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Modeling and analysis of the shear capacity of adhesive anchors post-installed into uncracked concrete

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This paper reports the results of an analytical study to predict the edge breakout shear capacity of single adhesive anchors post-installed into uncracked hardened concrete. For this purpose, an experimental database for the adhesive anchors compiled by the ACI Committee 355 was obtained and utilized to construct training and test sets so as to derive the closed-form solution by means of gene expression programming (GEP). The independent variables used for development of the prediction model were anchor diameter, type of anchor, edge distance, embedment depth, clear clearance of the anchor, type of chemical adhesive, method of injection of the chemical, and compressive strength of the concrete. The generated prediction model yielded correlation coefficients of 0.98 and 0.92 for training and testing data sets, respectively. Moreover, the performance of the proposed model was compared with the existing models proposed by American Concrete Institute (ACI) and Prestressed/Precast Concrete Institute (PCI). The analyses showed that the proposed GEP model provided much more accurate estimation of the observed values as compared to the other models.
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Modeling and analysis of the shear capacity of adhesive anchors
post-installed into uncracked concrete
Mehmet Gesog
˘lu
a,
, Esra Mete Güneyisi
a
, Erhan Güneyisi
a
, Muhammet Enes Yılmaz
a
,
Kasım Mermerdasß
b
a
Gaziantep University, Civil Engineering Department, 27310 Gaziantep, Turkey
b
Hasan Kalyoncu University, Civil Engineering Department, 27410 Gaziantep, Turkey
article info
Article history:
Received 16 January 2013
Received in revised form 18 November 2013
Accepted 9 January 2014
Available online 17 January 2014
Keywords:
A. Metal–matrix composites (MMCs)
B. Adhesion
B. Strength
C. Analytical modeling
abstract
This paper reports the results of an analytical study to predict the edge breakout shear capacity of single
adhesive anchors post-installed into uncracked hardened concrete. For this purpose, an experimental
database for the adhesive anchors compiled by the ACI Committee 355 was obtained and utilized to con-
struct training and test sets so as to derive the closed-form solution by means of gene expression pro-
gramming (GEP). The independent variables used for development of the prediction model were
anchor diameter, type of anchor, edge distance, embedment depth, clear clearance of the anchor, type
of chemical adhesive, method of injection of the chemical, and compressive strength of the concrete.
The generated prediction model yielded correlation coefficients of 0.98 and 0.92 for training and testing
data sets, respectively. Moreover, the performance of the proposed model was compared with the exist-
ing models proposed by American Concrete Institute (ACI) and Prestressed/Precast Concrete Institute
(PCI). The analyses showed that the proposed GEP model provided much more accurate estimation of
the observed values as compared to the other models.
Ó2014 Elsevier Ltd. All rights reserved.
1. Introduction
Anchorages to concrete include cast-in-place and post-installed
anchors. Post-installed anchors are either mechanical or adhesive
(bonded) anchors [1]. As a result of advancement in high strength
bonding agent technology, using adhesive anchors have soared
considerably [2]. An adhesive anchor can be defined as a reinforc-
ing bar or a threaded rod inserted into a drilled hole in hardened
concrete with a structural adhesive acting as a bonding agent be-
tween the concrete and the anchor steel [3]. The hole is filled with
the adhesive that bonds the steel to the concrete. Currently
available structural adhesives can be classified as several types of
thermosetting plastics including epoxies, polyesters, vinylesters
as well as a few hybrids of organic and inorganic binders [4]. Adhe-
sive anchors have three possible failure mechanisms: yield and
fracture of failure anchor steel, formation of an adhesive cone,
accompanied by pullout of an adhesive core; and pullout of
adhesive core [5].
In the literature, there have been elaborate studies regarding
the mechanical behavior of post installed anchors [1–9]. Gesog
˘lu
et al. [6] examined the behavior of such anchors in normal and
high-strength concretes with and without steel fiber reinforce-
ment. They stated that the maximum capacity of the anchors
increased with increasing the concrete strength. They also revealed
that the use of steel fibers in concrete did not importantly affect
the pullout capacity of the anchors but the failure type altered
from cone to pullout. Fujikake et al. [9] carried out an experimental
study to investigate the behavior of the adhesive anchors subject to
rapid pullout loading. They reported that the dynamic ultimate
pullout resistance increased with the loading rate. In the study of
Cook and Konz [4], the factors affecting the bond strength of the
polymer based adhesive anchors were investigated. The findings
of this study demonstrated that the reliable prediction of adhesive
anchor performance was only practical by extensive product and
condition specific testing.
The studies regarding the modeling shear capacity of the anchor
bolts has not yet found adequate attention in the literature. Lee
et al. [10] carried out an experimental study on shear behavior of
headed anchors with large diameters and deep embedments. They
compared the experimental shear capacities with those computed
from existing formulas proposed by ACI 349 and ACI 318 design
codes. They concluded that the existing methods yielded less shear
capacities for the specimens dealt with the study. Bickel and Shaik
[11] compared the prediction of performance of the model speci-
fied in PCI Design Handbook and CCD model from ACI 318-02,
for shear capacity of the headed and adhesive anchors. They re-
ported that PCI Design Handbook method and CCD method, with
proper adjustments, can be used for predicting the shear capacities
of adhesive anchors with similar accuracy.
1359-8368/$ - see front matter Ó2014 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.compositesb.2014.01.015
Corresponding author. Tel.: +90 342 3172404; fax: +90 342 3601107.
E-mail address: mgesoglu@gantep.edu.tr (M. Gesog
˘lu).
Composites: Part B 60 (2014) 716–724
Contents lists available at ScienceDirect
Composites: Part B
journal homepage: www.elsevier.com/locate/compositesb
In the prediction models, various parameters are used for
estimating the shear capacity of the adhesive anchors. Compres-
sive strength of the concrete and edge distance parallel to the
loading direction are included in all of the shear capacity predic-
tion models. The models proposed by ACI 349 and ACI 318 (CCD
model) also contain anchor diameter and embedment depths.
However, type of anchor bolt (rebar or threaded rod), type of
chemical adhesive, method of injection and clear clearance of the
drilled hole are not taken into account when computing the shear
or pullout capacities of the post installed anchors. For example, the
hole diameter is typically 10 or 25 percent larger than the inserted
anchor bolt or bar diameter [3]. Therefore, the effect of clear clear-
ance can be considered as a factor influencing the mechanical
behavior of anchor.
This study aims at providing a handful tool for prediction of the
edge breakout shear capacity of post-installed anchors with a rea-
sonable degree of accuracy. For the purpose of explicit formula-
tions for the shear capacity of the single adhesive anchors, the
worldwide database compiled by the ACI Committee 355 was
employed. A total of 98 adhesive single anchor tests were selected
regarding shear load testing in uncracked concrete. Soft computing
technique of genetic programming was applied for constructing
the model proposed in this study. Moreover, the derived model
was compared with the existing formulas given in ACI 318, ACI
349 and Prestressed/Precast Concrete Institute (PCI) as well as
Modified CCD method proposed by Hoffman [12].
2. Estimating edge breakout shear capacity of a single anchor as
specified in current design codes
In order to estimate the concrete edge breakout strength of an-
chor bolts under shear loading various relations have been pro-
posed in the literature as given in Eqs. (1)–(8) [11–18]. The ACI
shear resistance formula assumes the concrete failure surface to
be a semicone of height equal to edge distance and a contact incli-
nation angle of 45°with respect to the contact edge [13]. The shear
resistance of anchor bolt is calculated on the basis of the tensile
strength of the concrete acting over the projected area of the sem-
icone surface. According to ACI 349-97 [14], the design shear
strength is given by the formula below (Eq. (1) in U.S. customary
units). Ueda et al. [13] presented the same relation in SI units
(Eq. (2)). The concrete capacity design method is based on K-meth-
od developed by University of Stuttgart (Germany) in the late
1980s [11]. For ACI 349-06 [15], the value of k=7 was valid for
cracked concrete while the tests selected herein were performed
in uncracked concrete. Assuming a ratio of uncracked to cracked
strength of 1.4, a value k=9.8 (k=71.4) was utilized for the
evaluation of predicted capacities [10]. In ACI 349-06, edge break-
out shear capacity of bolt was presented by Eq. (3) [15]. The mod-
els based on Concrete Capacity Design (CCD) [16,17] and Modified
CCD [12] were given in Eqs (4)–(6). The capacity of a single anchor
in uncracked structural member under shear loading toward the
free edge is also described in Precast/Prestressed Concrete Institute
(PCI) Design Handbook (fifth edition) [18]. Eqs. (7) and (8) of PCI
method are given in US customary and SI units below.
ACI 349-97 (in U.S. customary units) [14]:
V
U
¼2
p
c
2
1
ffiffiffi
f
0
c
qðlbÞð1Þ
SI equivalent of this formula [13]:
V
U
¼0:522c
2
1
ffiffiffi
f
0
c
qðNÞð2Þ
ACI 349-06 (in U.S. customary units) [15]:
V
U
¼9:8ðl=d
0
Þ
0:2
ffiffiffiffiffi
d
0
pffiffiffi
f
0
c
qc
1:5
1
ðlbÞð3Þ
Concrete Capacity Design (CCD method) (in U.S. customary units)
[16,17]:
V
U
¼13ðl=d
0
Þ
0:2
ffiffiffiffiffi
d
0
pffiffiffi
f
0
c
qc
1:5
1
ðlbÞð4Þ
SI equivalent of this formula [11]:
V
U
¼1:1ðl=d
0
Þ
0:2
ffiffiffiffiffi
d
0
pffiffiffiffiffi
f
0
cc
qc
1:5
1
ðNÞð5Þ
Modified CCD method (in SI units) [12]:
V
U
¼3d
0:1ðh
ef
=c
1
Þ
0
h
0:1ðd
0
=c
1
Þ
0:2
ef
ffiffiffiffiffi
f
0
cc
qc
1:5
1
ðNÞð6Þ
PCI method (in U.S. customary units) [18]:
V
U
¼12:5c
1:5
1
ffiffiffi
f
0
c
qðlbÞð7Þ
SI equivalent of this formula [18]:
V
U
¼5:2c
1:5
1
ffiffiffi
f
0
c
qðNÞð8Þ
where V
U
is the ultimate shear capacity of an adhesive anchor in un-
cracked concrete (lb for the equation in U.S. customary units N for
the equation in S.I. unit); f
0
c
is concrete compressive strength (psi
for the equations in U.S. customary units MPa for the equations in
S.I. unit) to be verified using cylinders; f
0
cc
is concrete compressive
strength (MPa) to be verified using 200 mm cubes; h
ef
is embed-
ment depth (mm); d
o
is diameter of anchor (in. for the equations
in U.S. customary units mm for the equations in S.I. unit); lis load
bearing length of anchor (in. for the equations in U.S. customary
units mm for the equations in S.I. unit); and c
1
is anchor edge
distance (in. for the equations in U.S. customary units mm for the
equations in S.I. unit).
3. An overview of genetic programming (GP) and gene
expression programming (GEP)
A genetic algorithm (GA) is a search technique that has been
used in computing for finding precise or approximate solutions
to optimize or search problems. Genetic algorithms can be catego-
rized as global search heuristics. They are a particular class of evo-
lutionary computation. The techniques used by GA are inspired by
evolutionary biology such as; inheritance, mutation, selection,
crossover (recombination).
Genetic programming (GP), proposed by Koza [19] is essentially
an application of genetic algorithms to computer programs. GP has
been applied successfully to solve discrete, non-differentiable,
combinatory, and general nonlinear engineering optimization
problems [20]. It is an evolutionary algorithm based methodology
inspired by biological evolution to find computer that performs a
task defined by a user. Therefore, it is a machine learning technique
used to construct a population of computer programs according to
Fig. 1. Schematic presentation of typical post installed single adhesive anchor
under shear loading.
M. Gesog
˘lu et al. / Composites: Part B 60 (2014) 716–724 717
a fitness landscape determined by a program’s ability to perform a
given computational task. Similar to GA, the GP needs only the
problem to be defined. Then, the program searches for a solution in
a problem-independent manner [19,21].
Table 1
Training data base used for development of the prediction model.
Sample
no
Independent variables (X
i
) Dependent
variable (Y)
X
1
: Diameter
(mm)
X
2
: Type of
injection
*
X
3
: Chemical
type
**
X
4
: Anchor
type
***
X
5
: Embedment
depth (mm)
X
6
: Clear
clearance (mm)
X
7
:fc
(MPa)
X
8
: Edge
distance (mm)
Shear capacity
(kN)
1 12.70 1 1 1 114 1.04 23.52 114.30 42
2 12.70 1 1 1 114 1.04 23.52 114.30 35
3 15.88 1 1 1 144 1.98 23.52 133.35 67
4 15.88 1 1 1 144 1.98 23.52 133.35 76
5 19.05 1 1 1 171 2.10 23.52 171.45 86
6 19.05 1 1 1 171 2.10 23.52 171.45 94
7 22.23 1 1 1 199 2.06 23.52 200.03 139
8 22.23 1 1 1 199 2.06 23.52 200.03 122
9 25.40 1 1 1 226 2.15 23.65 228.60 189
10 12.70 1 1 1 115 1.04 40.89 114.30 49
11 15.88 1 1 1 144 1.98 40.89 142.88 79
12 15.88 1 1 1 144 1.98 40.89 142.88 71
13 19.05 1 1 1 173 2.07 40.89 171.45 112
14 22.23 1 1 1 199 2.06 40.89 200.03 149
15 22.23 1 1 1 199 2.06 40.89 200.03 138
16 9.53 1 1 1 86 1.08 23.52 38.10 6
17 9.53 1 1 1 86 1.08 23.52 38.10 6
18 15.88 1 1 1 86 1.98 23.65 63.50 16
19 15.88 1 1 1 86 1.98 23.65 63.50 15
20 9.53 1 1 1 86 0.79 13.48 85.73 19
21 12.70 1 1 1 114 0.79 13.48 85.73 36
22 12.70 1 1 1 117 0.79 13.48 85.73 42
23 15.88 1 1 1 117 1.59 13.48 142.88 60
24 15.88 1 1 1 117 1.59 13.48 142.88 57
25 19.05 1 1 1 169 1.59 13.48 171.45 64
26 19.05 1 1 1 168 1.59 13.48 171.45 65
27 19.05 1 1 1 175 1.59 13.48 171.45 72
28 19.05 1 1 1 173 1.59 13.48 171.45 72
29 22.23 1 1 1 201 4.76 13.48 200.03 103
30 25.40 1 1 1 225 1.59 13.28 228.60 107
31 25.40 1 1 1 230 1.59 13.28 228.60 114
32 12.70 0 1 0 113 1.59 31.57 107.95 54
33 15.88 0 1 0 133 1.59 31.57 127.00 69
34 15.88 0 1 0 135 1.59 31.57 127.00 57
35 19.05 0 1 0 178 1.59 31.57 155.58 87
36 19.05 0 1 0 168 1.59 31.57 155.58 90
37 22.23 0 1 0 164 3.18 31.57 168.28 120
38 22.23 0 1 0 165 3.18 31.57 168.28 100
39 25.40 0 1 0 210 3.18 31.57 203.20 145
40 25.40 0 1 0 208 3.18 31.57 203.20 151
41 12.70 0 1 0 103 1.59 13.13 107.95 38
42 12.70 0 1 0 114 1.59 13.13 107.95 38
43 15.88 0 1 0 133 1.59 13.13 127.00 44
44 15.88 0 1 0 127 1.59 13.13 127.00 49
45 19.05 0 1 0 162 1.59 13.13 171.45 78
46 19.05 0 1 0 175 1.59 13.13 171.45 65
47 22.23 0 1 0 165 3.18 13.34 177.80 75
48 22.23 0 1 0 175 3.18 13.34 177.80 87
49 25.40 0 1 0 203 3.18 13.34 203.20 126
50 25.40 0 1 0 187 3.18 13.34 203.20 103
51 8.00 0 0 1 80 1.00 15.00 40.00 9
52 8.00 0 0 1 80 1.00 43.00 40.00 12
53 10.00 0 0 1 90 1.00 43.00 45.00 17
54 12.00 0 0 1 110 1.00 15.00 55.00 19
55 16.00 0 0 1 125 1.00 16.00 62.50 24
56 16.00 0 0 1 125 1.00 16.00 125.00 53
57 20.00 0 0 1 170 2.50 16.00 85.00 44
58 20.00 0 0 1 170 2.50 16.00 170.00 70
59 24.00 0 0 1 210 2.00 16.00 262.50 115
60 24.00 0 0 1 210 2.00 28.00 105.00 62
61 12.00 0 0 1 110 1.00 14.00 55.00 45
62 12.00 0 0 1 110 1.00 14.00 165.00 68
63 12.00 0 0 1 110 1.00 36.00 110.00 106
64 12.00 0 0 1 110 1.00 36.00 137.50 106
*
1 for cartridge injection, 0 for glass capsule.
**
1 for epoxy and 0 for unsaturated polyester.
***
1 for steel rebar, 0 for threaded bars.
718 M. Gesog
˘lu et al. / Composites: Part B 60 (2014) 716–724
Gene expression programming (GEP) is a natural development
of genetic algorithms and genetic programming. GEP, introduced
by Ferreira [22], is a natural development of GP. GEP evolves com-
puter programs of different sizes and shapes encoded in linear
chromosomes of fixed-length. Its algorithm begins with the
random generation of the fixed-length chromosomes of each indi-
vidual for the initial population. Then, the chromosomes are
expressed and the fitness of each individual is evaluated based
on the quality of the solution it represents [23].
4. Construction of the prediction model
When loaded in shear, anchor’s adhesive layer bears on the
concrete. With enough force this will cause the edge of the con-
crete to break out [11].Fig. 1 shows a typical edge breakout failure
of a single adhesive anchor. The test setup generally consists of a
loading frame, loading plate, jack assembly, and load cell. The load
is applied to the anchor under force control in increasing percent-
ages of the estimated capacity (such as 5%, 10%, 20%, and so forth to
failure) with loading frames oriented parallel to the concrete
surface of each specimen. Load is continuously and very slowly
increased to avoid abrupt failure. The clear distance between the
supports is arranged to allow for unrestricted formation of a con-
crete breakout. Moreover, in some of these experimental studies
the displacement of the anchor in the direction of load at the level
of concrete top surface and axial strains in the anchor bolt are
observed as well as the failure load [10].
The models given in the design codes basically depend on the
compressive strength of the concrete and edge distance. Some
models also consider embedment depth and diameter of the
anchor bolt. However, clearance distance (see Fig. 1), type of the
anchor, type of adhesive and method of injection have not yet been
considered in the formulation of shear capacity of the anchor. For
this, anchor diameter, type of anchor (threaded bar or rebar), edge
distance, embedment depth, clear clearance of the anchor, type of
chemical adhesive (epoxy or unsaturated polyester), method of
injection of the chemical (glass capsule or cartridge injection),
and compressive strength of the concrete with the experimental
results of shear capacity of the anchors were arranged to obtain
a data set.
The adhesive anchors dealt with this study are steel anchors
either threaded rod or deformed bar inserted into a drilled hole
in normal strength concrete. A structural adhesive was utilized to
achieve proper bonding between the concrete and anchor. For
adhesive anchors, the diameter of the drilled hole is typically not
larger than 1.5 times the diameter of the steel element [1]. Adhe-
sive anchors are available in glass capsules or in injection systems
using organic or inorganic compounds. The adhesives utilized in
this study are epoxy and unsaturated polyester. For example,
test-1 in Table 1 has the following experimental parameters: steel
rebar of 12.70 mm diameter was post installed into a concrete
Table 2
Testing data base used for evaluating the performance of the prediction model.
Sample
no
Independent variables (X
i
) Dependent
variable (Y)
X
1
: Diameter
(mm)
X
2
: Type of
injection
*
X
3
: Chemical
type
**
X
4
: Anchor
type
***
X
5
: Embedment
depth (mm)
X
6
: Clear
clearance (mm)
X
7
:fc
(MPa)
X
8
: Edge
distance (mm)
Shear capacity
(kN)
1 9.53 1 1 1 85.73 1.08 23.52 85.73 26
2 12.70 1 1 1 114.30 1.04 23.52 114.30 41
3 15.88 1 1 1 143.94 1.98 23.52 133.35 77
4 19.05 1 1 1 171.45 2.10 23.52 171.45 91
5 25.40 1 1 1 226.49 2.15 23.65 228.60 170
6 25.40 1 1 1 226.49 2.15 23.65 228.60 149
7 15.88 1 1 1 143.94 1.98 40.89 142.88 78
8 19.05 1 1 1 172.52 2.07 40.89 171.45 105
9 9.53 1 1 1 85.73 1.08 23.52 38.10 8
10 15.88 1 1 1 85.73 1.98 23.65 63.50 16
11 12.70 1 1 1 117.37 0.79 13.48 85.73 41
12 15.88 1 1 1 114.33 1.59 13.48 142.88 59
13 19.05 1 1 1 170.92 1.59 13.48 171.45 83
14 22.23 1 1 1 206.45 4.76 13.48 200.03 93
15 22.23 1 1 1 200.05 4.76 13.48 200.03 92
16 25.40 1 1 1 226.70 1.59 13.28 228.60 116
17 15.88 0 1 0 133.35 1.59 31.57 127.00 71
18 19.05 0 1 0 168.28 1.59 31.57 155.58 96
19 22.23 0 1 0 168.28 3.18 31.57 168.28 128
20 25.40 0 1 0 206.38 3.18 31.57 203.20 126
21 12.70 0 1 0 106.38 1.59 13.13 107.95 41
22 15.88 0 1 0 142.88 1.59 13.13 127.00 48
23 19.05 0 1 0 155.58 1.59 13.13 171.45 78
24 22.23 0 1 0 177.80 3.18 13.34 177.80 83
25 25.40 0 1 0 177.80 3.18 13.34 203.20 98
26 10.00 0 0 1 90.00 1.00 15.00 45.00 14
27 10.00 0 0 1 90.00 1.00 15.00 67.50 19
28 12.00 0 0 1 110.00 1.00 43.00 55.00 27
29 16.00 0 0 1 125.00 1.00 36.00 62.50 37
30 20.00 0 0 1 170.00 2.50 36.00 85.00 63
31 24.00 0 0 1 210.00 2.00 16.00 105.00 51
32 12.00 0 0 1 110.00 1.00 14.00 110.00 57
33 12.00 0 0 1 110.00 1.00 38.00 55.00 60
34 12.00 0 0 1 110.00 1.00 38.00 110.00 91
*
1 for cartridge injection, 0 for glass capsule.
**
1 for epoxy and 0 for unsaturated polyester.
***
1 for steel rebar, 0 for threaded bars.
M. Gesog
˘lu et al. / Composites: Part B 60 (2014) 716–724 719
having a compressive strength of 23.52 MPa using cartridge
injection system in which epoxy being the bonding agent.
The derivation of a genetic-programming-based explicit formu-
lation is accomplished by means of the training data containing
input and output variables. Besides, in order to examine and test
the performance of the developed model, a supplementary data
set containing the same number and sequence of input and output
variables is used. Therefore, in the current study, the ensemble of
available experimental data was arbitrarily divided into two parts
to obtain the training and testing databases. Approximately 2/3 of
the total data samples were used as the training set while the rest
was employed as the testing set, as shown in Tables 1 and 2,
respectively. Thus, a total of 64 and 34 samples from training
and testing sets, respectively were utilized in constructing the
model.
A software, named GeneXproTools.4.0 was employed for
derivation the mathematical model presented in Eq. (9). The mod-
els developed by GEP in its native language can be automatically
parsed into visually appealing expression trees, permitting a quick-
er and more complete comprehension of their mathematical/logi-
cal intricacies. Fig. 2 demonstrates the expression tree for the
terms used in the formulation of the GEP model which has the
Fig. 2. Expression tree for the GEP model [d
0
: Anchor diameter (mm), d
1
: Injection type (1 for cartridge injection, 0 for glass capsule), d
2
: Chemical type (1 for epoxy and 0 for
unsaturated polyester), d
3
: Type of anchor (1 for steel rebar, 0 for threaded bars), d
4
: Embedment depth (mm), d
5
: Clear clearance (mm), d
6
: Concrete compressive strength
(MPa), d
7
: Edge distance (mm) c
0
and c
1
: constants (c
0
=7.08139 for Sub-ET3, c
0
= 5.801483 for Sub-ET4, c
0
= 5.138702 for Sub-ET5, c
0
=6.909272 for Sub-ET6,
c
1
=6.227142 for Sub-ET1, c
1
= 3.883179 for Sub-ET3, c
1
= 8.091003 for Sub-ET5)].
720 M. Gesog
˘lu et al. / Composites: Part B 60 (2014) 716–724
parameters given in Table 3. As seen in Table 3 that various math-
ematical operations were included to provide a reliable model.
V
U
¼V
1
V
2
V
3
V
4
V
5
V
6
ð9Þ
V
1
¼ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6:227142 þX
8
X
3
X
6
þtanðX
4
1
Þ
3
q

2
ð9aÞ
V
2
¼ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
2
7
þX
1
qÞ
sin
2
ðX
7
Þ
þX
1
3
s
2
6
43
7
5ð9bÞ
Table 3
GEP parameters used for the proposed model.
P1 Function set +, ,
,/,p, ^, ln, exp, sin, tan
P2 Number of generation 940,695
P3 Chromosomes 40
P4 Head size 10
P5 Linking function Multiplication
P6 Number of genes 6
P7 Mutation rate 0.044
P8 Inversion rate 0.1
P9 One-point recombination rate 0.3
P10 Two-point recombination rate 0.3
P11 Gene recombination rate 0.1
P12 Gene transposition rate 0.1
Fig. 3. Performance of the proposed GEP model: (a) train set and (b) test set.
Fig. 4. Predicted shear capacity values from ACI 349-97.
Fig. 5. Predicted shear capacity values from ACI 349-06.
Fig. 6. Predicted shear capacity values from CCD method.
Fig. 7. Predicted shear capacity values from modified CCD method.
M. Gesog
˘lu et al. / Composites: Part B 60 (2014) 716–724 721
V
3
¼ln½X
7
sin½ðX
5
þ3:883179Þ
3
þtanð7:08139 X
5
Þ ð9cÞ
V
4
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
7
þX
6
½tanðsin X
7
ÞX
4
þ5:801483
6
qð9dÞ
V
5
¼ln½lnðX
1
tanðX
8
þ8:091003 X
4
Þtanð5:138702
X
1
ÞÞ ð9eÞ
V
6
¼½lnðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
tanðln X
8
Þþð6:909272Þ
X
2
þX
8
þX
4
3
qÞ
3
ð9fÞ
where V
U
is the ultimate shear capacity of adhesive anchor in un-
cracked concrete (kN); X
1
: Anchor diameter (mm); X
2
: Injection
type (1 for cartridge injection, 0 for glass capsule); X
3
: Chemical
type (1 for epoxy and 0 for unsaturated polyester); X
4
: Type of an-
chor (1 for steel rebar, 0 for threaded bars); X
5
: Embedment depth
(mm); X
6
: Clear clearance (mm), X
7
: Concrete compressive strength
(MPa), and X
8
: Edge distance (mm).
5. Performance of the prediction model
Performance of the proposed GEP prediction model in Eq. (9)
was depicted in Fig. 3 for both train and testing data sets. More-
over, the correlations between experimental and predicted shear
capacities for the existing models were also given in Figs. 4–8 for
the entire data. Fig. 3 revealed that a high estimation accuracy
was accomplished for both training and testing data sets. The cor-
relation coefficient of training set was 0.98 while that of testing
was 0.92. It is seen in Fig. 6 that despite having lower R
2
value,
CCD method presented similar trend as GEP model. As a result of
uniform scatter of the data, the correlation coefficients calculated
for the other models also appeared to be very close to each other.
However, the shear capacities computed from ACI 349-97, ACI 349-
06, and PCI models underpredicted the actual values while
Modified CCD method provided overprediction. Some statistical
parameters were also given in Table 4 for comparing the tendency
of the distribution of the predicted values. The proposed GEP
model had the lowest errors such that MAPE (mean absolute
percentage error) was about 10 and 14% for the train and test sets,
respectively. However, when the existing models in the literature
were considered, MAPE ranged from 19% to 66%, depending on
the prediction capability of the model. Therefore, this absolute
error of the proposed GEP model seemed to be fairly reasonable
when the noisy nature of the experimental results of adhesive an-
chors was taken into account [24,25]. Of all the existing formulas,
CCD method appeared to be the most reliable one attributed to its
relatively lower prediction error.
Figs. 9–12 indicate the variations of the normalized shear
capacity found by dividing the predicted over experimental values,
versus compressive strength of the concrete, edge distance, diam-
eter of anchor and embedment depth of the anchor, respectively.
Since compressive strength of the concrete and edge distance are
the fundamental factors as being available in all of the prediction
models, Figs. 9 and 10 contained all of the prediction models dealt
with this study. Nevertheless, in Figs. 11 and 12, ACI 349-97 and
Fig. 8. Predicted shear capacity values from PCI method.
Table 4
Statistical parameters of the proposed model as well as existing ones.
Parameters GEP model CCD model Modified CCD model ACI 349-97 ACI 349-06 PCI model
Training data set Testing data set
Mean Square Error (MSE) 36.9 168.7 261.3 3858.9 556.7 1051.5 1327.6
Mean absolute percent error (MAPE) 10.0 14.2 18.9 66.0 33.0 42.2 41.5
Root Mean Square Error (RMSE) 6.1 13.0 16.2 62.1 23.6 32.4 36.4
Correlation coefficient (R
2
) 0.98 0.92 0.88 0.89 0.87 0.88 0.89
Fig. 9. Prediction performance of the GEP, CCD, modified CCD, ACI 349-97, ACI 349-
06, and PCI models for different concrete compressive strengths.
Fig. 10. Prediction performance of the GEP, CCD, modified CCD, ACI 349-97, ACI
349-06, and PCI models for different edge distances.
722 M. Gesog
˘lu et al. / Composites: Part B 60 (2014) 716–724
PCI formulas are excluded because they do not include the diame-
ter and embedment depth of the anchor (Eqs. (2) and (8)). It can be
seen from the figures that CCD method and proposed GEP model
revealed a very close trend in terms of prediction performance.
For example, considering the overall 98 normalized values, 60
points for GEP model and 38 points for CCD model fell between
±10% limits while 10, 5, 3 and only 1, point were observed for
ACI 349-97, modified CCD, PCI, and ACI 349-06 models, respec-
tively. Modified CCD model gave the highest normalized values
for all of the factors considered. The range of the normalized values
for modified CCD model was observed to be 0.50-2.67. However,
the range for the proposed GEP model was 0.45-1.69. The lowest
upper limit for the normalized values was observed for both ACI
349-06 and PCI models as 0.91. As seen in Figs. 9 and 10, these
two models exhibited similar trend in underpredicting the shear
capacities for a given compressive strength and edge distance.
Fig. 9 showed that the normalized values tend to approach to
almost 1 for the compressive strengths of 24–32 MPa whereas
beyond that a divergence was obtained. Tendency of clustering
the data was observed for the highest edge distance in Fig. 10,
for the highest diameter in Fig. 11, and for the highest embedment
depth in Fig. 12. In the study of Gesog
˘lu and Güneyisi [25], predic-
tion models were developed to estimate the pullout capacity of
adhesive anchors through soft computing methods. They also re-
ported that the prediction capability of the proposed models and
the CCD method were increased for deeper embedment depth
and larger diameter anchors. It can be seen from Figs. 9–12 that
the normalized shear capacity results tend to be divergent as the
compressive strength of concrete, the diameter of anchor, the
embedment length and the edge distance decrease. In the study
of Bickel and Shaik [11] it was stated that the shear capacity exhib-
its lesser increase in strength with concrete compressive strength.
Moreover, the predicted shear capacity values were scattered for
lower embedment depth and edge distances. This situation may
be attributed to the number of experimental data used for deriva-
tion of the models. The higher the number of data, the more robust
and repeatable the derived model. For example, in the proposed
GEP model, few number of data for the anchors having bar diame-
ter less than 12 mm generated both underestimation and overesti-
mation performance. However, due to the fact that most of the
anchors have bar diameter of higher than 12 mm, the predicted
results for such anchors fell between ±10% of the actual results.
6. Conclusions
Based on the analyses presented above, the following conclu-
sions may be drawn:
Breakout shear capacity prediction model was developed by
genetic programming considering the chemical characteristics
and method of the placing of the adhesive anchors. The model
provided reasonable predicted values with significantly high
accuracy. The empirical formulation was generated through
gene expression programming (GEP) with correlation coeffi-
cient of 0.98.
Although the database for testing were not utilized for training,
a high level of estimation was obtained for both training and
testing data sets associated with low mean absolute percentage
of error and high coefficients of correlation. This indicates the
generalization capability of the developed model.
The proposed model was compared with the existing formulas
available in ACI 349-97, ACI 349-06, ACI 318-08 (CCD method),
PCI-98 design handbook as well as the model proposed by Hoff-
man [12], namely, modified CCD method. The statistical analy-
sis revealed that the proposed GEP model had relatively lower
errors than the others. The closest prediction tendency to the
GEP model was demonstrated by CCD method.
Normalization of the predicted values was performed to evalu-
ate the performance of the existing and proposed prediction
models. It was observed that ACI 349-97, ACI 349-06, PCI
method, and CCD method underpredicted while modified CCD
method overpredicted the shear capacity. The values obtained
from PCI model, ACI 349-97 and ACI 349-06 models appeared
to be close to each other. However, the values obtained from
GEP model were observed to be more uniform and much closer
to the actual results.
Acknowledgement
The authors would like to thank to Professor Ashour for provid-
ing the database of adhesive anchors. Professor Cook maintains
this database on behalf of the ACI Committee 355.
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Fig. 12. Prediction performance of the GEP, CCD, modified CCD, and ACI 349-06
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724 M. Gesog
˘lu et al. / Composites: Part B 60 (2014) 716–724
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A user-friendly, highly transparent model for the design of post-installed steel anchors or cast-in-place headed studs or bolts, termed the concrete capacity design (CCD) approach, is presented. This approach is compared to the well-known provisions of ACI 349-85. The use of both methods to predict the concrete failure load of fastenings in uncracked concrete under monotonic loading for important applications is compared. Variables included single anchors away from and close to the edge, anchor groups, tension loading, and shear loading. A data bank including approximately 1200 European and American tests was evaluated. The comparison shows that the CCD method can accurately predict the concrete failure load of fastenings for the full range of investigated applications. On the other hand, depending on the application in question, the predictions of ACI 349 are sometimes unconservative and sometimes conservative. The CCD method is more user-friendly for design. Based on this, the CCD method is recommended as the basis for the design of fastenings.
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In this study, single cast-in-place anchors and retrofit anchors (expansion, undercut, and adhesive) were exposed to five simulated environmental exposure conditions: 1) ultraviolet light; 2) freezing and thawing; 3) corrosion in a pH-neutral salt solution; 4) wetting and drying with simulated acid rain; and 5) combined freezing and thawing, corrosion in a pH-neutral salt solution, and wetting and drying. To evaluate the effects of different environmental exposures, the tensile load-deflection behavior of the exposed anchors was compared with that of otherwise identical unexposed anchors. Ultraviolet light exposure did not affect the anchors studied. Freeze-thaw exposure reduced the preload and decreased the initial stiffness of torque-controlled expansion anchors. Salt (corrosion) exposure did not adversely affect the behavior of expansion anchors. Wetting and drying exposure to simulated acid rain did not significantly affect the anchors studied. The combination exposure reduced the stiffness of some torque-controlled expansion anchors. These results show that when expansion anchors are used in concrete subjected to cycles of freezing and thawing, their preload should be checked regularly. The results also imply that in concrete subject to cycles of freezing and thawing while wet, water should be prevented from entering the drilled holes of mechanical anchors.