Engineering Structures 93 (2015) 129–141
Contents lists available at ScienceDirect
Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
Mechanical behavior of electrical hollow composite post
insulators: Experimental and analytical study
Siamak Epackachi a,⇑, Kiarash M. Dolatshahi b, Nicholas D. Oliveto a, Andrei M. Reinhorn a
a
b
Department of Civil, Structural and Environmental Engineering, University at Buffalo, Buffalo, NY 14260, USA
Department of Civil Engineering, Sharif University of Technology, Tehran, Iran
a r t i c l e
i n f o
Article history:
Received 14 October 2014
Revised 3 March 2015
Accepted 4 March 2015
Keywords:
Electrical post insulators
Hollow-core composite insulator
Cyclic test
Experimental test
Analytical model
Dynamic behavior
a b s t r a c t
Electrical post insulators are important components of electrical substations since any type of failure in
such insulators leads to the breakdown of the local network. Although the electrical substations are often
in service condition, any horizontal excitation due to the earthquake, or any extreme event, may cause lateral deformation and damage to the post insulators. Hollow composite post insulators, a new and evolving
technology, have a very complex mechanical behavior due to their materials and connections. To date, the
design of such post insulators has been based on the limited test results available in the literature. Most of
experiments have been conducted on small-scale specimens focusing on the elastic response. This study
presents a series of experiments conducted on a full-scale electrical hollow composite post insulator to
investigate the static and dynamic mechanical behaviors, while a computational model is derived. The test
series comprise, pull and cyclic quasi-static tests in addition to impact hammer tests, to assess the
mechanical behavior of the insulators subjected to the lateral forces at different stages of damage. The
key experimental results include the pre-peak force–displacement relationship, the cyclic response, the
stiffness and strength deteriorations, and failure modes. The modal frequencies and the corresponding viscous damping ratios for the undamaged and damaged post insulator are calculated using the results of
impact hammer tests. An analytical model is derived from the mechanical behavior to simulate the
response of the un-damaged and damaged post insulator, and is verified by the test results.
Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction and background
Electric power supply is recognized as one of the most important
services after an earthquake. A survey of 200 hospital employee
including doctors and administrative personnel revealed that the
power supply has the first priority after an extreme event [1].
Despite this fact, observations from past earthquakes show that
electrical systems are among the least reliable services [1–4]. For
example, after the Kocaeli (Turkey) earthquake in 1999 [1], Kobe
(Japan) earthquake in 1995 [2–4], and Northridge (USA) earthquake
in 1994 [2] most of the hospitals were not fed for few days. The
damage to electrical equipment by Loma Prieta and Northridge
earthquakes resulted in more than $200 million worth of losses
[5]. In the last twenty years, many studies have tried to raise reliability of the electrical power systems either in urban level [6,7] or
by studying the seismic behavior of its components [8–12].
⇑ Corresponding author. Tel.: +1 7168660584.
E-mail addresses: siamakep@buffalo.edu (S. Epackachi), dolatshahi@sharif.edu
(K.M. Dolatshahi), noliveto@buffalo.edu (N.D. Oliveto), reinhorn@buffalo.edu
(A.M. Reinhorn).
http://dx.doi.org/10.1016/j.engstruct.2015.03.013
0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.
An electric substation consists of several parts such as, electrical
transformers, bushings, and insulators. The voltage level continuously varies during the generation, transmission and distribution
of electric energy. The electrical transformer is used to pass from
medium voltage of the generation to the high voltage of transmission and back to the low voltage for distribution. Insulators are
used to separate electrically metal parts with different voltage
levels to avoid short-circuits and breakdown in the network.
Although insulators have been used for decades, the introduction of composite hollow-core insulators is relatively new and the
technology is evolving [13–18]. Like other structural systems,
earthquake-excitation is one type of loading that can cause severe
damages to the insulators. Under seismic loads, insulators can be
subjected to substantial lateral (‘‘cantilever’’) loads, simultaneously with complex axial compression and tension loads.
Reinhorn et al. [19] tested hollow-core insulators to investigate
the behavior of tube-flange connection and its failure modes.
They conducted two types of tests: (1) pull tests using a series of
loads with increasing magnitude, and (2) snap-back test performed
after each pull load test. The cyclic response of insulators was not
addressed in their study, which reported three major types of
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S. Epackachi et al. / Engineering Structures 93 (2015) 129–141
failures, namely cracking of the lower flange, failure of the bond
between the flange and the tube, and the failure of the tube adjacent to the bonding material in the flange [20]. On the basis of
these test results, it was shown that the Specified Mechanical
Load (SML), which is the lateral load capacity corresponding to
the flexural strength of the insulator and was assumed as 2.5 times
the maximum mechanical load specified by the manufacturer, was
much lower than the cantilever failure load measured from the
tests. Roh et al. [21] and Cimellaro et al. [22] developed analytical
models aimed to predict the response of insulator at different
stages of the damage, using linear and nonlinear springs, viscous
and frictional dampers, and inertial mass.
Due to the lack of information associated to the mechanical
behavior of the column insulators in sustained dynamic motion,
this paper addresses the static and dynamic characteristics of the
column insulators through an organized test series. Results
obtained from the pull, cyclic, and impact hammer tests are presented. An elaborated analytical model is proposed to simulate
the structural response of the column insulator at different stages
of the damage. The performance of the developed analytical model
is compared to test data.
2. Experimental program
A full-scale electrical hollow composite post insulator was
tested under a set of force-controlled (pull) loading and a
displacement-controlled cyclic loading at the Structural
Engineering and Earthquake Simulation Laboratory (SEESL) at
University at Buffalo. The following sub-sections of the paper
describe the testing program and present key experimental results.
2.1. Test specimen description
The specimen consisted of a 6-mm thick tube of fiber glass reinforced polymer and metal caps at both ends of the tube (see Fig. 1).
The tube was connected to the metal caps using a bonding material
between the tube and metal caps. The height of the specimen was
1530 mm with the exterior and interior diameters of 210 mm and
198 mm, respectively. The mass of the tube and each metal cap
was 35 kg and 6.8 kg, respectively.
Fig. 2 presents the details of the connection between the specimen and test frame. The bottom flange of the specimen was connected to a 38-mm thick steel plate using 16 number equally
spaced M12 bolts. Two bottom adopter plates (see Fig. 2(c) and
(e)) were secured to the steel beam support using 4 number M
25 headed bolts. The top tube flange was connected to the top
adopter plate using 12 number M12 bolts. A 26-mm diameter
threaded stud attached to the center of the top adopter plate was
used to connect the actuator to the top of the post insulator. The
locations of the adopter steel plates at the top and bottom connections are presented in Fig. 3. It should be noted that the post insulator was attached to a rigid base using typical connections used in
( a) S peci men
(b) Elevation view of the specimen
(c) Top and bottom flanges
Fig. 1. Test specimen.
S. Epackachi et al. / Engineering Structures 93 (2015) 129–141
(a) Bottom connection
(b) Plan view of the metal flange
(c) Bottom adopter plate#1
(d) Top adopter plate
(e) Bottom adopter plate#2
Fig. 2. Details of the connection between the specimen and test frame.
Fig. 3. Test setup.
131
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S. Epackachi et al. / Engineering Structures 93 (2015) 129–141
practice. The choice of a rigid connection rather than a flexible connection as found in practice was made to allow to study the post
insulator stiffness and strength without being influenced by the
base properties which vary in situ.
2.2. Test setup and instrumentation
The schematic drawing of the test frame is shown in Fig. 3. A
horizontal actuator was used to apply the quasi-static lateral load
at the top of the specimen. Hinges at both ends of the actuator
allow rotation about a horizontal axis perpendicular to the actuator to accommodate any rotation at the top of the insulator during
the loading.
Strain gages, accelerometers, linear potentiometers, linear variable displacement transducers (LVDT) were used to monitor the
response of the insulator during the loadings. Strain gages were
attached to the inside and outside of the tube to directly measure
vertical and horizontal (hoop) strains at discrete locations of the
bottom of the post insulator (see Fig. 4). Three accelerometers were
installed at the top of the specimen to measure two orthogonal
horizontal and one vertical accelerations. The lateral displacement
at top of the specimen was measured using the displacement
transducer (LVDT) of the actuator. The lateral displacement profile
was measured using three string potentiometers attached to the
specimen at different levels along the height. The movement of
the tube relative to the bottom metal cap was monitored using four
equally spaced linear potentiometers. Locations of the strain gages
and linear potentiometers on specimen are presented in Fig. 4.
2.3. Loading protocol
Table 1 presents the loading (sequence) protocol designed to
investigate the static and dynamic behavior of the composite post
insulator and to identify the extent of damage to the tube-flange
connection under monotonic and cyclic loadings. As shown in
Table 1, the testing protocol consists of three types of tests, namely
(1) a pull test comprising four load steps of 15%, 40%, 60%, and
100% of the manufacturer-provided Specified Mechanical Load
(SML, see Table 1) and a subsequent unloading per load step, (2)
a cyclic test comprising 11 load steps with two cycles per load step,
and (3) a series of impact hammer tests conducted using a rubber
mallet to produce free vibrations.
The impact hammer tests were conducted before and after each
pull test and after each cyclic test to correlate the extent of damage
with the changes in the damping and frequency of the test specimen. Since the maximum lateral load in pull tests was limited to
SML, the tests were conducted in force-control. However, the cyclic
test of the damaged specimen was conducted in displacementcontrol to avoid any significant displacement of the specimen after
the failure. Note that, Dmax in Table 1, is the maximum displacement corresponding to SML measured from pull (PU100) test.
2.4. Test results
2.4.1. Pull test
A pull test consisting of four increasing amplitude loading–
unloading cycles were conducted (see Table 1) and the measured
(a) Schematic drawings of the vertical section (left) and elevation view (right)
(b) Photographs of the inner (left) and outer (right) views of the specimen
Fig. 4. Instrumentation of the post insulator.
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S. Epackachi et al. / Engineering Structures 93 (2015) 129–141
Table 2
Pull test results.
Test type
Load step
⁄⁄
Peak force or
displacement
per cycle
Impact hammer test
Pull test
IHX1
PU015
PU040
PU060
PU100
–
LS1
LS2
LS3
LS4
–
–
–
–
–
–
15%SML⁄
40% SML
60% SML
100% SML
Impact hammer test
IHX2
–
–
–
Cyclic test
CY040
CY060
CY100
CY130
CY140
CY150
CY170
CY200
CY250
CY300
CY350
LS1
LS2
LS3
LS4
LS5
LS6
LS7
LS8
LS9
LS10
LS11
2
2
2
2
2
2
2
2
2
2
2
40% Dmax ⁄⁄
60% Dmax
100% Dmax
130% Dmax
140% Dmax
150% Dmax
170% Dmax
200% Dmax
250% Dmax
300% Dmax
350% Dmax
IHX3
–
–
–
Impact hammer test
⁄
Number
of cycles
Test no.
Max pull load (kN)
Max displacement (mm)
PU015
PU040
PU060
PU100
2.0
5.3
8.0
13.0
2.8
9.7
15.7
30.0
Displacement ratio [%]
-400
15
-200
-100
0
100
200
300
400
Bond failure
9
6
3
0
-3
-6
-9
-12
SML = 13 kN.
Dmax ¼ 30 mm.
-300
12
Lateral load [kN]
Table 1
Loading protocol.
-15
-120
Bond failure
-90
-60
Hysteretic response
Backbone curve
-30
0
30
60
90
120
Lateral displacement [mm]
Fig. 6. Cyclic force–displacement relationship.
Drift ratio [%]
0
0.5
1
1.5
2
2.5
15
Lateral load [kN]
12
9
6
PU015
PU040
PU060
PU100
3
0
0
5
10
15
20
25
30
35
40
Lateral displacement [mm]
Fig. 5. Force–displacement relationship of pull load tests.
load–displacement curve is presented in Fig. 51. No stiffness degradation was observed for loading up to SML. However, as the pull load
increased, the energy dissipation represented as the area enclosed by
the loading–unloading path raised showing an increase in damping.
Pull test results are presented in Table 2. The maximum displacement corresponding to SML is 30 mm which is equivalent to
a drift ratio of 1.9%. The maximum displacement corresponding
to SML, measured from the pull test, is considered as the reference
displacement in the cyclic test which is conducted in displacement-control.
2.4.2. Cyclic tests
Cyclic tests with 11 load steps were conducted after the pull
test, to investigate the hysteretic response of the post insulator.
Fig. 6 presents the cyclic force–displacement relationship (solid
1
For interpretation of color in Fig. 5, the reader is referred to the web version of
this article.
black line) and the backbone curve (dashed blue line).
Displacement ratio, in Fig. 6, is defined as the ratio of the lateral
displacement applied at the top of the specimen to the displacement corresponding to SML.
As seen in Fig. 6, the specimen exhibited almost linear elastic
behavior up to SML. However, the response changed significantly
after the specimen reached SML. A significant pinched curve and
loss of stiffness and strength, occurred at displacements greater
than that corresponding to SML, are attributed to the bond-slip
failure at the flange-tube connection shown in Fig. 6. At a displacement corresponding to SML, a loud popping sound was heard
indicative to the breakdown of the connection and loss of bond
between the flange and tube.
The intra-cycle stiffness and strength reductions in the
post-peak-strength region were not substantial. However, in
the post-peak response, the reloading stiffness deteriorated as the
displacement increased whereas the unloading stiffness remained
unchanged, almost identical to the initial elastic stiffness of the
specimen. Fig. 6 indicates that the post-peak strength has not been
deteriorated up to a displacement ratio of 350%. The energy dissipation capacity in the pre-peak-strength response is insignificant.
However, in the post-peak-strength response, the large areas
enclosed by the hysteresis loops are indicative of the significant
energy dissipation and equivalent (viscous) damping due the
friction-slip mechanism of the damaged connection.
Fig. 7 presents the secant stiffness calculated using the maximum displacement of the first cycle in each load step and the
corresponding force. The secant stiffness deteriorates as the displacement increases. The secant stiffness suddenly drops at the
displacement corresponding to SML, where the bond between
the composite tube and metal cap fails.
Fig. 8 presents the vertical strain distribution at the displacements corresponding to the first and fourth steps of the pull test
and the first, third, and eleventh load steps of the cyclic test. The
vertical strains were measured using the seven strain gages (strain
gages# 1–7 in Fig. 4(a)) attached to the east and west sides of the
inner face of the post insulator. The first four strain gages measured the vertical strains of the tube inside of the flange and the
strain gages 5–7 measured the vertical strains of the tube outside
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S. Epackachi et al. / Engineering Structures 93 (2015) 129–141
Displacement ratio [%]
Displacement ratio [%]
350
Secant stiffness [kN/mm]
400
0.6
300
250
200
150
100
0
50
50
100
150
Loading towards the East direction
200
250
300
350
400
Loading towards the West direction
0.45
0.3
0.15
0
-120
-90
-60
0
-30
30
60
120
90
Lateral displacement [mm]
Lateral displacement [mm]
Fig. 7. Variation of the secant stiffness during cyclic loading.
7
Strain gage No.
6
East side of
the inner face
West side of
the inner face
5
4
3
2
1
7000
6000
5000
4000
3000
2000
1000
Micro strain
0
1000
2000
3000
4000
5000
6000
7000
Micro strain
Fig. 8. Vertical strain profiles at inner side of the post.
of the flange. Highly nonlinear distribution of the vertical strain at
displacements greater than that corresponding to SML is attributed
to the damage to the bottom of the post insulator.
2.4.3. Impact hammer test
During the impact hammer tests, the actuator was disconnected
from the post insulator and the top of the specimen was hit by the
hammer with a soft rubber tip. The first impact hammer test was
preformed to identify the natural frequencies and the corresponding damping ratios for the intact post insulator. The second and
third impact hammer tests were conducted after the pull and
cycling tests, respectively, to identify any change in the fundamental frequencies and the damping ratios due to the damage occurred
in the specimen.
The histories of the acceleration measured in the in-plane direction (East to West direction, see Fig. 3) at the top of the post insulator and the normalized transfer functions are presented in Fig. 9.
The impact hammer test results, shown in Fig. 9, are also summarized in Table 3. Fig. 9(a) indicates that the main frequency of the
post is 17.8 Hz. The equivalent viscous damping ratio corresponding to the main frequency is 0.84%, where the equivalent viscous
damping ratio was calculated using the logarithmic decrement
method (Chopra [23]) representing the natural log of the ratio of
two adjacent peak displacements in free vibration, or the nth fraction of the logarithmic rate of n cycles repeated displacement
amplitudes.
After the pull tests, the extent of the damage to the specimen
was measured by the changes in the main frequency and damping
ratio calculated from the second impact hammer test. Fig. 9(b)
shows two vibration modes after the specimen reached the peak
load in the pull test. The first and second vibration mode frequencies are 17.3 Hz and 124.7 Hz, respectively, with the corresponding
equivalent viscous damping ratio of 1.38% and 1.58%. The amplitude of the main frequency has decreased 4% indicating an increase
in the equivalent viscous damping. The equivalent critical viscous
damping ratio of the first mode, of 1.38%, is 65% more than that calculated using the impact hammer after the first test. The higher
rate of deterioration in the acceleration response (i.e. 3% reduction
in the main frequency) with an increase in the damping ratio, and
the appearance of the second mode with a low amplitude and a
high frequency are all attributed to the damage to the specimen
after the pull tests.
Fig. 9(c) presents the time history of the measured acceleration
and the transfer function of the third impact hammer test performed after the cyclic test. Two major modes with the frequencies
of 16.9 Hz and 123.6 Hz, respectively, can be seen after the failure
of the specimen. It is interesting to note that the second mode has a
higher amplitude than the first mode and is developed due to the
rocking of the cylindrical tube inside the cap (see Fig. 10). The
damping ratio of the first and second mode equals to 2.08% and
15.11%, respectively, are associated to the viscous and frictional
mechanisms, respectively. The changes in the frequency and
damping ratio of the damaged specimen after the pull and the cyclic tests indicate that the monitoring of the vibration mode frequencies can be successfully used to identify damage states
during testing.
2.4.4. Damage to the post insulator
Fig. 10 provides photographs of damage to the post insulator at
the end of the cyclic test. As Fig. 10 presents, the damage to the
post insulator concentrated near the base of the tube within the
zone of the connection between the composite tube and the bottom metal cap. Damage to the post insulator was initiated by a
bond failure at tube-flange connection (see Fig. 10(a)) and followed
135
6
1
4
0.8
Normalized amplitude
Acceleration [g]
S. Epackachi et al. / Engineering Structures 93 (2015) 129–141
2
0
-2
-4
0.6
0.4
17.8 Hz
0.2
0
-6
0
0.5
1
1.5
2
2.5
3
3.5
4
0
10 20 30 40 50 60 70 80 90 100 110 120 130
Time [sec.]
Frequency [Hz]
6
1
4
0.8
Normalized amplitude
Acceleration [g]
(a) First impact hammer test
2
0
-2
-4
0.6
0.4
17.3 Hz
0.2
124.7 Hz
0
-6
0
0.5
1
1.5
2
2.5
3
3.5
4
0
10 20 30 40 50 60 70 80 90 100 110 120 130
Time [sec.]
Frequency [Hz]
6
1
4
0.8
Normalized amplitude
Acceleration [g]
(b) Second impact hammer test
2
0
-2
-4
0.6
0.4
16.9 Hz
0.2
123.6 Hz
0
-6
0
0.5
1
1.5
2
2.5
3
3.5
4
0
10 20 30 40 50 60 70 80 90 100 110 120 130
Time [sec.]
Frequency [Hz]
(c) Third impact hammer test
Fig. 9. Impact hammer test results; acceleration time history (left) and Fourier amplitude (right).
Table 3
Summary of impact hammer test results.
Test no.
IHX1
IHX2
IHX3
Equivalent viscous damping
ratio (n) (%)
Frequency (Hz)
First mode
Second mode
First mode
Second mode
0.84
1.38
2.08
–
1.58
15.11
17.8
17.3
16.9
–
124.7
123.6
by buckling and crushing of the tube inside of the flange (see
Fig. 10(b)). No cracking or crushing was observed in the tube outside of the connection region.
3. Analytical study
An analytical model is developed for the dynamic behavior of
the electrical composite insulators. As shown in Fig. 11, the insulator is modeled as a cantilever beam with distributed mass (mðzÞÞ,
elasticity (EIðzÞÞ, and a lumped mass (M 0 Þ at the top, subjected to
arbitrary external dynamic forces pðz; tÞ. A frictional spring is introduced at the base to account for energy dissipation due to sliding of
the tube inside the flange and an additional elastic spring is used to
account for stiffness reductions due to damage of the base flange
itself. The stiffness of the frictional spring is indicated by k0 and
its associated friction rotation is denoted as q0 . The stiffness of
the additional elastic spring is ke and its associated elastic rotation
is denoted as qe .
Based on the test results, confirming that all the nonlinearities
in these system are concentrated at the base, two different base
moment-rotation constitutive models are used for the slip-friction
spring, depending on the state of damage of the insulator. A symmetric bilinear model shown in Fig. 12(a) is used to describe the
behavior of the system prior to breaking of the base flange. The
flexural-base yielding moment, M y , represents the bending
moment at the onset of sliding of the tube along the walls of the
flange, due to breakage of the bonding glue. An asymmetric model
shown in Fig. 12(b) describes the behavior of the device once the
base flange has been irreversibly damaged.
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S. Epackachi et al. / Engineering Structures 93 (2015) 129–141
(a) Bond failure
(b) Crushing and buckling of the tube
(c) Slipping of the tube
Fig. 10. Damage to the post insulator at the end of the cyclic test.
(a) Insulator
(b) Kinematics
Fig. 11. Proposed analytical model.
Fig. 13. Cantilever beam with flexible support.
M
My
M
My
k0
rotation increases or decreases. In the following sections, the equations of motion are derived separately for the two phases.
k0
q0
q0
-M y
(a) Symmetric model
3.1. Plastic or sliding response
When the system is in a plastic or sliding phase, the displacement function can be expressed as follows:
(b) Asymmetric model
Fig. 12. Proposed constitutive models.
When triggered into motion, the system alternates phases of
elastic behavior in which the plastic rotation at the base remains
constant, and plastic or sliding phases in which the plastic base
uðz; tÞ ¼ q0 ðtÞz þ
1
X
/i ðzÞqi ðt Þ
ð1Þ
i¼1
where /i ðzÞ are the modes of vibration of the system, as shown in
Fig. 13, and qi ðtÞ are the corresponding modal coordinates. For the
special case of a uniform cantilever beam, mðzÞ ¼ m and EIðzÞ ¼ EI,
a lumped mass M0 at the top and a rotational spring ke at the bottom, the frequencies of vibration can be obtained by solving the
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S. Epackachi et al. / Engineering Structures 93 (2015) 129–141
following frequency equation, derived by analytical methods that
are well established in the literature (e.g., Chopra [23]):
If N modes of vibration are included in the analysis, Eq. (7) may
be written as:
að1 þ cos bH cosh bH þ rbH sin bH cosh bH þ rbH cos bH sinh bHÞ
þ cos bH sinh bH sin bH cosh bH 2rbH sin bH sinh bH ¼ 0
m00
mT1
m1
m
"
€0
q
k
þ 0
€
q
0
0T
#
q0
q
k
¼
p00 ðtÞ My sgn ðq_ 0 Þ
pðtÞ
ð9Þ
ð2Þ
where H is the height of the insulator and parameters a; b; q, and r
are given by:
b4 ¼
x2 m
a¼
;
EI
1
;
qbH
q¼
M0
;
mH
EI
ke H
r¼
ð3Þ
The equilibrium equation of the fixed base cantilever (ke ¼ 1Þ
is obtained by simply setting r ¼ 0 in Eq. (2). Eq. (2) can be solved
numerically and its roots bj H used in the first of Eq. (3) to obtain
the natural frequencies of the beam. For each value bj H, the expression for the corresponding natural mode of vibration is:
aðcS sC Þ 2sS
/n ðzÞ ¼ C 1 sinbn z þ
cosbn z
að1 þ cC þ sSÞ þ 2cS
að1 þ cC sSÞ 2sC
2sS aðcS sC Þ
sinhbn z þ
cosh bn z
þ
að1 þ cC þ sSÞ þ 2cS
að1 þ cC þ sSÞ þ 2cS
ð4Þ
where C1 is an arbitrary constant.
The equations of motion of the system may be derived using
Hamilton’s Principle:
Z
t2
dðT U Þdt þ
t1
Z
t2
dW NC dt ¼ 0 8t1 ; t2
ð5Þ
m00
mT1
m1
m
Z
H
1
2
mðzÞ½u_ ðz; t Þ dz þ M0 ½u_ ðH; t Þ
2
0
Z
Z H
1
1 H
U ¼ k0 q20 ðtÞ þ
EIðzÞu00 ðz; tÞu00 ðz; t Þdz pðz; tÞuðz; t Þdz
2
2 0
0
W NC ¼ M y q0 ðt Þ sgn ðq_ 0 Þ
q0
q
k
¼
M y sgn ðq_ 0 Þ
0
ð10Þ
Since the system of Eqs. (11) is linear, it can be solved by modal
may be expanded as:
decomposition. The unknown vector q
ðt Þ ¼
q
Nþ1
X
ai ðtÞwi
ð12Þ
i¼1
~ i are
The eigenvectors wi and the corresponding eigenvalues x
obtained by solving the following matrix eigenvalue problem:
~ 2i M wi ¼ 0
Kx
ð13Þ
Substituting Eq. (12) into Eq. (11) leads to the following set of
uncoupled equations in the modal coordinates, aj :
~ 2j aj ¼
a€ j þ x
ej
P
ej
M
ð14Þ
where
e j ¼ wT Mw ;
M
j
j
e j ¼ wT P
P
j
ð15Þ
For classically damped systems Eq. (15) becomes:
~ j a_ j þ x
~ 2j aj ¼
a€ j þ 2~fj x
ej
P
ej
M
ð16Þ
The solution to Eq. (16) is given by:
"
n
o
~
~
aj ðtÞ ¼ exp fj xj t
að0Þ e
ð7Þ
€0 þ mjj q
€j þ kjj qj ¼ pjj ðt Þ ðj ¼ 1; 2; . . . ; N Þ
mj0 q
!
ej
P
~ jD t
cos x
ej
~ 2j M
x
!
3
~ j að0Þ P j
a_ ð0Þ þ ~fj x
7
ej
~2M
ej
x
7
P
j
~ jD t 7
sin x
þ
þ
7
~ jD
ej
x
~ 2j M
5 x
If Eq. (1) and its derivatives are substituted into Eq. (6), then Eq.
(5) leads to:
j¼1
#
ð11Þ
2
N
X
€j þ k0 q0 þ M y sgn ðq_ 0 Þ ¼ p00 ðtÞ
m0j q
0T
€ þ Kq
¼P
Mq
ð6Þ
€0 þ
m00 q
"
€0
q
k
þ 0
€
q
0
Eq. (10) may be written in compact form as:
t1
where T is the kinetic energy, U is a potential function, sum of the
elastic energy and the work done by the external loads on the system, and W NC is the work done by the non-conservative forces. The
equations of motion are derived considering the symmetric constitutive model shown in Fig. 12(a), but the derivation can be easily
extended to the asymmetric model of Fig. 12(b). The kinetic energy
T, the potential function U, and the non-conservative work W NC are
given:
1
T¼
2
3.1.1. Solution for free vibration
In the case of free vibration, the equations of motion (9)
become:
ð17Þ
where
qffiffiffiffiffiffiffiffiffiffiffiffiffi
~ jD ¼ x
~ j 1 ~f2j
x
ð18Þ
where
m00 ¼
Z
H
0
m0j ¼ mj0 ¼
Z
p00 ðtÞ ¼
Z
0
Z
Z
mjj ¼ M j ¼
kjj ¼ K j ¼
3.2. Elastic response
mðzÞz2 dz þ M0 H2
0
H
0
H
When no change in rotation occurs at the base, the system
behaves like a simple cantilever beam. In this case, the uncoupled
equations of motion are given by the second of Eq. (9), that is:
H
0
H
mðzÞzuj ðzÞdz þ M0 Huj ðHÞ
h
i2
h
i2
mðzÞ uj ðzÞ dz þ M0 uj ðHÞ
h
i2
2
EIðzÞ /00j ðzÞ dz þ ke ½/0 ð0Þ
pðz; t Þzdz;
pjj ðt Þ ¼ P j ðtÞ ¼
Z
0
ð8Þ
€ þ kq ¼ pðtÞ
mq
ð19Þ
Introducing damping, Eqs. (19) may be written as:
H
pðz; tÞ/j ðzÞdz
€j þ 2fj xj q_ j þ x2j qj ¼
q
P j ðt Þ
Mj
In the case of free vibration, the modal equations become:
ð20Þ
138
S. Epackachi et al. / Engineering Structures 93 (2015) 129–141
€j þ 2fj xj q_ j þ x2j qj ¼ 0
q
ð21Þ
4
3
n
o
q_ ð0Þ þ fj xj qj ð0Þ
~ j t qj ð0Þ cos xjD t þ j
qj ðtÞ ¼ exp ~fj x
sin xjD t
2
xjD
ð22Þ
where
qffiffiffiffiffiffiffiffiffiffiffiffiffi
xjD ¼ xj 1 f2j
Acceleration [g]
The solution to Eq. (21) is given by:
1
0
-1
-2
-3
ð23Þ
Test
Model
-4
0
1
0.5
In the following, the analytical model developed above is used
to simulate the results of the hammer tests, presented in the previous sections.
3.3.1. First impact hammer test
The frequency, f, measured in the first hammer test performed
before the static pull tests when the system was completely
undamaged, can be used to evaluate the bending rigidity EI of
the insulator. A first approximation can be obtained via the
Southwell–Dunkerley method applied to the fixed base cantilever
ðke ¼ 1Þ (Newmark 1971). As shown in Fig. 14, the system is
decomposed as the sum of a cantilever with uniformly distributed
mass m and one with a lumped mass M 0 ¼ qmH at the top, both
systems having the same stiffness.
The fundamental frequency f of the combined system can then
be evaluated as follows:
1
f
2
¼
1
2
f distributed
þ
1
ð24Þ
2
f lumped
where f distributed and f lumped are the frequencies of the systems shown
in Fig. 14(b) and (c) respectively. The squares of these frequencies
are given by:
2
f distributed ¼
ð3:516Þ2 EI
ð2pÞ2
mH4
2
f lumped ¼
1 3 EI
ð2pÞ2 q mH4
ð25Þ
Substituting Eq. (25) into Eq. (24), and solving for the square of frequency f, gives:
2
f ¼
1
3
EI
ð2pÞ2 q þ 1k mH4
ð26Þ
where
k¼
ð3:516Þ2
¼ 4:12
3
ð27Þ
2
2.5
3
Fig. 15. Measured and calculated histories of acceleration (first impact hammer
test).
Eq. (26) can then be used to provide an estimate of the bending
rigidity EI, that is
EI ¼
ð2pÞ2
1
q þ mH4 f 2
k
3
Substituting
ð28Þ
the
35 kg), M0 ¼ 2:71 105 kN s2 =mm (for the mass of 27 kg) and
f ¼ 17:78 Hz (measured in the experiment) into Eq. (28) gives
EI ¼ 5:25 108 kN=mm. Using this value as the flexural rigidity,
the computed frequency is 17.83 Hz, which is slightly higher than
the measured one. A trial and error procedure converges to the
exact frequency f ¼ 17:78 Hz for EI ¼ 5:22 108 kN=mm. The analytical model, including just one mode of vibration for the fixed
base cantilever ðke ¼ 1Þ and assuming 0.82% damping ratio, produces the top acceleration response shown in Fig. 15. Note that
the value taken for the damping ratio is merely the one that gives
the best match between analytical and experimental results.
3.3.2. Second impact hammer test
The second hammer test, performed after the pull tests, shows a
slight reduction of the first mode frequency and the appearance of
a higher frequency in the acceleration response. The reduction of
the first mode frequency ðf ¼ 17:3 HzÞ, probably due to local damage at the base, is considered in the analytical model by introducing an elastic spring of stiffness ke ¼ 1:98 107 kN=mm, while a
second mode of vibration is also introduced to account for the
higher frequency component. The symmetric constitutive model
shown in Fig. 12(a) is used to simulate the test. Assuming a bilinear
static force–displacement relationship, as shown in Fig. 16(a), the
ρmH
=
m, EI
(a) uniformly distributed and
lumped mass
data:
H ¼ 1524 mm; m ¼ 2:29 108 kN s2 =mm2 (for the total mass of
ρmH
H
1.5
Time [sec.]
3.3. Simulation of the dynamic experiments
m, EI
(b) uniformly distributed mass
only
Fig. 14. Fixed base system decomposition.
+
EI
(c) lumped mass only
139
S. Epackachi et al. / Engineering Structures 93 (2015) 129–141
F
M
Fy
k2
My
k1
k0
u
uy
q0
-M y
-Fy
( a) For ce-d ispl ace me nt
(b) Moment-plastic rotation
6
6
4
4
Acceleration [g]
Acceleration [g]
Fig. 16. Constitutive models.
2
0
-2
-4
0
0.25
0.5
0.75
1
1.25
1.5
1.75
0
-2
-4
Test
Model
-6
2
Test
Model
-6
2
0
0.25
0.5
Time [sec.]
1
0.75
1.25
1.5
Time [sec.]
Fig. 17. Measured and calculated histories of acceleration (second impact hammer
test).
Fig. 19. Measured and calculated acceleration response at top of insulator (third
impact hammer test).
FH ¼ F y H þ moment-rotation relationship can be derived as follows. The base
moment is given by:
M ¼ FH ¼ F y þ k2 u uy H
ð29Þ
k2 H2
q0 or M ¼ M y þ k0 q0
3
2H
1 k3EI
a
ð33Þ
where
where
u ¼ q0 H þ
FH
FH3
Hþ
;
ke
3EI
uy ¼
FyH
F y H3
Hþ
ke
3EI
My ¼ F y H;
ð31Þ
The values of F y and k2 , needed to determine M y and k0 by Eq.
(34), are obtained by idealizing the force–displacement response
exhibited in the static pull tests (Fig. 5) by the bilinear relation
shown in Fig. 16(a). Given F y ¼ 2:2 kN and k2 ¼ 0:4 kN=mm results
Expressions (30) may be written as:
FH3
u ¼ q0 H þ
a;
3EI
F y H3
uy ¼
a
3EI
where
3EI
a¼1þ
ke H
model ðke ¼ 1:98 107 kN=mmÞ yields the acceleration at the top
shown in Fig. 17. Rayleigh damping was assumed with damping
Substituting Eq. (31) into Eq. (29) then leads to:
9000
450
Test
Model
400
Amplitude [g-sec.]
6000
3000
0
-3000
-6000
-9000
-2E-005 -1E-005
Test
Model
0
1E-005
Plastic rotation [rad]
ð34Þ
in M y ¼ 3390 kN=mm and k0 ¼ 2:36 107 kN=mm. Including two
modes of vibration of the flexible base cantilever into the analytical
ð32Þ
Moment [kN.mm]
k2 H 2
;
k0 ¼ 3
2H
1 k3EI
a
ð30Þ
2E-005
350
300
250
200
150
100
50
0
0
10 20 30 40 50 60 70 80 90 100 110 120 130
Frequency [Hz]
Fig. 18. Measured and calculated moment-plastic rotation relationships (left) and Fourier spectrums of acceleration (right) (second impact hammer test).
S. Epackachi et al. / Engineering Structures 93 (2015) 129–141
9000
200
6000
160
Amplitude [g-sec.]
Moment [kN.mm]
140
3000
0
-3000
-6000
-9000
-0.015 -0.01 -0.005
120
Test
Model
0
Test
Model
0.005 0.01 0.015
80
40
0
0
10 20 30 40 50 60 70 80 90 100 110 120 130
Plastic rotation [rad]
Frequency [Hz]
Fig. 20. Measured and calculated moment-plastic rotation relationships (left) and Fourier spectrums of acceleration (right) (third impact hammer test).
ratios f1 = 1.3% and f2 = 0.5% in the two elastic modes. Again these
values are taken to obtain the best match between analytical and
experimental results.
Fig. 18 shows the response in terms of the moment-plastic rotation behavior and the Fourier spectrum of the acceleration. Since
the sampling frequency of the signals is 256 Hz, the actual second
mode frequency of the model ðf 2 ¼ 165 HzÞ cannot be captured by
the Fourier spectrum, which shows an erroneous peak at 91 Hz.
Consequently, even the second mode frequency of the experimental signal given by the Fourier spectrum is not reliable since this
could be higher than that shown.
3.3.3. Third hammer test
The third and last hammer test was carried out after permanently breaking the base flange during the cyclic tests. The first
mode frequency shows a further reduction which is considered in
the model by reducing the stiffness of the elastic base spring to
ke ¼ 8:81 106 kN=mm. The asymmetric constitutive model of
Fig. 12(b) is used to simulate the test. The stiffness k2 , needed to
compute k0 , is taken in this case as the first quadrant slope of the
force–deformation response in the last cycle of the tests shown in
Fig. 6. Given k2 ¼ 0:11 kN=mm, Eq. (34) yields k0 ¼ 3:55
105 kN=mm. The analytical model, including two modes of vibration
of the flexible base cantilever ðke ¼ 8:81 106 kN=mmÞ, results in
the acceleration at the top shown in Fig. 19.
Rayleigh damping was used with damping ratios f1 = 2.0% and
f2 = 0.3% in the two elastic modes, in order to successfully match
the history of the measured acceleration. Fig. 20 shows the
moment-plastic rotation behavior and the Fourier spectrum of
the acceleration.
4. Summary and conclusions
A series of pull, cyclic, and impact hammer tests were conducted on a full-scale electrical hollow composite post insulator
using the test facilities of the Structural Engineering and
Earthquake Simulation Laboratory (SEESL) of the University at
Buffalo. The experimental study aimed to assess the mechanical
dynamic behavior and failure modes of the insulator. The frequency and equivalent viscous damping ratio of the intact post
insulator were estimated using the results of the impact hammer
test conducted prior to the pull and cyclic tests. The initial stiffness
and the displacement corresponding to the SML, suggested by the
manufacturer, were measured using the pull test data. The second
impact hammer test was conducted to determine the source and
extent of the damage to the specimen and its correlation with
the stiffness deterioration. The slight difference between the first
and second impact hammer test results indicated that only minor
damage occurred during the pull test.
The effects of cyclic response of the post-insulator and the
deteriorations of strength and stiffness were assessed conducting
a tests with amplitudes lower, equal and larger than the peak
specified manufacturer load (SML). The major failure mode of the
specimen was the flange–tube bond failure. Such failure significantly affected the hysteretic response. Pinched hysteretic
response was observed at lateral displacements greater than that
corresponding to SML indicating slip and stiffness recovery. After
the significant drop in strength at the bond failure, no further
strength deterioration was observed up to 350% of the displacement corresponding to SML. At the end of the test, the part of composite tube inside the bottom metal cap was significantly buckled
and crushed with no signs of cracking and crushing in other parts
of the tube.
The results of the third impact hammer test, conducted after the
cyclic test, showed a significant change in the dynamic responses
due to the damage occurred in the specimen. After the bond failure
and damage progression in tube–flange connection, a second mode
of vibration appeared, representing the rocking of the insulator
inside the metal cap.
On the basis of the test results, an analytical model was developed to simulate the mechanical behavior of the post insulator
before and after damage. The hollow composite post insulator
was idealized as a cantilever beam with distributed mass and elasticity and a lumped mass at the top. The bonded connection of the
tube to the lower flange was modeled using two springs; a frictional spring to consider the energy dissipated due to sliding of
the tube inside the flange and an elastic spring to consider the stiffness reduction due to damage of the base flange itself. The cyclic
response and dynamic characteristics of the post insulator were
successfully calculated using the proposed model. The computational model can be used as a base analytical model to predict
the behavior of other hollow composite post insulators. However,
further studies are needed to determine the ranges of variation
of the modeling parameters.
Acknowledgement
The authors acknowledge the participation of the insulator
manufacturers that provided the test specimens and insight on
the performance of composites. The technical staff of the SEESL
at the University of Buffalo are commended for their skillful contributions to the project.
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