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Pull-out and shear-strength models for FRP spike anchors
Villanueva Llauradó, Paulaa1; Ibell, Timb; Fernández Gómez, Jaimea; González Ramos, Francisco J. a
a School of Civil Engineering, Technical University of Madrid. Prof Aranguren, s/n, 28040, Madrid, Spainb Department of Architecture and Civil Engineering, University of Bath, BA2 7AY, UK
Abstract
Spike anchors are a promising way to enhance the maximum capacity and post-peak
load-strain response of externally-bonded fibre-reinforced polymer (FRP) materials for
retrofitting of concrete structures. Although laboratory testing has proven the
effectiveness of these spike anchors (also known as fan anchors), little work has been
conducted to provide an analytical basis on which to rely on such anchorage systems.
For instance, which parameters govern the behaviour of these anchors? Without an
analytical or predictive basis for the behaviour of such anchors, their use will be limited.
Therefore, this paper presents an analytical model to be able to predict the behaviour of
such anchors, by including important geometrical and installation parameters. This
model will, for the first time, allow engineers to specify, with confidence, the use of
spike anchors as a method to anchor FRP sheets and plates to structural concrete
members.
KEY WORDS
Spike anchors, fan anchors, concrete retrofitting, fibre reinforced polymer (FRP), shear
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NOTATION
Aa cross section of FRP anchor (mm2)
Panc nominal capacity of FRP anchor (kN)
Pbend maximum bend apacity of FRP anchor (kN)
Pcc maximum capacity of FRP anchor in case of concrete cone failure (kN)
Pcb maximum capacity of FRP anchor in adherent of mixed failure (kN)
Pdb bond strength of FRP joints, maximum capacity of unanchored joints (kN)
P joint maximum capacity of anchored bonded FRP joints (kN)
Pu tensile strength of FRP anchor (kN)
da anchor diameter
d0 hole diameter
f FRP tensile strength of FRP composite (MPa)
f ' c concrete cylinder compressive strength (MPa)
f fb effective bend strength of internal FRP reinforcements (MPa)
f fib tensile strength of carbon or glass fibre sheet (MPa)
f fu direct tensile strength of internal FRP reinforcements (MPa)
hemb embedment length (mm)
rb bend ratio
t fib thickness of carbon of glass fibre sheet for construction of FRP anchors (mm)
w fib width of carbon or glass fibre sheet for construction of FRP anchor (mm)
α dowel angle of FRP anchors
γ reduction coefficient for anchor’s tensile strength
γ ' reduction coefficient for anchor’s capacity on anchored FRP joints
γ 'd design reduction coefficient for anchor’s capacity on anchored FRP joints
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τ ave average shear strength (MPa)
1.INTRODUCTION
Fibre-reinforced polymer (FRP) composites are broadly employed as externally-bonded
reinforcements for concrete structural members, in the form of sheets or plates. The
advantages of these materials include: high strength/weight ratio, flexibility of design,
ease of installation, light weight and durability. A considerable body of knowledge
about FRP reinforcements has been accumulated to date [1-3]. However, the
effectiveness of FRP reinforcements depends on their (usually limited) adherence to the
concrete substrate, commonly leading to an unfortunate under-exploitation of the
mechanical properties of the composite material. Extensive research has been
undertaken to understand the adherent mechanism and the effective bonded length that
controls the maximum bond strength. This has resulted in the development of numerical
and empirical models for adhesion between FRP and concrete [4-8].
In an attempt to make fuller use of the FRP, research has also focused on finding ways
to prevent or delay debonding and, with this in mind, a range of anchorage systems has
been developed as external or embedded systems [9, 10]. External systems mainly
consist of U-wrap jackets, broadly used for simultaneous shear and flexural
strengthening, or patch anchors, such as the ones developed by Kalfat and Al-Mahaidi
[11,12]. Embedded systems generally involve greater strength increase as they work in a
similar manner to traditional chemical anchors or steel mechanical fasteners. These
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systems include: FRP spike anchors, end anchorage, steel angle, bolted plates, U-
anchors and special systems such as π−¿anchors [9,10] [13]. Among the embedded
systems, FRP spike anchors (also referred to as fan anchors, fibre anchors, fibre bolts or
FRP dowels) show great promise; they can be applied to flexural and shear reinforced
members alike, and they guarantee maximum compatibility with both the reinforcement
and the substrate [9, 10].
A typical FRP spike anchor is shown in Figure 1. It can be formed from rolled fibre
sheet or from bundles of fibres, and can be of any fibre type. Spike anchors consist of a
dowel region, which is inserted in a hole drilled in the substrate, and a splay region or
fan which is bonded to the FRP reinforcement. Until now, the majority of anchors have
been hand-made, following one of two techniques, namely hardened (pre-impregnated)
anchors or wet anchors; for the hardened installation, the anchor dowel is typically
impregnated with resin at least 24 hours before embedding it in the drilled hole, whereas
wet application entails impregnation immediately prior to insertion. For single plies
(including plate installations), the fan is placed on the outer surface of the FRP [9],
whereas for sheet applications involving multiple plies, the fan of the anchor can be
placed onto the surface of the reinforcement or between ply levels [14].
Figure 1. FRP spike anchor
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The fan itself is often formed into a specific geometric angle in order to ensure highest
effectiveness [15]. Figure 2 shows typical configurations involving multiple anchors;
multiple anchors can be arranged longitudinally, transversally or a combination of both,
and spacing between anchors in a row is generally derived from the fan angle and fan
length so as to guarantee that the entire width of the reinforcement is covered by the
fans.
Figure 2. Configurations of multiple anchors for externally-bonded FRP sheets
The use of spike anchors in structural concrete members such as beams and slabs,
including ways to provide continuity in frames, has been reported in a number of
experimental works [16-20]. Thus, it is generally accepted that spike anchors can
provide beneficial anchorage, being able to delay debonding in flexural and shear
applications. However, the importance of each parameter affecting their performance is
far from fully understood.
A lack of reliable design rules to be able to predict the effectiveness of spike anchors
has forced existing guidelines to stipulate that the implementation of anchorage devices
must be substantiated by representative testing [9]. To assist with this approach, Grelle
and Sneed [10] suggested a verification process which any proposed anchorage system
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should undergo prior to being implemented. The process includes independent
anchorage testing, performance of representative tests on anchored reinforcements, and
verification of tests and design procedures. According to this design approach, FRP
anchors must satisfy two levels of testing and modelling prior to being included in
design guidelines. The first level is that of an isolated anchor’s strength; isolated
anchors tested in tension and shear are indispensable in helping to analyse the behaviour
and load bearing capacity across various configurations. Once enough data have been
collected, analytical models should be created to make predictions of the level of
improvement achieved with the use of FRP anchors in reinforced concrete members.
Then, shear tests and beam tests on anchored FRP-to-concrete members are needed to
judge the adequacy of the analytical model.
In this paper a predictive model for FRP anchors is presented, utilising experimental
data from both isolated anchors and anchored sheets, following the implementation
procedure proposed by Grelle and Sneed [10]. The parameters affecting the behaviour
of the anchors have been identified and the model predictions have been compared with
the experimental database in order to validate the model and to obtain a comprehensive
assessment of the adequacy of the test methods utilised to date. The proposed model
aims to be apt for design for anchors mainly subjected to pull-out or shear forces. The
major novelty of the paper is the adaptation of an expression for internal reinforcement
for its use on the problem associated with spike anchors in shear, considering the
particular nature of these anchors; to date, there has existed no design model for these
anchors in shear. The proposed model for bend strength is related in a comprehensive
way to existing pull-out models and tensile strength expressions, so as to provide
solutions for all potential failure modes of the anchors.
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2.PULL-OUT AND SHEAR TESTS ON ISOLATED ANCHORS
Spike anchors can be subjected to predominantly pull-out forces or shear forces,
depending on the geometrical configuration and load pattern of the reinforced member.
Anchors subjected to pull-out can be also referred to as 180º anchor spikes [10], as they
are installed in-plane with the anchored FRP; this configuration is used, for instance, in
T-beams in which there is an inner corner where anchor dowels can be inserted. In
most cases, however, the geometry of the existing member prevents anchors from being
installed at 180º, and thence dowel angles lower than 180º, typically raging from 90º to
135º, are used with consequent shear and stress concentration in the curved region
(referred to as the ‘bending’ region). Figure 3 shows spike anchors mainly subjected to
shear (A) and pull-out forces (B), respectively.
Figure 3. Spike anchors in different constructive configurations
Given the potential for complex stress states in the anchors, as presented in Figure 3, it
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is considered that tests on the anchorage device alone must include both pull-out and
shear tests. Accordingly, a database has been constructed by including the pull-out
results obtained by Özdemir [21], Kim and Smith [22], Özbakaloglu and Saatcioglu
[23], Eshwar et al. [24] and a series of experimental tests of carbon fibre ropes in shear
by Villanueva et al. [25-26].
2.1 Pull-out experimental data
The direct tensile pull-out behaviour of post-installed anchors has been extensively
investigated for steel anchors and chemical anchors in concrete [16-18]. Aligned with
this literature, failure modes and pull-out strength of CFRP anchors were reported in
Özdemir [21], Kim and Smith [22], and Özbakaloglu and Saatcioglu [23]. Eshwar et al.
[24] reported results on GFRP anchors. The key parameters presented in these papers
were: concrete strength, embedment length, fibre tensile strength, fibre Young’s
Modulus, anchor fibre content, anchor diameter and hole diameter. A range of
embedment depths were tested between 20 mm and 150 mm; diameters of the anchors
ranged from 12 mm to 20 mm. The failure modes observed in the investigations were
concrete cone failure (CC), mixed failure (CB), adherent failure (BF), bending failure
(BD) and fibre rupture (FR); these failure modes are shown graphically in Figure 4.
Figure 4. Failure modes for FRP anchors in pull-out and shear
From the available results of pull-out tests [21-24], it can be concluded that concrete
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strength and embedment length are the most relevant parameters affecting the failure
mode and pull-out strength.
2.2 Shear experimental data
A crucial limitation for the assessment of effectiveness of a particular FRP anchor is
that most of the tests conducted on shear to date have been carried out under conditions
of the FRP being bonded to the concrete in tandem with spike anchors being used. This
means that separation of the effects of adhesion and anchorage is complex.
Consequently, for the modelling of the anchor’s unitary capacity in shear the database
was constructed only using available results for isolated anchors.
The shear capacity of an anchor is tested when the main stress resultant is not parallel
with the anchor dowel (see Figure 3). As a result, some specimens of Ozbakaloglu and
Saatcioglu [23], which had varying inclinations of the anchor dowel, may be regarded
as shear tests; the main parameter for this consideration is the dowel angle α , which
changes the stress state.
Villanueva et al. [25-26] conducted single-shear tests on isolated carbon ropes
embedded in concrete specimens, comparing three dowel angles (90º, 120º and 135º)
and different smoothing of the hole edge. From the results it was concluded that the
strength of the anchors in shear is governed by a reduction in tensile strength in the
bending-zone region where the dowel protrudes and turns into the fan. In this series, a
minimum embedment depth of 50 mm was used, and no concrete cone failures were
observed; the failure modes were adherent failure, bending failure and anchor rupture.
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Dowel angle and embedment length were found to be the most significant variables
concerning shear strength, immediately followed by the smoothing technique.
According to the results, there is a relationship between the hole diameter, dowel angle
and smoothing technique, and it was hypothesised that this relationship was related to
the inner bending radius of the anchor.
2.3 Database with parameters considered
The complete database used to underpin the analytical model, including pull-out and
single-shear tests, is presented in Tables 1 and 2. The angle between the anchor dowel
and the free length of the spike anchors, α , was considered equal to 180º in pull-out
tests where no angle was specified. In such tests there is no reduction in capacity due to
bending, and the model only considers adherent failure and anchor tensile rupture.
The main parameters to evaluate the effectiveness of the anchor are the ratio Pmax /Pu
and the failure mode. Pu is the tensile capacity of the anchor considering its nominal
tensile strength and cross sectional area, and Pmax is the maximum load attained in each
test. Data are classified in terms of their failure modes, as presented in Figure 4. Tables
1 and 2 also specify the concrete strength of the substrate; for the anchor, the variables
are the dowel angle α , the hole diameterd0, the embedment length hemb, the nominal
tensile strength of the fibres or of the impregnated anchor, and the bend ratio rb defined
by rb=R/df, where R is the inner bend radius and df is the diameter of the FRP dowel,
assuming a circular cross section of the reinforcement. For shear tests, rb was obtained
from the geometry of the hole depending on the smoothing (if any) of the hole edge. For
non-smoothed specimens, rb is a function of d0 and da; for smoothed holes the inner
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radius of bending was calculated according to the geometry of the radial saw and drill
tool. Variation of inner radius R with different smoothing techniques and dowel angles
is shown in Figure 5.
Figure 5. Different relationships of R and α for FRP anchors in shear
Table 1. Database of pull-out tests on FRP anchors
Reference Specimen identification
f'c (MPa)
α rb d0
(mm)hemb
(mm)Tensile strength (MPa)
Pu
(KN)P
max/Pu
Failure mode
Özdemir (2005) [21]
w120h50f10d20 10 180º - 20 50 3430 67.91 0.21 CCw120h50f10d20 0.22 CCw120h50f10d20 0.21 CCw120h70f10d20 70 0.38 CBw120h70f10d20 0.33 CBw120h70f10d20 0.30 CBw120h100f10d2 100 0.52 CBw120h100f10d2 0.43 FRw120h100f10d2 0.45 CBw120h50f16d20 16 50 0.24 CCw120h50f16d20 0.23 CCw120h70f16d20 70 0.41 CBw120h70f16d20 0.39 CBw120h70f16d20 0.39 CBw120h100f16d2 100 0.61 CBw120h100f16d2 0.52 CBw120h100f16d2 0.52 CBw120h150f10d2 0.44 FRw120h150f10d2 0.46 FRw120h150f10d2 0.47 FRw120h150f16d2 0.55 FRw120h150f16d2 0.51 FRw120h150f16d2 0.53 FR
Kim & Smith
(2009) [22]
PF-20-12-1 33,6 180º - 11.9 18.3 3800 26.68 0.25 CCPF-20-12-2 12.1 19.4 0.22 CCPF-20-12-3 12.1 17.5 0.23 CCPF-20-14-1 14.2 20.8 0.27 CCPF-20-14-2 14.1 25.1 0.32 CCPF-20-14-3 16.1 21.9 0.27 CC
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Reference Specimen identification
f'c (MPa)
α rb d0
(mm)hemb
(mm)Tensile strength (MPa)
Pu
(KN)P
max/Pu
Failure mode
PF-20-16-1 16.1 23.5 0.30 CCPF-20-16-2 15.3 21.4 0.27 CCPF-20-16-3 14.2 25.0 0.32 CCPF-40-12-1 12.2 41.5 48.91 0.39 CBPF-40-12-2 11.8 44.6 0.32 CBPF-40-12-3 11.8 44.8 0.44 CBPF-40-14-1 14.7 44.8 0.42 CCPF-40-14-2 14.8 44.6 0.37 CBPF-40-14-3 14.7 44.9 0.47 FRPF-40-16-1 16.3 41.3 0.42 CCPF-40-16-2 16.6 41.5 0.38 CCPF-40-16-3 16.4 41.3 0.23 BFPF-60-12-1 11.8 66.3 57.80 0.56 FRPF-60-12-2 11.8 65.6 0.56 CBPF-60-12-3 11.7 66.3 0.56 BFPF-60-14-1 14.6 64.5 0.56 FRPF-60-14-2 14.6 65.7 0.56 FRPF-60-14-3 14.7 65.6 0.56 FRPF-60-16-1 16.3 64.5 0.56 FRPF-60-16-2 16.4 64.5 0.56 FRPF-60-16-3 16.5 64.5 0.56 FR
Ozbakaloglu & Saatcioglu (2009) [23]
HD12.7L25T1 53 180º - 12.7 24 3800 50.16 0.16 CCHD12.7L25T2 54 12.7 26 0.19 CCHD12.7L25T3 56 12.7 27 0.20 CCD12.7L25T4 50 12.7 18 0.12 CC
HD15.9L25T1 57 15.9 26 0.21 CCHD15.9L25T2 57 15.9 26 0.21 CCHD15.9L25T3 60 15.9 28 75.24 0.16 CCHD19.1L25T1 57 19.1 24 50.16 0.21 CCHD19.1L25T2 60 19.1 22 75.24 0.13 CCHD19.1L25T3 60 19.1 28 0.18 CCND15.9L25T1 27 15.9 24 0.11 CCND15.9L25T2 27 15.9 26 0.12 CCND15.9L25T3 27 15.9 24 0.12 CCHD12.7L50T1 56 12.7 47 87.78 0.23 CBHD12.7L50T2 54 12.7 51 0.27 CBHD12.7L50T3 55 12.7 46 0.24 CBHD12.7L50T4 55 12.7 51 0.28 CBHD15.9L50T1 53 15.9 50 94.05 0.28 CBHD15.9L50T2 48 15.9 50 0.28 CBHD15.9L50T3 49 15.9 49 0.28 CBHD15.9L50T4 49 15.9 48 0.27 CBHD15.9L50T5 53 15.9 56 0.33 CBHD15.9L50T6 48 15.9 51 0.30 CBHD19.1L50T1 52 19.1 59 0.42 CBHD19.1L50T2 57 19.1 50 0.32 CBHD19.1L50T3 57 19.1 49 0.30 CBND12.7L50T1 27 12.7 48 87.78 0.24 CBND12.7L50T2 27 12.7 49 0.25 CBND12.7L50T3 27 12.7 53 0.28 CBHD12.7L75T1 56 12.7 78 0.40 CBHD12.7L75T2 54 12.7 75 0.38 CBHD12.7L75T3 50 12.7 72 0.37 CBHD15.9L75T1 53 15.9 77 125.40 0.32 CBHD15.9L75T2 52 15.9 75 0.31 CBHD15.9L75T3 52 15.9 74 0.33 CBHD15.9L75T4 57 15.9 78 0.36 CBHD19.1L75T1 48 19.1 72 0.32 CB
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Reference Specimen identification
f'c (MPa)
α rb d0
(mm)hemb
(mm)Tensile strength (MPa)
Pu
(KN)P
max/Pu
Failure mode
HD19.1L75T2 57 19.1 74 0.35 CBHD19.1L75T3 52 19.1 76 0.38 CBHD19.1L75T4 57 19.1 75 0.36 CBHD19.1L100T1 48 19.1 99 188.10 0.31 CBHD19.1L100T2 50 19.1 100 0.32 CBHD19.1L100T3 57 19.1 100 0.32 CBHD12.7L50T8 54 12.7 50 50.16 0.10 FRHD19.1L50T8 54 19.1 50 0.50 FRHD12.7L75T7 54 12.7 75 0.41 FRHD12.7L75T8 50 12.7 75 75.24 0.38 FRHD12.7L100T3 50 12.7 100 0.38 FRHD12.7L100T4 53 12.7 100 0.30 FRHD12.7L100T5 56 12.7 100 87.78 0.44 FRHD15.9L100T2 53 15.9 100 137.94 0.31 FRHD12.7L50I45 53 135º 0.3 12.7 50 75.24 0.12 BF+BDHD15.9L75I15 53 165º 15.9 74 125.40 0.29 BFHD15.9L75I15 53 15.9 77 0.34 BFHD15.9L75I15 53 15.9 79 0.33 BDHD15.9L75I30 53 150º 15.9 75 0.25 BDHD15.9L75I30 53 15.9 77 0.19 BDHD15.9L75I45 53 135º 15.9 75 0.14 BF+BDHD15.9L75I45 53 15.9 78 0.17 BF+BD
Eshwar et al. (2008)[24]
pull-out 1 in 35,2 180º - 13 25 1836 144.20 0.15 BFpull-out 2 in 35,2 50 0.20 BF
Table 2. Database of shear tests on FRP anchors
Reference Specimen identification
f'c (MPa)
α rb d0
(mm)hemb
(mm)Tensile strength (MPa)
Pu
(KN)P
max/Pu
Failure mode
Villanueva el al
(2016) [25-26]
NA/F/90º/16/100-1 42.6 90º 0.3 16 100 1900 (anchor)
49.23 0.17 BDNA/F/90º/16/100-2 0.17 BDNA/F/90º/16/100-3 0.33 FRNA/F/90º/16/100-4 0.32 FRNA/F/90º/20/100-1 0.5 20 0.26 BDNA/F/90º/20/100-2 0.29 BDNA/F/90º/20/100-3 0.37 FRNA/F/90º/20/100-4 0.35 BDR/F/90º/16/100-1 2.0 16 0.36 FRR/F/90º/16/100-2 0.29 FR+BDR/F/90º/16/100-3 0.33 FRR/F/90º/16/100-4 0.47 --R/F/120º/20/75-1 120º 2.0 20 75 0.25 FRR/F/120º/20/75-1 0.24 FRR/F/120º/20/75-3 0.29 FRR/F/120º/20/75-4 0.36 --R/F/120º/20/100-1 100 0.49 FRR/F/120º/20/100-2 0.24 BDR/F/120º/20/100-3 0.32 BDR/F/120º/20/100-4 0.32 BDAV/F/90º/20/75-1 90º 2.5 20 75 0.26 FRAV/F/90º/20/75-2 0.30 BDAV/F/90º/20/75-3 0.37 FR
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Reference Specimen identification
f'c (MPa)
α rb d0
(mm)hemb
(mm)Tensile strength (MPa)
Pu
(KN)P
max/Pu
Failure mode
AV/F/90º/20/75-4 0.27 FRAV/E/90º/20/75-1 0.35 FRAV/E/90º/20/75-2 0.29 FRAV/E/90º/20/75-3 0.24 BDAV/E/90º/20/75-4 0.35 BDAV/F/90º/20/50-b 50 0.12 BFAV/F/90º/20/50-b2 0.15 BFAV/E/90º/20/50-1 0.24 BFAV/E/90º/20/50-2 0.32 FRAV/E/90º/20/50-3 0.21 BFAV/E/90º/20/50-4 0.31 BFAV/F/90º/20/50-1 0.32 BFAV/F/90º/20/50-2 0.31 BDAV/F/90º/20/50-3 0.20 BFAV/F/90º/20/50-4 0.12 BFAV/F/90º/20/100-1 100 0.33 FRAV/F/90º/20/100-2 0.34 FRAV/F/90º/20/100-3 0.33 FRAV/F/90º/20/100-4 0.33 BDAV/E/90º/20/100-1 0.35 FRAV/E/90º/20/100-2 0.38 FRAV/E/90º/20/100-3 0.29 BDAV/E/90º/20/100-4 0.29 BDAV/E/90º/20/125-1 125 0.33 FRAV/E/90º/20/125-2 0.39 BDAV/E/90º/20/125-3 0.45 FRAV/E/90º/20/125-4 0.38 FRAV/F/90º/20/125-1 0.45 FRAV/F/90º/20/125-2 0.29 FRAV/F/90º/20/125-3 0.32 BDAV/F/90º/20/125-4 0.40 FRR/F/90º/20/100-1 3.0 20 100 0.34 BD R/F/90º/20/100-2 0.42 FRR/F/90º/20/100-3 0.36 FRR/F/90º/20/100-4 0.44 FRR/F/135º/20/75-1 135º 2.5 20 75 0.33 FRR/F/135º/20/75-2 0.37 FRR/F/135º/20/75-3 0.42 FRR/F/135º/20/75-4 0.37 FRR/F/135º/20/100-1 100 0.44 FRR/F/135º/20/100-2 0.39 FRR/F/135º/20/100-3 0.33 FR+BDR/F/135º/20/100-4 0.43 FR
From the database, it is clear that the ratio Pmax /Pu is always considerably lower than
unity across all tests, implying that this relatively poor performance is a consequence of
the stress concentration in the bending region, despite the apparent nature of failure,
which often includes fibre rupture. This effect has already been pointed out by Orton
[27] and by Zhang and Smith [28], who also observed that the response of spike anchors
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in shear was influenced by the dowel angle. For pull-out tests on inclined anchors
included in the database the trend is similar, being the ultimate load enhanced by larger
values of α .
3. EXISTING PULL-OUT MODEL
To the best knowledge of the authors, the only published model for predicting the pull-
out strength of FRP spike anchors is by Kim and Smith [29]. This model considers pull-
out resistance as being the minimum predicted for concrete cone failure, anchor rupture,
or a mixed failure including the adherent. The transition from concrete cone to
combined failure is determined by the embedment length of the anchor; the anchor
rupture failure depends on the sectional area of the anchor and on the characteristic
tensile strength of the FRP. The model was calibrated using the results reported in
Özdemir [21], Kim and Smith [22] and selected data from Ozbakaloglu and Saatcioglu
[23]. Based on this statistical study, the final model is as follows:
Pu=min ( P cc , Pcb , Pu ) (1a)
Pcc=9.68· hemb1.5 ·√ f 'C (cone failure) (1b)
Pcb=τave · π · d0 · hemb (mixed/adherent failure) (1c)
Pu=γ · wFRP · tFRP · f FRP (anchor failure) (1d)
where τ ave , the average shear strength in the adhesive-to-concrete interface, is calibrated
as 4.62 MPa for f’c <20 MPa and 9.07 for f’c ≥20 MPa; the factor γ is introduced to
reflect the reality of a reduction in capacity when such rolled and bundled fibres are
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tested, and is given a value of 0.59.
This model works acceptably well for pull-out tests with embedment lengths up to 100
mm. However, the prediction for concrete cone failure has limitations: considering a
concrete strength of 30 MPa and a hole diameter of 20 mm, adherent failure would not
occur according to the model, as the prediction for concrete cone failure would be lower
than that for adherent failure up to the predicted value for Pu; similarly, for a hole
diameter of 16mm the transition from concrete cone to adherent failure is predicted to
occur at an embedment depth of 75 mm. In the database a great number of anchors with
large hole diameters exhibited adherent failure. Furthermore, in some cases concrete
cone failure occurred at similar loads to adherent failure, such as in PE40 14-1, PE40
16-1 and PE40 16-2 from Table 1. This, together with the fact that many anchors
displaying adherent failure were predicted to exhibit concrete cone failure, makes it
necessary to refine the model. It is hypothesized that concrete cone failure, mixed
failure and adherent failure can be grouped and predicted with a single expression, in
the same way in which mixed and adherent failure are united in the Kim and Smith
model [29].
4. STRENGTH OF THE ANCHOR IN THE BENDING ZONE
The existing tests on FRP anchors in shear reveal that there exists a reduction in pull-out
strength due to bending of the fibres. This has not been studied specifically for anchors
to date. Nonetheless, the equivalent effect in which reductions in capacity are found in
FRP stirrups used for shear reinforcement in concrete structures is comprehensively
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understood, with validated predictive expressions well documented [30-38].
As a consequence of the anisotropy of FRP materials, their strength in shear and
bending is much weaker than their tensile strength. At the bend in a spike anchor, the
embedded FRP experiences shear action against the surrounding concrete combined
with normal stresses in the direction of the fibres [30, 31]. The reduction in strength
which FRP stirrups embedded in concrete undergo due to a similar combination of
effects is primarily dependent on the bend ratio rb. The existing expression to estimate
the ratio between the effective bend strength ( f fb)and the direct tensile strength (f fu¿ of
the FRP making up the stirrup indicates a linear increase in bend strength with bend
radius:
f fb
f fu=( 0.05 · rb+0.3 )/F s (2)
The factor Fs is given different values in existing codes and guidelines, such as Fs=1.3
in JSCE [32], and Fs= 1.5 in ACI440.1R-06 [33] and ISIS-M03-07 [34]. This
formulation has been validated empirically, leading to some further considerations such
as the minimum embedment depth for testing [35], the influence of the cross-sectional
form for fixed fibre content [31], and the validity of the test methods [36]. The lack of
efficiency of bent FRP bars has led to the adoption of equation (2) in some models for
internal shear reinforcement with FRP bars, such as by Oller et al. [37], or alternatively
to the inclusion of high safety factors in other models such as by Lignola et al. [38].
Some researchers working on spike anchors have highlighted the importance of the
bend radius, though not necessarily in a systematic way. Orton et al. [27] introduced the
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necessity of smoothing the hole edge to avoid stress concentration, recommending a
radius at the edge of at least four times the anchor diameter, according to Morphy et al.
[30]. This requirement has been regarded as unrealistic as for a typical 10mm-diameter
anchor, rounding of 40mm around the hole is impractical if several anchors are being
installed [20]. Consequently, the ACI 440.2R [1] approach for externally bonded plates
in bond-critical applications is usually employed, namely that the radius around which
fibres are wrapped should be a minimum 13 mm radius, and smoothed, in order to limit
stress concentration. It is worth noting, however, that most of the published
investigations do not specify the bend radius but just the fact that some smoothing was
provided.
Although the existing recommendations for bend-related reductions in capacity for FRP
stirrups have clearly not been developed for FRP anchors, behaviour in terms of
combinations of stress appears to be similar between the two situations. For the model
proposed here, the expression for reduction in capacity at a bend found for internal FRP
reinforcement has been adopted for spike anchors, suitably modified. In particular, the
anchor fan is not embedded in concrete, and subsequently the bending region cannot be
equally confined over all its length, and the angle of bending may not be 90º. These
modifications are described below as part of the full analytical model.
5. DEVELOPMENT OF THE ANALYTICAL MODEL
Based on test results, it is clear that the analytical model needs to be able to predict three
distinct possible failure mechanisms. These are:
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1.Mixed or adherent failure, related to insufficient embedment depth of the dowel,
and governed by the shear strength of the resin-substrate interface within the
anchor dowel (as concrete cone failure only occurred for very low embedment
depths in the pull-out tests, it is incorporated into this failure mode);
2.Bending zone failure of the anchor, in which combinations of stress are important;
and
3.Tensile failure of the anchor, related directly to its cross-sectional area and tensile
strength.
5.1 Adherent failure
Concrete cone failure is only likely to occur for very low embedment depths. For mixed
concrete-cone and adherent failure, the depth of the concrete pull-out cone decreases
when the embedment length is increased; for anchors with 25 mm embedment, the pull-
out depth is nearly equal to the embedment depth, whilst for 75 mm embedment anchors
the average cone depth drops to 10 mm [21]. Moreover, concrete cone failure is an
undesirable failure when it comes to structural retrofitting, as it would mean that the
original member is substantially damaged by the reinforcement. Taking this into
consideration, and for design purposes, the concrete cone failure is treated here as a
failure to be avoided, with a minimum depth of 50 mm dowel embedment being
sufficient to ensure this for concrete strengths equal or greater than 20 MPa.
Embedment lengths greater than 40 mm also guarantee that stress transfer is not taking
place within the concrete cover [39].
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The expression predicting the adherent failure, therefore, must be able to adequately
predict the maximum load capacity of anchors tested in pull-out or in shear, undergoing
substrate and mixed failures. Existing bond models for post-installed anchors have been
considered. All models have the same general form, entailing bond shear strength and
interfacial area. The bond shear strength is often reported by the manufacturer in
relation to installation instructions [40], and depends on the cleanliness of the hole and
the drilling method, amongst other parameters [41-44]. The general equation is equal to
(1c) from the model by Kim and Smith [26] adopting different values of τ ave.
In spite of the applied bond shear stress being non-uniform along the embedment depth
in reality, the expression considers an average stress, which is likely to only be
applicable to small embedment lengths. However, as bending and tensile failure govern
behaviour of deep anchors, this assumption appears to be acceptable for modestly-
embedded dowels. Table 3 summarises the values of τ ave found in the literature, as well
as the average experimental values from the database for anchors undergoing adherent
failure.
Table 3. Values of τ avefor the existing bond models and database adherent failure tests
Existing models for post-installed anchors
Reference τ ave(MPa)
CEB (1994, adhesive metallic anchors) [45] 8
Cook and Kunz (2001) [46] 12
Kim and Smith (2010) [29] 4.62 for f ' c<20 MPa
9.07 for f ' c ≥ 20 MPa
ACI 318-14 (2014) [47] 13.8-17.3 (13 to 19 mm anchors)
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397
Database
Reference f ' c (MPa) τ ave(MPa)
Özdemir (2005) [21] 10 4.5
16 5.2
Ozbakaloglu and Saatcioglu (2009) [23] 50.5 8.0
Kim and Smith (2009) [22] 33.6 9.5
Eshwar et al. (2008) [24] 35.2 17.5
Villanueva et al. (2016) [25, 26] 42.6 8.6
With the exception of the Kim and Smith model [29], the existing models from
literature were based on the behaviour of metal anchors and post-installed reinforcing
bars. In most cases, neither the quality of the surface of the hole nor the bond strength of
the adhesive is defined. It is clear from existing results that a specific design value for
τ ave is not easy to determine. However, it appears from the data that the lower the
concrete strength, the lower the bond strength. In this spirit, Kim and Smith [29]
provided two values for τ ave, depending on the compressive strength of the concrete.
The proposed model here retains the bond strength of 4.62 MPa associated with low-
strength concrete (less than 20 MPa) used by Kim and Smith [29]. For concrete stronger
than 20 MPa, τ ave is selected to be 9.5 MPa, as previously found by Kim and Smith [22]
in the context of spike anchors.
5.2 Failure in the bend zone
This aspect of the model corresponding to rupture in the bending region is based on the
capacity-reduction expression at a bend in an internal FRP shear stirrup. The
formulation has been modified to take into account the influence of hemband that of α ,
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parameters known to be important in defining an anchor’s strength in shear [23, 25-26,
28].
Even though the database reveals an almost linear increase in strength with embedment
length, this does not imply that capacity will continue to increase indefinitely as the hole
deepens. It has been hypothesized, therefore, that there must be an effective embedment
length similar to the effective bond length in externally-bonded FRP reinforcement.
This hypothesis seems to be confirmed by an increase in maximum load and initial
stiffness observed by Ehsani et al. [35] for FRP internal stirrups; they found a
proportional increase in capacity with embedment lengths up to 12 times the bar
diameter, and recommended a straight embedment length equal to 16 times the bar
diameter for 90º hooks.
Even though the behaviour of FRP spike anchors is not identical to that of internal FRP
stirrups, as the bend region itself is never completely covered by concrete, a similar
effect was observed by Villanueva et al. [25]. For the proposed model here, an
embedment length of 150 mm is considered to provide sufficient stiffness and
development length. In other words, it provides enough additional bond capacity to the
hook for spike anchors of common diameter between 10 and 13 mm.
The effect of the dowel angle is closely associated with the bend radius, due to
interlinked geometry, because both parameters affect the arc length that is subject to
kinking action (see Figure 5). The beneficial effect of dowel angles greater than 90º was
reported in previous publications. Zhang and Smith [28] reported an almost linear
increase in capacity for angles ranging between 45º and 157.5º, whereas Ozbakaloglu
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440
and Saatcioglu [23] concluded that angles close to 180º produce a similar response to
pure pull-out.
Based on the assumptions made above, the expression for failure in the bend zone is
defined as follows:
Pbend=[0.3 ·hemb
150+0.05 · rb · α
( π2 ) ] · Pu (4)
Equation (4) modifies equation (2), including the specific parameters affecting the spike
anchor’s performance (embedment length and dowel angle). This is a novel
contribution, as no equation has existed to date for the capacity prediction of FRP
anchors. The bend strength can be expressed as a fraction of the tensile capacity of the
anchor Pbend / Pu.
5.3 Tensile capacity of the anchor
The tensile capacity of the anchor itself is clearly the upper-bound limit of capacity. The
characteristic tensile strength of a spike anchor may be determined using coupon tests.
However, as most of the FRP anchors from the literature are hand made, the tensile
strength must be determined according to whether the anchor was formed from bundle
fibres or from a sheet. The tensile strength of an anchor formed from a rolled sheet can
be calculated according to equation (1d); unless specifically calibrated against flat
coupon tests, the value of 0.59 suggested by Kim and Smith [29] can be used. In this
case the resulting cross section is irrelevant, as the contribution of the resin to the tensile
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strength is disregarded.
Alternatively, the tensile strength can be calculated from the circular section in cases
where a manufacturer is able to provide commercial bundles containing a fixed quantity
of fibres. It can also be estimated by considering the ratio of fibres and of matrix, and
the characteristic tensile strength of each component. Once the homogenized tensile
strength f FRP is calculated, the diameter of the anchor da defines the anchor’s capacity in
terms of its cross section:
Pu=π ·( da
2 )2
· f FRP=Aa · f FRP (5b)
It is worth noting that the tensile strength of a spike anchors is rarely achieved in
testing, even when the recorded failure mode is fibre rupture in the free length [21-
23,25-26].
In the database, failure rupture did not occur at significantly higher values than adherent
(in the case of pull-out tests) or bend-zone failure (in shear tests); consequently, instead
of using a reduction factor, the tensile strength of the anchor has been kept as nominal,
and the tests exhibiting this failure have fallen into the categories of adherent or
bending-zone failure. In the proposed model, the capacity associated with each possible
failure mode is expressed in terms of the ratio between its capacity prediction and the
tensile strength of the anchor, leading to a maximum value of 1. For design purposes,
the maximum strength of the anchor is defined as the minimum value amongst the
adherent strength, bending-zone strength and tensile strength. Therefore, the general
form of the model is:
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Panc=min {Pcb , Pbend , Pu } (6)
6. CALIBRATION OF THE MODEL
The expressions presented above have been calibrated against experimental data. Three
approaches have been considered: raw model, best-fit model and design model. The raw
model derives from the general formulations above. The best-fit model was obtained
from a linear regression optimised using a least-squares approach. The design model
was optimised to obtain a 75% probability of exceedance, given the considerable scatter
of results from the database caused by the high number of variables considered in the
database; a constraint was imposed to keep the ratio of the average test value to
prediction value as close as possible to 1, but greater than 1.
Equation (6) above is employed, where Pu is the tensile capacity of the anchor estimated
using equation (5a) or (5b); Pcb, correspondent to adherent or mixed failure, can be
predicted according to (1c) with:
τ ave={4.62 for f ' c<20 MPa9.50 for f ' c ≥ 20 MPa (7b)
Whereas expression for bending failure is as follows:
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Pbend=a ·hemb
150+b · rb ·[ α
( π2 ) ]
c
(7c)
where the parameters a, b and c are the factors defining the influence of embedment
length, bend ratio and dowel angle. Table 4 summarises the values of a, b, and c which
are used or discovered (through calibration against 175 results in the database) to define
each of the raw model, best-fit model and design model. Interestingly, the raw-model
results are quite similar in accuracy to those of the best-fit model, which indicates that
the originally-specified parameters are valid, such that an approximation to behaviour
associated with internal stirrups seems to be a reasonable analogy to the behaviour of
spike anchors. From this it is concluded that the proposed effective embedment length is
enough to make the anchor work as if completely confined by surrounding concrete, as
would be the case for internal reinforcement. The subsequent suggested design values
for a, b, and c provide acceptably safe statistics.
Table 4. Calibration factors and statistical performance of the models
Calibration factors Statistics
a b c AV SD PE
Raw model 0.3 0.05 1 1.09 0.28 0.68
Best-fit model 0.391 0.032 1.575 1.05 0.27 0.62
Design model 0.32 0.03 1 1.13 0.28 0.75
Note: a, b and c are the calibration factors for embedment length, bend ratio and dowel angle,
respectively; AV = average ratio of test results to predictions; SD = standard deviation; PE =
percentage of exceedance (%)
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The actual test results in terms of Panc
Pu(vertical axis) versus the predicted values
(horizontal axis) for the raw, best-fit and design cases are shown in Figures, 6, 7 and 8,
respectively.
Figure 6. Test results of Panc
Puversus predicted for the raw model
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536
Figure 7. Test results of Panc
Puversus predicted for the best-fit model
Figure 8. Test results of Panc
Puversus predicted for the design model
7. PARAMETRIC ANALYSIS
This section assesses the capacity of the best-fit and design models to predict both
Pmax /Pu and the failure mode of anchors. Two fixed relationships have been considered
as they are most representative from the existing database: rb=2.5, with α=90 º , across
various embedment depths (Figure 9), andhemb=100 mm across various values of rb · α
(π2 )
(Figure 10). The parameters rb and α have been grouped in this way because for the
design model the factor c affecting the dowel angle α is equal to 1, and so for any dowel
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angle the strength evolves proportionally to rb · α
(π2 ) ; for the best-fit model, on the
contrary, for a given rb · α
(π2 ) the bend-zone capacity does depend on the angle, as
represented in Figure 10.
Figure 9 shows the predictive capacity of the best-fit model and the design model across
varying embedment lengths. The models have been plotted for adherent and bend-zone
failure and the specified rb · α
( π2 )
=2.5, so as to observe the general form of the models
and the transition from adherent to bend-zone failure. This leads to bi-linear solutions
corresponding to the two regions of the models to the adherent and bending capacity,
respectively; as for adherence there is only one approach, best-fit and design models
share the first region and they diverge at the point where the failure mode changes.
The models have been plotted together with the results from the database [21-26]. It can
be observed that, in spite of the scatter of the results, the trend when increasing the
embedment length is predicted reasonably well by the models. Notwithstanding this, the
reliability of the predictions drops in the transition zone from adherent to bend-zone
failure. The best-fit model locates the transition from adherent to bend-zone failure at an
embedment length of 56 mm, whereas according to the design model this transition is
predicted to occur at 40 mm. These predictions are consistent with the observed failure
modes of 50 mm embedment anchors tested in shear, as both adherent and bend-zone
failures were found. Because bend-zone failure is more brittle than adherent failure, a
model in which no tests failing in the bend zone are predicted to have adherent failure is
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safer, justifying the use of the proposed design model.
Figure 9. Predictive capacity of best-fit and design models for rb=2.5 and α = 90º
Figure 10 represents the variation in bend-zone strength with the bend ratio and dowel
angle. As previously discussed, the design model has only one prediction because the
value of the parameter c is equal to 1, while the best-fit model presents slightly different
predictions forα=¿90º, α=¿120º and α=135º.
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Figure 10. Best-fit and design model predictions for hemb=100 mm
The high scatter of results for α=¿120º was the underlying reason for the stronger
reliance on the dowel angle in the best-fit model. It appears from Figure 10 that the
design model achieves a suitable degree of safety, especially for α=¿90º, by assuming c
=1.
8. PERFORMANCE OF SPIKE ANCHORS IN ANCHORED JOINTS
The efficacy of spike anchors to enhance the response of externally bonded plates has
been reported in the literature [14, 28, 48-51], but the parameters of the anchor dowel
governing the anchor’s performance are rarely optimised. To date, much more attention
has been paid to the fan angle, since Kobayashi et al. [15] highlighted that it plays a key
role in the stress-transfer mechanism; they also suggested that the fan angle should be
limited to less than 90º. Subsequently, many authors have employed fan anchors
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598
ranging from 60º to 75º [27-28, 50-51]. Further investigation on the influence of the fan
angle was conducted by Zhang and Smith [28] through a series of tests on single-fan
and bow-tie (double-fan) anchors with 60º fan angle. It was found that no significant
differences existed between these configurations provided that the principal stress
direction did not vary, demonstrating the suitability of 60º fans. Other investigations,
however, have been carried out using anchors having a fan angle of 360º, intended to
cover all possible changes in the direction of principal stress [14, 24, 48-49]. Regardless
of the angle of the anchor fan, it is generally agreed that the length of the anchor fan
must be designed to completely cover the reinforcement width. Accordingly, the
research community has employed various geometrical combinations of fan length and
fan angle, covering the full width of the FRP in the case of single anchors, or spaced to
have tangent or secant fans in the case of multiple anchors.
Tables 6 and 7 presents the results from the most relevant shear tests on anchored joints
in concrete. They include the results of tests conducted by Eshwar et al. [24] and Zhang
and Smith [28, 50] with one anchor (Table 6), and those by Niemitz [48], Niemitz et al.
[14], Breña and McGuirk [49] with two anchors transversally arranged (Table 7).
To assess the applicability of the proposed model for externally bonded FRP plates with
spike anchors, the contribution of the anchors (γ ') has been evaluated through the
expression:
P joint−Pdb
Panc=γ ' (8)
where P joint is the capacity of the anchored joint, Pdb is the capacity achieved by the
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601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
equivalent control (unanchored) specimens of each reference, and Panc is calculated
according to the proposed design model. The expression is consistent with the findings
of Breña and McGuirk [49] and Eshwar et al. [24], who observed that the capacity of
the bond alone plus the capacity of the anchors alone was very close to the capacity of
the bonded anchored joints, by testing unbonded anchored sheets as well as bonded
sheets with and without anchors.
Table 6. Anchored joints’ database for specimens with one anchor
Eshwar et al. (2008) [24]
T-2(50) 10 50 -- 66.7 360 90 53.4 0.83T-2(75) 10 75 -- 66.7 360 90 53.4 0.56
Zhang & Smith (2011-2013)
[28][50]
SF-200 12 40 50 32.5 60 90 15 1.00SF-200R 12 40 50 25.4 -60 90 15.9 0.57BF-200 12 40 50 28.8 ± 60 90 15.9 0.77BF-300 12 40 50 29.9 ± 60 90 15.9 0.84BF-400 14 40 50 32.8 ± 60 90 15.9 0.89DA-45 12 40 50 19.7 60 45 15.9 0.23DA-67 12 40 50 21.4 60 67.5 15.9 0.33DA-90 12 40 50 32.5 60 90 15.9 1.00
DA-101,3 12 40 50 30.4 60 101.3 15.9 0.87DA-112,5 12 40 50 35.7 60 112.5 15.9 1.19DA-123,8 12 40 50 35.3 60 123.8 15.9 1.16DA-135 12 40 50 39.0 60 135 15.9 1.38
DA-157,5 12 40 50 40.4 60 157.5 15.9 1.47CD-134 12 40 50 24.8 60 90 17.9 0.41CD-200 12 40 50 27.7 60 90 17.9 0.58CD-259 12 40 50 26.4 60 90 17.9 0.50CD-134 12 40 50 29.0 60 90 17.9 0.66CI-200 12 40 50 31.1 60 90 17.9 0.79CI-259 12 40 50 30.1 60 90 17.9 0.73PL-100 12 40 50 17.5 60 90 15.9 0.10PL-125 12 40 50 24.0 60 90 15.9 0.39PL-150 12 40 50 24.9 60 90 15.9 0.54PL-175 12 40 50 26.2 60 90 15.9 0.62PL-200 12 40 50 26.8 60 90 15.9 0.66PL-225 12 40 50 25.4 60 90 15.9 0.57PL-250 12 40 50 32.5 60 90 15.9 1.00PL-275 12 40 50 36.1 60 90 15.9 1.21PL-300 12 40 50 35.3 60 90 15.9 1.16PL-325 12 40 50 32.0 60 90 15.9 1.16PL-350 12 40 50 35.4 60 90 15.9 1.17PW-50 12 40 50 28.5 60 90 15.9 0.76PW-75 12 40 50 36.9 60 90 25.0 0.72PW-100 12 40 50 42.8 60 90 32.9 0.59PW-125 12 40 50 51.3 64 90 40.4 0.65PW-150 12 40 50 59.5 74 90 52.1 0.44
PT-2 12 40 50 26.2 60 90 14.4 0.71PT-3 12 40 50 28.5 60 90 16.6 0.71PT-4 12 40 50 33.9 60 90 18.9 0.90
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625
626
627
628
629
630
631
PT-5 12 40 50 34.9 60 90 20.0 0.89PE-227 12 40 50 33.9 60 90 19.0 0.90PE-95 12 40 50 24.8 60 90 13.4 0.681FA 12 40 50 31.1 60 90 15.5 0.931RA 12 40 50 40.9 60 90 15.5 1.52
Mean value 0.83
Table 7. Anchored joints’ database for specimens with two anchors transversally arranged
Reference Specimen da
(mm)hemb
(mm)
Fan length (mm)
P joint(kN)
Fan angle
Dowel angle
Pdb(kN)
γ '
Niemitz (2010)two anchors
[48]
B-Y-2-5-4 13 51 51 55.3 360 90 35.6 0.90B-X-2-5-4 13 51 51 60.6 360 90 35.6 1.15C-Y-4-10-6 19 51 102 96.6 360 90 50.9 1.35C-X-4-10-6 19 51 102 87.6 360 90 50.9 1.09
Niemitz et al. (2010)[14]
BII-13-1.3-5 13 51 25.5 55.3 360 90 35.6 0.90BIIS-13-1.3-5 13 51 25.5 60.6 360 90 35.6 1.15BII-25-1.9-10 19 51 51 96.6 360 90 50.9 1.35
BIIS-25-1.9-10 19 51 51 87.6 360 90 50.9 1.09Breña and
McGuirk (2013)[49]
S1-2a-24 13 51 32 77.9 360 90 43.4 2.71F1-2a-24 13 51 32 80.6 360 90 49.8 2.34F2-2a-24 13 51 32 150.5 360 90 69.8 6.13S2-2a-24 13 51 32 117.0 360 90 -- --
Mean value 1.83
It should be noted that there is considerable scatter in the results in Tables 6 and 7.
However, it appears that, particularly for a single anchor, the design model provides
reasonable accuracy.
The fan angle does not appear to significantly influence the effectiveness of the anchor.
In most cases involving two anchors and a full 360º anchor fan the effectiveness of each
anchor is lower than the average value for one anchor, whereas in tests with one anchor
and various fan angles, no definite trend is observed (see Table 6, specimens PW-100,
PW-125 and PW-150). Accordingly, it is concluded that the fan angle should be
designed according to the stress pattern, as it appears to plays a minor role in the spike
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anchor’s performance.
Given the scatter of results, and in line with the design model for isolated anchor
capacity, a design value of γ 'd has been chosen to provide a 75% probability of
exceedance. Thus, for design purposes, it turns out that γ 'd = 0.58 for sheets with one
anchor.
The scarcity of results for anchored joints with two anchors in one row hinders the
optimisation of multiple anchors, and consequently a conservative equation is needed.
For designs with multiple anchors transversally arranged, the mean value of γ ' is 1.83,
equivalent to an individual contribution of each anchor with γ '=0.91. Thus, accepting a
value of γ 'd = 0.58 represents a conservative approach, as proposed in equation (9):
P joint=min {(Pdb+0.58 ·n· Panc) ,1.58 · Pdb } (9)
where P joint is the strength of the anchored joint, Pdb is the pull-off bond capacity
between FRP and concrete, n is the number of FRP spike anchors, and Panc is the
individual anchor’s capacity. The expression is equivalent to limiting the contribution of
the anchors to 0.58 times the bond strength of the reinforcement, regardless of the
anchors’ capacity
9. DESIGN RECOMMENDATIONS
From the results reported in the previous sections, the design model recommendations
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are given as follows:
The general model is applied to any anchor, so that the maximum attainable strength is
defined as in equation (6). The tensile strength of the anchor can be calculated using
either equation (5a) or (5b) depending of the fabrication method for the anchor. The
adherent or mixed failure must be evaluated using equation (1c) and (7a) or (7b)
according to the concrete strength. For its part, the resulting equation for bend strength
is:
Pbend=[0.32 ·hemb
150+0.03 · rb · α
( π2 ) ]· Pu (10d)
The design model has been plotted in Figures 11 and 12, showing the variation in
bending strength with rb · α
(π2 )and that of the bending and adherent strength with hemb,
respectively. By comparing the two figures, the most relevant conclusion is that bend-
zone failure leads to a value of Panc which is considerably lower than Pu, regardless of
the dowel angle and the bending radius, while adherent failure leads to greater ratios of
Panc /Pu. In other words, higher capacities can be reached by anchors in pull-out than by
anchors subjected to kinking in the bend zone.
The influence of hemb is most important for both adherence and bend-zone behaviour.
For bend-zone strength, a 16% enhancement in Panc /Pu is obtained with hemb=150 mm
compared with hemb=50 mm. This influence is even greater in pull-out, as can be seen in
Figure 12, where the strength is proportional to hemb. For hemb=150 mm, the predicted
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pull-out strength is twice that for bend-zone failure in which rb · α
( π2 )=2.5.
Notwithstanding, the enhancement due to an increasing bend radius is significant. For
instance, for α = 150º with an inner radius equal to 36 mm (which leads to rb · α
(π2 ) = 6),
the bending strength is enhanced by 11% compared with non-smoothed holes with α =
90º (characterised by rb · α
(π2 ) = 0.3-0.5, according to Figure 5).
Figure 11. Model for bending failure as a function of rb · α
(π2 )
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Figure 12. Model for adherent and bending failure as a function of hemb
In figure 12 the variation in maximum capacity is plotted against the embedment length
for varying values of rb · α
( π2 ) . The point where the failure mode shifts from adherent to
bending failure depends on the arc length of the bending region. Pull-out response (
α=180 º) represents the limit on the capacity of anchors in shear. For greater ratios of
rb · α
(π2 ) , apart from the enhancement in bend-zone capacity, adherent failure governs up
to larger embedment depths; as adherent failure is more ductile than bend-zone failure
[25], shallow anchors could be designed when such ductility is necessary.
The plots presented above can be used to estimate the capacity of the anchor, Panc. Once
this value is known and accepted as the design load-bearing capacity of the anchor in
pull-out or shear, it can be added to the debonding strength Pdb for FRP reinforcement.
The joint strength of anchored reinforcement is then determined as follows:
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P joint=Pdb+0.58 · Panc (11a)
where Pdb may be calculated according to the existing guidelines.
For multiple anchors there are not enough data to establish an optimum spacing and
further investigation is needed. For the time being, it is recommended that anchor fans
are designed so as to fully cover the width of the reinforcement, and that the anchors’
contribution to the strength is limited to:
P joint≤ 1.58 · Pdb (11b)
10. CONCLUSIONS
The pull-out and shear resistance of FRP anchors in concrete have been investigated
analytically. A model comprising three different failure modes is proposed, based on an
existing model for post-installed anchors in pull-out and an expression for reduction in
capacity at the bend in internal FRP reinforcement. A test database was employed to
calibrate the model; the database includes results from various previous works, covering
various construction and installation techniques.
The best-fit model provides reasonably good predictions of capacity. However, it is
regarded as somewhat unsafe given the scatter in experimental data. The proposed
design model, on the other hand, provides sufficient accuracy at a safer level for the
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capacity of isolated FRP anchors.
According to the proposed model, and in accordance with the experimental results from
the database, it is concluded that the embedment length is the most critical value for
both pull-out capacity and bend strength of spike anchors. Nevertheless, the dowel angle
and the smoothing technique are also crucial when it comes to evaluating the anchor’s
capacity in shear.
Spike anchors mainly subjected to shear forces have a limited efficacy due to the
reduction in strength in the bending region. This reduction, which can be estimated with
the proposed model, can be partially mitigated by a proper design of dowel angle and
smoothing of the hole edge; for a given embedment length, an enhancement of up to
11% of the anchor’s tensile strength can be achieved by a combination of these
parameters.
For the application of FRP anchors to externally bonded plates, parameters associated
with hole diameter, depth of embedment, anchor diameter, bend radius and angle of
dowel embedment have been found to play a more significant role in the efficiency of
the anchors than the fan angle.
This finding has underpinned the analytical model in this paper, which now allows
designers to specify spike anchors with confidence, for the first time.
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