Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 2311-9020; ISBN: 978-972-752-165-4
An Integrated Ice-Shedding Model of Electric Transmission Lines with
Consideration of Ice Adhesive/Cohesive Failure
Ghyslaine McClure 1, Kunpeng Ji2, Xiaoming Rui 2
Department of Civil Engineering and Applied Mechanics, Faculty of Engineering, McGill University,
Montreal, Canada
2
Department of Power Machinery Engineering, School of Energy Power, Mechanical Engineering,
North China Electric Power University, 102206, Beijing, China
email:, [email protected]; [email protected]; [email protected]
1
ABSTRACT: This research is an attempt to propose an integrated accreted ice failure model for iced overhead line conductors
that will lead to more realistic nonlinear dynamic analysis of the ice-shedding phenomenon of transmission lines, by taking into
account the adhesive / cohesive strength of ice deposits.
Ice shedding induced by sudden mechanical forces is understood to be a two-stage process. First, the continuous ice
deposits along the conductor span are broken into smaller separate ice chunks and fragments (ice fracture failure), and then these
fragments detach from the cables and fall off due to insufficient cohesive strength within the ice or adhesive strength at the icecable interface (ice detachment failure). Two recent successive studies have developed computational models using ice deposit
failure criteria based on the maximum effective plastic strain and the maximum bending stress. These models have yielded
reasonably accurate results in predicting cable tensions and mid-span displacements, by comparing their numerical results with
experimental data from tests carried out on a 4m reduced-scale model span with varying cable diameters and ice thickness,
following sudden mechanical shock loads. However, there is still about 20% disparity in ice fracture rates between the
computational results and experimental data, and it is deemed necessary to refine the ice failure model to introduce the effects of
adhesive / cohesive forces.
Therefore, as the first step towards the development of an integrated two-tier ice shedding criterion, the authors have
improved the previous FE models, in terms of mesh size, load types and locations, material models and so on, to provide a better
description of the experiment results. Then, the refined FE models of the reduced-scale span tests are used to check the newly
proposed ice adhesive /cohesive failure criterion. The idea of this criterion is to simply compare the inertia forces acting on the
fractured ice segments, and the ice adhesive strength or cohesive strength. The process is done automatically by a subroutine
interacting with the nonlinear dynamic analysis commercial software ADINA. Although there is no satisfactory model to
calculate the adhesive and cohesive strengths of glaze ice - especially for atmospheric ice, only several representative pairs of
values are selected. Validation is underway to ascertain that the proposed two-tier glaze ice shedding criterion provides a more
realistic description of the ice-shedding phenomenon of transmission lines than in the previous studies.
KEY WORDS: Nonlinear dynamics; Shock loads; Computational models; Ice failure.
1
INTRODUCTION
Atmospheric icing is one of the major threats to the security of
overhead electric transmission lines in cold regions. These
threats can be classified into two categories: electric ones,
(such as flashovers, outages), and mechanical ones (such as
galloping, overload icing, uneven icing, ice shedding, etc.).
During the Great Ice Storm of January 1998 in North America,
the losses on the Hydro Québec transmission grid alone
included: 600 steel tower collapses and 100 damaged towers,
and 16 000 line components failure (poles, cross-arms,
hardware, cables) in the distribution network [1]. An even
more destructive ice storm hit South China in 2008, which
resulted in more than 140 000 collapsed towers on 10~110 kV
lines and 1 500 towers on the transmission grid (above 220 kV)
operated by the State Grid company [2].
After these events, a large number of anti-icing (referring to
methods which are used before or during the early stage of
icing to prevent the accumulation of ice on cables) and deicing techniques (referring to methods which are employed
during or after icing to remove accreted ice from cables) have
been proposed and tested by researchers and engineers from
all over the world. A technical brochure edited by CIGRE
Working Group B2.29 [3] gives a comprehensive review on
the operational and potential anti-icing and de-icing methods,
which can be divided into four categories: passive methods,
active coating methods, mechanical methods and thermal
methods. Among these methods, mechanical methods shows
clear advantages in de-icing ground wires and short sections
of strategic lines as timely and fast intervention, because they
are easy to apply and cost effective. A portable mechanical
de-icing device, DAC (De-icer Actuated by Cartridge),
proposed by Hydro Quebec, is the focus of this research.
DAC takes advantage of the brittle characteristics of glaze
ice at high stain rates (>10-3/s). So, when the shock wave
travels along the span, it can break the ice accreted on cables
into small fragments by releasing the energy of shock waves.
Details of this device can be found in [4]. A reduced-scale
single span physical model was tested in the CIGELE
laboratory at UQAC, Chicoutimi, Canada, to test the effect of
this device and to evaluate its impact on supports and the
jump heights of midpoint, which may give useful information
on the design and optimization of DAC[5].
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Two recent successive studies have developed
computational models using ice deposit failure based on the
maximum effective plastic strain and the maximum bending
stress, to model the “unzipping” process when shock loads
were applied to iced cables [5-8]. These models have yielded
reasonably accurate results in predicting cable tensions and
mid-span displacements, by comparing their numerical results
with experimental data from tests carried out on the 4m
reduced-scale model span with varying cable diameters and
ice thickness. However, there is still about 20% disparity in
ice fracture rates between the computational results and
experimental data, and it is necessary to refine the ice failure
model to introduce the effects of adhesive/ cohesive forces.
This research is an attempt to propose an integrated accreted
ice failure model that will lead to more realistic nonlinear
dynamic analysis of the ice shedding phenomenon of
transmission lines, by taking into account the adhesive /
cohesive strength of the ice deposits.
2
Four scenarios are studied in the tests, though only the
scenario with 1 mm of equivalent radial ice thickness on a
cable with the diameter of 4.1 mm will be presented in this
study. Since the round cross section cannot be used for the
iso-beam element, the accreted ice is modeled using an
equivalent iso-beam with rectangular section with the same
area and second moment of area as the idealized round cross
section: The resulting dimensions are a width of 6.36 mm and
height of 2.52 mm.
The impact force used for this scenario is shown in Figure 2.
SENSITIVITY STUDY
Ice shedding induced by sudden mechanical forces is
understood to be a two-stage process. First, the continuous ice
deposits along the conductor span are broken into smaller
separate ice chunks and fragments (ice fracture failure), and
then these fragments detach from the cables and fall off due to
insufficient cohesive strength within the ice or adhesive
strength at the ice-cable interface (ice detachment failure). So,
as the first step, a series of sensitivity study will be done on
the basis of previous studies.
Figure 2. Characteristics of the impact force [5]
Two cable material models (Figure 3) are used: one is the
theoretical tension-only model (MAT1) with a constant
Young’s modulus of 172.4 GPa, and the other one (MAT2) is
a multi-linear tension-only model whose values are got from
static tensile test of the cable specimen.
General modeling assumptions
2.1
Since previous research using the nonlinear dynamic
analysis commercial software ADINA [6, 8-11] has yielded
good simulation results compared with experiments, , it is also
used in this study using similar modeling assumptions and
computational methods. The iced cables were modeled by
paralleling ice elements and cable elements which share the
same end nodes, as shown in Fig. 1. [5, 6]. The conductor is
modeled as a mesh of 3-D two-node iso-parametric truss
elements with tension-only material properties, and the ice
accretion is modeled with a parallel mesh of general 3-D isobeam elements, with bilinear plastic material model. The total
Lagrangian formulation is used for this large displacement but
small strains problem in ADINA [12, 13].
Aerodynamic damping is neglected, and an equivalent
viscous damping of 2% critical is used to model the structural
damping of the iced conductor, by using a nonlinear axial
dashpot element in parallel to each cable element. More
details about the selection of damping constant are discussed
by Roshan Fekr et al.[9] and McClure and Lapointe[11].
a)
b)
Figure 1. FE model of the 4 m single span tests [5]
Figure 1 a) shows the whole FE model , and b) shows the
diagram of the iced cable FE model at the element level.
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a) MAT1
b) MAT2
Figure 3. Material models of the cable [5]
The ice material model has a Young’s modulus of 10 GPa,
Poisson’s ratio 0.33, initial yield stress 2 MPa, and maximum
allowable effective plastic strain 10-10 (for the maximum
normal stress failure criterion) or maximum allowable
effective plastic strain 9.756×10-5 (for the maximum normal
strain failure criterion).
2.2
Mesh size analysis
The motivation for a mesh size analysis stems from the
following: 1) It is essential to achieve convergence and
stability when the number of elements are large to achieve
convergence and stability. This was not examined in previous
studies as the meshes were relatively coarse with only 20 to
30+ cable elements per span; 2) the experiments showed that
the ice accretion remaining on the cable was broken into small
fragments with an average length of around 5 mm, which
means that fracture did occur within every ice element and
between its integration points, thus leading to a rate of ice
shedding (RIS) of 100% for the mesh size of 25 (element
length of 160 mm); 3) the impact pulse was applied at the
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
midpoint of the span in these tests, and the measured
displacement values are also for the midpoint, but there was a
80 mm drift from the mid span when the shock loads were
applied at node 13 in the 25-element model. The mesh size
sensitivity analysis is conducted from 25 to 4000 cable
elements per span. The ice element number changes to
accommodate the fact that there is a 30 cm length of cable
with no ice, at the far ends of the spans owing to the limitation
of the spraying system[5].
observed when the 1000 elements FE model was used, as
shown in Figure 5.
a)
Figure 4. Comparisons of cable tensions
with different mesh densities
As is shown in Figure 4, the computational results of
tension of the left end cable element begin to converge until
no less than 400 elements were used. The comparison of mid
span displacements showed the same trend, although not
shown here due to space limitations. Clearly, 25 cable
elements (21 ice elements) are not sufficient for this research,
and the mesh with 1000 cable elements (850 ice elements)
will be used throughout this study. It is also seen in Figure 4
that the finer mesh models better predict the trend after the
first peak, and that the finer the mesh model is, the first peak
occurs earlier in time.
Besides, the time step is set to 0.05ms for the first 2000
steps (100 ms) and then to 0.25 ms for the remainder. The
reasons are: 1) the maximum peak of the shock load arrives at
approximately 0.20-0.25ms, so if the time step is set to be
0.25ms as in previous studies, the shock load will be applied
to the system from 0 to maximum in a single step, which may
cause computational instability of the nonlinear model; 2)
after the duration of shock load, a larger time step can help to
accelerate the computation and save time.
2.3
Material characteristics
In previous studies, different cable material models were used
to get results of cable tension and mid span displacement
separately in the same scenario, that is the MAT1 cable
material model with greater tensional rigidity (EA=
2,275,680 N) was used to generate the time history of midspan displacement, and the MAT2 material model with less
tensional rigidity (EA= 346,500 N) was used in its initial
linear section (0-105MPa) to obtain the time history of cable
tension [5, 7, 8]. The flexible cable model can result in a
decrease of 52% of the maximum cable tension compared
with the rigid one [5]. By doing this, the simulation results can
better agree with the experimental data. This was also
b)
Figure 5. Comparisons of different FE models
a) cable tension at left support b) Displacement
at mid point
However, the numerical models still overestimate the
maximum value of both the cable tension and mid-span
displacement. This discrepancy may be explained as follows:
1) The amount of ice detached from the cable as predicted
by the FE simulation (rate of ice shedding R.I.S =100%) and
that observed in the test (R.I.S =8%) are very different. This is
partially validated by assuming a portion of 8% ice elements
near the midpoint to “undergo element death” in ADINA
when the shock load peak arrives (t=1.05025s), and setting
the maximum allowable strain of ice to be an unrealistic large
value (e.g. 1.0 ). Therefore, only these ice elements are
removed at the early stage of the simulation, while the others
will remain on the cable throughout the analysis. As a result,
the computational maximum cable tension decreases to the
level of values measured in the test, and the overall
displacement tendency seems in accordance with test data, as
shown in Figure 5. Also, the ice shapes in the tests are
irregular, which is not considered in the present study.
2) The real tensile rigidity (EA) of a stranded cable is not
constant but varies with time and cable deformation, which
cannot be considered in the present FE model.
3) The bending rigidity (EI) of the cable is totally neglected
in the FE model by using truss elements. However, the
bending rigidity of the cable is about 4.42 （ with MAT1
model, EI=2.3913 m2 ·N） or 0.67 （ with MAT2 model,
EI=0.3641 m2 · N ） times the bending rigidity of ice
(EI=0.5409 m2 ·N). This simplification may be useful and
reasonable in simulating the global motion of cables (e.g.
galloping and Aeolian vibration), but not for localized shock
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
loading which involves dramatic local cable bending motion
and initiates wave propagation after the shock load. This
model refinement is still ongoing.
4) The flexibility of supports (assumed perfectly fixed in the
FE model) may also play an important role in the disparity of
computational and experimental results.
Besides, in the attempt to get more accurate numerical
simulation of the tests, another way of applying the shock load
is also simulated in the FE model. It divides the original force
into three equal lumped forces distributed on a length of
about 1 cm (i.e. on nodes 500, 501 and 502 of the model),
which is equal to the diameter of the piston end. As shown in
Figure 5 (with the legend ‘1000E_MAT2_1/3 load’), the
results are similar to those obtained in previous model and the
refinement is not deemed necessary.
3
PRELIMINARY STUDY OF ADHESIVE/COHESIVE ICE
FAILURE CRITERIA
Since the experiments showed that large amounts of ice were
sticking on the cable after the shock loads, it becomes
necessary to consider the de-icing process as a two-stage
process. First, the continuous ice deposits along the conductor
span are broken into smaller separate ice chunks and
fragments (ice fracture failure), and then these fragments
detach from the cables and fall off due to insufficient cohesive
strength within the ice or insufficient adhesive strength at the
ice-cable interface (ice detachment failure).
3.1
General concept of adhesive/cohesive ice failure
criterion
The main and simple concept of this criterion is to compare
the inertia forces acting on the fractured ice segments, and the
ice adhesive or cohesive stress resultants.
the length of the ice fragment,
the diameter of cable,
τ cohesive
a is the acceleration, Dcable is
tice is the ice thickness, τ adhesive and
are the adhesive and cohesive strengths of glaze ice.
The first difficulty for the implementation of this failure
criterion is to select realistic adhesive and cohesive strength
values for atmospheric glaze ice. Although the physical and
mechanical properties of ice have been studied for decades, a
theoretical model to calculate the adhesive and cohesive
strengths is still lacking, and experimental values reported in
literature are both scarce and with large variability as ice is a
highly complex natural material. The measured values vary
considerably with many factors, such as ice deposit types,
temperature, nature and texture of substrate, wind speed, test
methods, and so on [14, 15].
In spite of these difficulties, one can get some general
conclusions: 1) the adhesive strength values obtained in
tensile tests are at least 15 times larger than that in shear tests
[16, 17]; 2) the adhesive strength of ice-metal interface is
larger than the cohesive strength of ice, contrary to polymeric
materials [14, 18]; 3) the adhesive strength tested with high
strain rates or with dynamic test methods is significantly less
than that with low strain rates achieved in static or quasi-static
tests; and 4) brittle fracture happens when ice is subjected to
high tensile stress, and the adhesive strength is temperature
independent in this instance, while ductile failure occurs when
tensile stresses remain below a critical value, and the adhesive
strength increases linearly when temperature decreases below
0℃ [14].
Several tests results are summarized in a recent study
published in 2012, which shows the adhesive shear strength of
ice-Aluminum and ice-Stainless steel interfaces varying
between 0.002 to 1.96 MPa [19]. On the basis of these
published test results, four pairs of adhesive and cohesive
strength values were chosen for this research (with dcable = 4.1
mm, tice = 1mm), and the critical vertical accelerations are
calculated as shown in Table 1. The calculated accelerations
are very large because the mass of ice per unit cable length is
very small; heavier deposits are easier to shed.
Table 1. Examples of Adhesion/Cohesion Strengths and
Critical Ice-shedding Accelerations
Figure 6. Schematic diagram of ice detachment
failure criteria
No.
As shown in Figure 6, if the inertia force of an ice segment
is larger than its adhesive and cohesive stress resultants, the
ice segment will detach from the cable. This can be described
by Equations (1)-(4).
(1)
Finertia ≥ Fadhesive + Fcohesive
πρ la 
( Dcable + 2tice ) − Dcable 2 

8
Fadhesive = Dcablel τ adhesive
Fcohesive = 2ticel τ cohesive
F=
inertia
2
(2)
(3)
(4)
Finertia , Fadhesive and Fcohesive are inertia ,adhesive and
cohesive force separately. ρ is the density of glaze ice, l is
where
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1
2
3
4
3.2
Adhesion
Strength
(MPa)
Cohesion
Strength
( MPa )
Critical
Acceleration
0.005
0.01
0.05
0.25
0.004
0.008
0.04
0.20
4.0
7.9
40
198
(  103m/s2 )
Acceleration distribution along the span
Before applying the adhesive/cohesive ice failure criterion
into the FE model, the maximum acceleration values of the
whole model calculated in each time step of analysis are
checked. The results show that the absolute maximum
acceleration occurs first at the mid-span where the shock load
is applied, and then moves to the two ends, as time increases.
After reaching the ends, the absolute maximum values go
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
from the ends to the midpoint. Besides, the absolute maximum
values decrease from the midpoint to each end before arrive
the ends as time increases, as shown in Figure 7.
It is worth noting that according to the ice fracture criterion,
all the ice elements “have died” at 1.05125s, when the
acceleration was 4,000 m/s2, and the absolute maximum
acceleration occurs near node 480 and 520.
A
in the test and simulation, the changing cable tensile rigidity,
and the neglecting of cable bending stiffness.
The general idea of the newly proposed ice detachment
failure criterion is presented, and several representative pairs
of adhesive and cohesive values are selected to calculate the
critical acceleration needed to shed off the broken ice
fragments induced by the shock wave. The analysis of
maximum acceleration values at each time step and each node,
and the effective shedding ice (8%) make it possible to
estimate the critical acceleration and the adhesive and
cohesive strengths of ice in the test, which in turn validates
the ice detachment failure criterion. The effort of integrating
the user-supplied subroutine containing the proposed ice
detachment failure criterion into ADINA is underway.
ACKNOWLEDGMENTS
a)
This work was supported by the Fundamental Research Funds
for the Central Universities of China (No. 12QX10) and the
China Scholarship Fund. The authors also acknowledge Dr. T.
Kálmán for providing of experimental data. The first author
also acknowledges financial support from the Natural
Sciences and Engineering Research Council of Canada.
REFERENCES
1.
2.
b)
Figure 7. Maximum/Minimum acceleration
values with respect to time
a) Long time duration b) Detail of A
So considering 8% ice shedding (i.e. 68 ice elements dead)
actuated by the proposed ice detachment failure criterion, the
critical acceleration should occur around nodes 467 and 535,
which is predicted to be approximately 3,000 m/s2.With this
critical value, the ice adhesive strength of this test can be
estimated with Equations (1)-(4) and is approximately 0.004
MPa, which falls into the reasonable range shown in Table 1
(assuming that τ adhesive =1.25 τ cohesive ). This validates the
feasibility of the proposed ice detachment failure criterion for
the case under study.
This process is to be programmed in a subroutine (written
by the second author) which will interact with the nonlinear
dynamic analysis commercial software ADINA. This
implementation is still ongoing.
4
3.
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7.
CONCLUSION
This research is an attempt to better understand and simulate
the conductor ice shedding phenomenon induced by shock
waves. The 1000 cable elements FE model is proved to be
adequate and able to simulate the nonlinear dynamic response
of the physical reduced-scale tests. The discrepancies between
the numerical results and experimental data are believed to be
“reasonable” before considering ice detachment failure, and
are analyzed in terms of the different amounts of ice shedding
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