data available.

MAHRENHOLTZ AND ELIGEHAUSEN ON ANCHOR DUCTILITY 377

cracked concrete than in uncracked concrete and increases with

increasing crack width, for otherwise constant boundary condi-

tions. However, the influence of cracked concrete compared to

uncracked concrete and of the crack width on s(Nu) is much less

pronounced than that on the peak strength Nu, and is overshad-

owed by the scatter of displacement data introduced by the variety

of available proprietary anchors. For this reason, and to allow

general conclusions regarding available displacement capacities,

the database was consolidated by combining displacement data

for all crack widths (0.0, 0.3, and 0.5 mm). The resulting range

of the combined displacement capacity data, as well as its median

are also given in Table 1.

Larger anchor diameters are generally installed with larger

embedment depths, and therefore develop larger displacements.

It is therefore interesting to evaluate the ratio of absolute displace-

ment to embedment depth. Dividing the absolute displacement s

(Nu) by the effective embedment depth hef gives a percentage dis-

placement with reference to the embedment depth. Applying this

approach to the data underlying Table 1 results in the percentage

displacements given in Table 2. In Fig. 8, the medians of the per-

centage displacements s(Nu)/hef are plotted as a function of an-

chor size for different anchor types. In contrast to the absolute

displacement, the median of the percentage displacements is rel-

atively constant for variable anchor sizes, with a slight decrease

for increasing anchor size.

The findings suggest the following analogy: if an anchor fail-

ing in any mode results in a percentage displacement that is as

large as the percentage elongation of an anchor tested in a

material tensile test with a gage length equal to the embedment

depth, it may be considered to provide equivalent ductility.

The required percentage elongation for an anchor to be classified

as ductile is according to EN 1992-4 [6] 12 % (Lo = 5d) and ACI

318 [9] 14 % (Lo = 4d), respectively. From Fig. 8 it can be seen

that bonded and screw anchors are far from meeting this require-

ment, however, expansion anchors are potentially close to it.

When taking the post-peak capacities into account, the maximum

available displacement capacity is increased to levels above 12 %.

TABLE 2 Data base evaluation: Range of percentage anchor displacement at ultimate load s(Nu)/hef in (%).

Anchor Type Crack Width

M10 (3/8 in.) hef:

60 ÷ 80 mm

M12 (1/2 in.) hef:

70 ÷ 90 mm

M16 (5/8 in.) hef:

80 ÷ 110 mm

M20 (3/4 in.) hef:

90 ÷ 120 mm

M24 (1 in.) hef:

100 ÷ 130 mm

Bonded Anchor 0.0 mm 0.9 ÷ 3.3 0.8 ÷ 3.1 0.6 ÷ 2.7 0.7 ÷ 0.8 0.7 ÷ 2.1

0.3 mm 1.4 ÷ 4.4 1.5 ÷ 4.0 0.9 ÷ 3.4 2.8 ÷ 3.0 0.7 ÷ 2.6

0.5 mm 1.3 ÷ 1.8 –a 1.3 ÷ 1.6 –a 0.7 ÷ 0.9

Combined 0.9 ÷ 4.4 0.8 ÷ 4.0 0.6 ÷ 3.4 0.7 ÷ 3.0 0.7 ÷ 2.6

Median 2.7 2.4 2.0 1.9 1.7

Screw Anchor 0.0 mm 1.6 ÷ 3.6 2.2 ÷ 2.6 1.3 ÷ 4.1 2.3 ÷ 2.7 –a

0.3 mm 1.5 ÷ 4.0 1.3 ÷ 4.8 2.9 ÷ 4.0 2.4 ÷ 2.6 –a

0.5 mm 2.3 ÷ 4.5 2.5 ÷ 4.8 2.1 ÷ 4.6 1.8 ÷ 3.7 –a

Combined 1.5 ÷ 4.5 1.3 ÷ 4.8 1.3 ÷ 4.6 1.8 ÷ 3.7 –

Median 3.0 3.1 3.0 2.8 –

Torque-controlled expansion

Anchor, sleeve-type

0.0 mm 5.7 ÷ 9.1 5.1 ÷ 12.9 5.4 ÷ 12.6 5.1 ÷ 8.7 6.0 ÷ 6.3

0.3 mm 6.0 ÷ 13.8 5.0 ÷ 11.4 3.7 ÷ 11.6 6.3 ÷ 7.8 6.7 ÷ 8.5

0.5 mm 6.6 ÷ 12.1 5.3 ÷ 11.3 4.1 ÷ 10.4 6.7 ÷ 10.3 –a

Combined 5.7 ÷ 13.8 5.0 ÷ 12.9 4.1 ÷ 12.6 5.1 ÷ 10.3 6.0 ÷ 8.5

Median 9.8 9.0 8.4 7.7 7.3

Torque-controlled expansion

Anchor, bolt-type

0.0 mm 10.7 ÷ 11.7 9.7 ÷ 15.0 8.1 ÷ 18.2 9.3 ÷ 11.8 –a

0.3 mm 12.5 ÷ 20.2 8.8 ÷ 18.0 8.8 ÷ 18.8 10.0 ÷ 14.4 –a

0.5 mm 7.2 ÷ 21.5 9.1 ÷ 19.1 12.6 ÷ 18.9 9.1 ÷ 13.5 –a

Combined 7.2 ÷ 21.5 8.8 ÷ 19.1 8.1 ÷ 18.9 9.1 ÷ 14.4 –

Median 14.4 14.0 13.5 11.8 –

a No or too few data available.

FIG. 8 Data base evaluation of anchor displacement at ultimate load:Percentage displacement s(Nu)/hef versus anchor size.

0.0

5.0

10.0

15.0

20.0

25.0

M10(3/8")

M12(1/2")

M16(5/8")

M20(3/4")

M24(1")

Per

cent

age

disp

lace

men

t s(N

υ)/h

ef (%

) Expansion anchors, taking post-peak branchinto account:

bolt-type

sleeve-type

Expansion anchors, bolt-type

Expansion anchors, sleeve-type

Screw anchors

Bonded anchors

378 Journal of Testing and Evaluation

The required percentage displacement has to be adjusted for an-

chor embedment depths that are different than the proportional

length by multiplying it by the conversion factor 2(S0.5o /Lo)0.4 given

in ISO 2566-1 [23]. So is the cross-section area, taken as π · d2/4where d is the diameter of the anchor, and Lo the gage length, taken

as the embedment depth hef of the anchor; e.g., for 8d, the required

percentage displacement would only be 10 %.

Based on a 12 % elongation requirement for 5d or equivalent

to classify the anchor as ductile, Fig. 9a depicts the required per-

centage displacement s(Nu)/hef for variable anchor diameter d as a

function of embedment depth hef. Fig. 9b transforms this relation

to the required absolute displacement s(Nu). The required per-

centage displacement for multiples of d form in these dia-

grams a line as shown for 5d and 8d. It can be seen that with

increasing embedment depth, more absolute displacement but

less percentage displacement is required for a given diameter.

With increasing anchor diameter, both required absolute dis-

placement and required percentage displacement increase.

Fig. 10 depicts the mean load-displacement curves from Fig. 7

and their idealization by an elasto-plastic system. The defined

yield displacement Δy and maximum displacement Δm allow

the evaluation of the relative displacement capacities. The yield

load was assumed as 75 % of the peak load and the yield displace-

ment capacity Δy was taken at this point s(Ny). The maximum

displacement capacity Δm was taken as the displacement at peak

load (s(Nu)). There are significant differences in the available rel-

ative displacement capacity (Δm/Δy) for each anchor type. For

bonded anchors, the ratio may be assumed to be just above

1.5; for screw anchors it is about 2.0; and for expansion anchors

it is also in the order of 2.0. Obviously, expansion anchors do not

benefit from their displacement capacities because of the soft as-

cending branch which also leads to large (pseudo) yield displace-

ments. Even when taking the maximum displacement capacity as

the post-peak displacement at 85 % of the peak load (s(Npp)), the

ratio Δm/Δy for expansion anchors is only increased to less than

2.5, which is relatively small, particularly in the context of local

ductility.

SHEAR DISPLACEMENT CAPACITIES

Historically, much fewer qualification tests have been carried

out on anchors loaded in shear than in tension. In consequence,

the shear database is much less populated. For this reason, the

evaluation of shear displacement capacities is not possible to the

same degree as for tension displacement capacities, and only the

following general considerations can be given. Anchors loaded

in shear do not generate material ductility by shear deformation

because the anchor cross-sectional area is too small. However,

the eccentricity between the acting shear load and the resulting

supporting force in the concrete creates a bending moment and

system shear ductility. Fixture lifting caused by the reaction to

the compressive force generated at the leading fixture edge or by

additional axial load, as well as lateral concrete compaction and

concrete spalling increase the available system ductility further

(Fig. 11a). As for tension tests, the envelope of the cyclic load-

displacement curve follows the monotonic mean curve (refer

to Fig. 5b), provided the anchor does not experience premature

low-cycle-fatigue failure, in which case the envelope of the cyclic

load-displacement curve may divert down from the monotonic

curve (Fig. 11b).

Since shear loaded anchors generally fail in steel, one may

expect less scatter in the test results compared to tension tests.

However, variable test boundary conditions, in particular re-

garding the prevention of fixture lifting and the extent of con-

crete spalling in front of the anchor, are the reason for relatively

large variations in the load-displacement data of shear load tests.

The analysis of the available data showed that, in general, the

absolute displacement capacity increases with increasing anchor

size, and is at peak load in the range of 5 to 30 mm for anchor

sizes 10 to 20 mm in diameter. This is in most cases a larger

displacement than the same anchor would show under tension

FIG. 9 Anchor displacement required to be classified as ductile as afunction of hef, based on a percentage elongation requirementof 12 % for 5d or equivalent; (a) s(Nu)/hef; (b) s(Nu).

(a)

(b)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 50 100 150 200 250 300

Req

uire

d s(

Nu)

/hef

(mm

)

hef (mm)

d = 10 mm

d = 16 mm

d = 24 mm

10% for hef = 8d

12% for hef = 5d

d = 20 mm

d = 12 mm

Exa

mpl

e:h e

f=

160

mm

; d

= 2

0 m

m

0.0

5.0

10.0

15.0

20.0

25.0

0 50 100 150 200 250 300

Req

uire

d s(

Nu)

(m

m)

hef (mm)

d = 24 mmd = 20 mmd = 16 mmd = 12 mmd = 10 mm

10% for hef = 8d

12% for hef = 5d

Exa

mpl

e:h e

f=

160

mm

; d

= 2

0 m

m

MAHRENHOLTZ AND ELIGEHAUSEN ON ANCHOR DUCTILITY 379

load. In addition, the relative displacement capacity may be

assumed to be larger for anchors loaded in shear than for an-

chors loaded in tension. Depending on the boundary conditions,

ductility factors Δm/Δy of 3.0 and more can be reached.

Shear ductility data published in Rieder [45] support these

findings. This means that anchors loaded in shear generally

exhibit more displacement capacity than the same anchors

loaded in tension.

Impact of Anchor Ductility onSeismic Anchor Design

The definition and quantification of anchor ductility based on the

displacement behavior of the installed anchor provides a mean-

ingful method to evaluate anchor ductility. As discussed, anchor

ductility is generally considered beneficial in seismic design.

However, anchor displacements developing during earthquakes

have different effects on the behavior of nonstructural and struc-

tural connections.

STRUCTURAL CONNECTIONS

In case of structural connections, the anchor forms an integral

part of the structure. Not inertial forces but displacements im-

posed by the deforming global structure due to strong ground

motions are the main demand for these connections. However,

current anchor design codes are strictly based on strength.

Predicting the exact displacement demand for a connection

and correlating this to anchor response characteristics is beyond

the capability of existing design models and codes [46]. A first

step towards recognizing displacement in design would be to

identify load versus displacement controlled design situations.

When interpreting anchor ductility as local ductility, the available

anchor ductility factor μ = Δm/Δy in the range of 1.0–2.5 is not

large taking into account that the local ductility has to be consid-

erably larger than the required global ductility [24,47]. The rel-

ative displacement and energy dissipation capacity of regular

post-installed anchors can only play a minor role and should

not be accounted for in seismic design.

Only tension loaded anchors failing in steel show adequate

resistance to load cycling. Structures which have been anchored

FIG. 10 Average load-displacement curves and extracted values for yield displacement Δy and max displacement Δm for anchors d = 12 mm;(a) bonded anchor; (b) screw anchor; (c) torque-controlled expansion anchor, dashed line for pull-through failure.

(a) (b)

-60

-40

-20

0

20

40

60

80

100

-20 -15 -10 -5 0 5 10 15 20 25 30 35 40L

oad

[kN

]Displacement [mm]

Monotonic mean

Concrete compaction

Sleeve

Bending

Fixture lifting

Shear loadGap

and spalling

FIG. 11

Anchor load-displacement behavior inshear; (a) anchor loaded in shear andillustration of shear system ductility;(b) example illustrating effect of low-cycle-fatigue on load-displacement curve(Mahrenholtz [37]).

380 Journal of Testing and Evaluation

with an increased stretch length, resulting in large relative and

absolute displacement capacities, showed adequate behavior in

past earthquakes [48,49]. This has been recognized by ACI

318 [7], which requires for ductile anchoring special anchor

configurations providing a free steel stretch length of 8d.

Furthermore, ACI 318 no longer recognizes anchor steel failure

in shear as a ductile failure mode. The absence of a meaningful

stretch length and the proneness to low cycle fatigue [50] supports

this change. Also EN 1992-4 [6] gives a requirement for sufficient

axial stretch length, the anchors may not be accounted for energy

dissipation, and anchors loaded in shear are considered as brittle.

However, further research is necessary to understand anchor dis-

placement demands resulting from deformations of the structure

under seismic loading.

NONSTRUCTURAL CONNECTIONS

In the case of nonstructural connections, the design methodology

given in ACI 318 [7] and EN 1992-4 [6] is principally the same as

for structural connections, although the 8d free stretch length re-

quirement for ductile anchor design may not be practical for most

nonstructural anchorages. In contrast to structural connections,

the displacement of anchors connecting nonstructural compo-

nents and systems to the primary structure is not directly related

to the deformation of the global structure. The anchor forces de-

velop according to the inertial response of the nonstructural com-

ponent and the acceleration of the structure it is connected to.

The resulting anchor displacement influences the anchored non-

structural component behavior and therefore the ductility of the

anchorage is one of the parameters influencing the behavior of

the nonstructural component (Wood and Hutchinson [51]).

Ductile behavior of the nonstructural component allows

reducing the seismic design loads obtained by linear-elastic analy-

sis as stipulated in, e.g., ASCE 7 [52] or EN 1992-4 [6]. If the

nonstructural component does not show beneficial ductile

behavior (qa = 1.0), a reduction factor to the anchorage itself

would help to reduce the load demand. For μ = Δm/Δy = 2.0,

the behavior factor can be theoretically calculated to qa = 2.0

for nonstructural components with fundamental period in the

long period domain (principle of equal displacement) or 1.73

for nonstructural components lying in the medium period do-

main (principle of equal energy) (Fig. 12a). However, it is arguable

whether the principles developed to recognize the beneficial effect

of ductility also apply to relative displacement capacities of an-

chors connecting nonstructural components to the structure

[45], in particular when the anchor does not have sufficient free

stretch length and does not fail in steel. Furthermore, large and

non-recoverable anchor displacements during seismic events may

be disadvantageous because they potentially lead to pounding ef-

fects [35] (Fig. 12b) and to an increase in component amplification

due to the elongation of the period [53,54]. Whether the anchor

displacement capacity is beneficial or adverse depends, therefore,

highly on the specific design situation as well as on the character-

istics of the component and motion spectral demand [55]. More

research is needed to quantify the effect of anchor ductility on the

behavior of anchorages of nonstructural components under seis-

mic loading.

Summary and Conclusions

A comprehensive study on the ductility of post-installed anchors

examined relevant aspects for testing and qualifying anchors

for ductility. The study reveals deficits of the current code prac-

tices in Europe and the US and discusses the importance of an-

chor displacement capacity in earthquake engineering. The key

findings are:

• Not the material ductility but the displacement behavior ofthe installed anchor system is critical for the definition of

FIG. 12 Specific ductility and displacements effects; (a) illustration of the principle of equal displacement and equal energy, and resulting reductionfactor (Paulay and Priestley [24]); (b) sequence of anchor failure, non-recoverable displacements, and resulting pounding effect, after Nuti andSantini [34].

MAHRENHOLTZ AND ELIGEHAUSEN ON ANCHOR DUCTILITY 381

anchor ductility. The percentage rupture elongation re-quired by EN 1992-4 (12 %) and ACI 318 (14 %), respec-tively, is founded in material science and needs to betransferred to the installed anchor system. A requirementconcerning the reduction in area of at least 30 % (ACI 318)is not meaningful since it is physically not needed.

• The characterization of anchor load-displacement curvesshowed that absolute and relative displacement capacitiesare the driving anchor ductility parameters which coverall relevant behavioral objectives. For the determinationof absolute and relative displacements, the yield andmaximum displacements were evaluated based on load-displacement curves of installed anchors. The maximumdisplacement may be conservatively taken as the displace-ment at peak load. Due to the lack of a pronounced yieldpoint, the determination of the (pseudo) yield displacementis required.

• The displacement capacities of post-installed anchors werequantified by means of a large database. The results showthat bonded and screw anchors exhibit axial displacementsthat are generally small; however, torque-controlled expan-sion anchors may develop displacements of a magnitudesimilar to the displacement of anchors which fail in steelwith a free strain length equal to the embedment depth.

• With regard to anchor qualification, the database evalu-ation enabled to come up with a new approach allowingthe evaluation of anchor ductility irrespective of the failuremode. This more general definition relates the maximumdisplacement capacity to the embedment depth. The an-chor is then qualified as ductile if the percentage displace-ment meets the current requirement of the percentageelongation of 12 % (for 5d). In particular, expansion an-chors showing pull-through failure may meet the criterionto be classified as ductile. The proposed criterion is moregeneral and substantiated than the definition given in cur-rent codes, but it is only satisfactory for static applications.

• For seismic design situation, requirements on anchor duc-tility capacities go beyond that which is required for staticapplications. Qualifying anchors as ductile based on axialpercentage elongation alone is not sufficient to ensure ben-eficial anchor behavior during earthquakes. The installedanchor has to provide an adequate free stretch length.Therefore, the anchor design codes ACI 318 and EN1992-4 exclude ductile shear design and give additional re-quirements for tension design like a free stretch length of 8dprimarily practicable for structural connections. For non-structural connections, large anchor displacements do notnecessarily result in beneficial behavior and the paradigmof generally beneficial anchor ductility in earthquake engi-neering should be reconsidered.

ACKNOWLEDGMENTS

This work was funded by the Hilti Corporation, which is grate-

fully acknowledged. The opinions, findings, and conclusions ex-

pressed in this paper are those of the authors and do not

necessarily reflect those of the sponsoring organization or of

the authors’ affiliations.

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