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The application of a uniform radial ice thickness to structural sections
K.F. Jones
a,
*, A.B. Peabody
b,1
a
Cold Regions Research and Engineering Laboratory, 72 Lyme Rd., Hanover, NH 03755, USA
b
Consulting Transmission Engineer, 12601 Turks Turn, Anchorage, AK 99516, USA
Received 26 November 2004; accepted 16 October 2005
Abstract
Equivalent uniform radial ice thicknesses accreted from freezing rain on wires, conductors, guys and cables with round
cross sections are determined for the design of ice sensitive structures from field measurements or from simulations using
historical weather data. Ice thicknesses on non-round structural shapes may also be required in the design of commu-
nication towers, towers for power transmission lines, and other ice-sensitive structures. In this paper the uniform radial ice
thickness on a wire is shown to be different from the uniform ice thicknesses on other structural shapes in the same
icing conditions. Consistent ice thicknesses are calculated for a variety of structural shapes including angles, bars, and
channels. Finally a simple method, which is used in ASCE Standard 7 Minimum Design Loads for Buildings and Other
Structures, is presented for determining ice loads on arbitrary structural shapes from the uniform radial ice thickness on a
wire.
D2005 Elsevier B.V. All rights reserved.
Keywords: Ice accretion; Ice loads; Power line icing; Transmission lines; Towers; Design criteria
1. Background
Equivalent uniform radial ice thicknesses accreted
from freezing rain on wires, conductors, guys, and
cables with round cross sections are determined for
the design of ice-sensitive structures from field mea-
surements or from simulations using historical weath-
er data. Ice thicknesses on non-round structural
shapes may also be required in the design of com-
munication towers, towers for power transmission
lines, and other ice-sensitive structures assembled
from components with non-round cross sections.
For two components that intercept the same depth
of freezing rain, the ice thickness will be less on the
section with the longer perimeter. For example, the
ratio of the uniform ice thickness on a square cross
section, oriented with one side perpendicular to the
rain drop trajectory, to the uniform ice thickness on a
cylinder is p/4 (Jones, 1996).
Design ice thicknesses from freezing rain are
often specified as uniform radial thicknesses. This
thickness has been determined for components with
essentially round cross sections, such as wires,
cables, conductors, or guys. In this paper we discuss
the application of this measure of the ice thickness to
structural sections with non-round cross sectional
shapes. We assume that the components are all ori-
ented horizontally, with axes perpendicular to the
direction of the wind accompanying the freezing rain.
0165-232X/$ - see front matter D2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.coldregions.2005.10.002
* Corresponding author. Fax: +1 603 646 4644.
E-mail addresses: Kathleen.F.Jones@erdc.usace.army.mil
(K.F. Jones), apeabody@acsalaska.net (A.B. Peabody).
1
Fax: +1 907 345 6819.
Cold Regions Science and Technology 44 (2006) 145 – 148
www.elsevier.com/locate/coldregions
2. Basic model
In many of the basic models for accreting ice
from freezing rain, the ice is assumed to accrete
uniformly thickly around the object, which is typi-
cally assumed to have a round cross section. This
accretion shape can occur in nature. The precipitation
that impinges on the upper windward quadrant of a
wire may not freeze immediately, instead flowing
part way around the circumference before finally
freezing. Or, if the impinging precipitation does
freeze immediately the eccentric weight may cause
a torsionally flexible wire to rotate under the eccen-
tric load. As the precipitation continues to fall, the
ice eventually covers the wire in a more-or-less
uniform layer. There are many exceptions to these
two scenarios, resulting in eccentric ice shapes, or a
significant fraction of the ice in the form of icicles
hanging from the wire. However, the assumption of a
uniformly thick layer of ice is a useful concept for
examining the important factors that determine the
accreted ice load. Here we extend that assumption to
structural shapes with non-round cross sections, even
though the corners and edges of these shapes and
their torsional stiffness, as structural elements in
lattice towers, for example, make the realization of
a uniform ice thickness less likely.
Consider a structural section with characteristic
dimension dand cross section perimeter p
0
(Fig.
1). Freezing rain is falling at an effective rate P,
with the drop trajectory perpendicular to d. The
effective rate includes the contribution from wind-
blown rain. What is the uniform thickness of ice on
the object at the end of the event at time T?Atan
arbitrary time in the storm there is a uniform layer of
ice with thickness x, and the perimeter of the ice-
covered section is p(x). In the increment of time dT,
an ice layer of thickness dxis added. Equating the
mass per unit length of impinging rain with the mass
per unit length of the ice layer results in
dþ2xðÞPdT¼cpxðÞdx;ð1Þ
where c= 0.9 is the specific gravity of ice. This can
be rearranged and integrated to give
ZT
0
P
c
dT¼Zt
0
pxðÞ
dþ2xdx;ð2Þ
where tis the uniform thickness of the ice on the
object at the end of the event. For a round cross
section p(x)=p(d+2x), which gives the familiar re-
lationship (Jones, 1998) between precipitation depth
and uniform radial ice thickness t
c
for a cylinder
pctc¼PT:ð3Þ
3. Calculation of consistent ice thicknesses
We can use the total precipitation amount in (3) that
results in t
c
on a cylinder and use (2) to calculate the ice
thickness that accretes on other cross sectional shapes
in the same conditions. Assume that these simple
shapes are made up of plates of zero thickness with n
sides of length L,m908outside corners, and q908
inside corners. Further, for convenience, assume that
the ice shape is round on the outside corners (Fig. 2)
and has square inside corners. If the shape is oriented
with the diagonal perpendicular to the rain drop trajec-
tory, the characteristic dimension d¼ffiffiffi
2
pL. The perim-
eter of the ice-covered shape is then
pxðÞ¼nL 2qx þm
2pxð4Þ
d
p(x)
po
x
P
Fig. 1. Freezing rain falling on a cross section with dimension d
perpendicular to the rain drop trajectory and perimeter p
0
, covered by
ice with thickness x.
characteristic length
L
t
Fig. 2. Layer of ice with thickness ton an angle with legs of length L.
There are four sides, five 908outside corners, and one 908inside
corner.
K.F. Jones, A.B. Peabody / Cold Regions Science and Technology 44 (2006) 145–148146
for a uniform ice thickness x. From (2)
ptc¼Zt
0
nL þmp=22qðÞx
ffiffiffi
2
pLþ2xdx¼t
2mp
22q
þL
2n1
ffiffiffi
2
pmp
22q
ln ffiffiffi
2
pt
Lþ1 ð5Þ
This transcendental equation cannot be solved ana-
lytically, but for a specified length Land shape given
by n,mand q, it can be solved iteratively to
determine the thickness t
0
of the uniform layer of
ice on this shape that is consistent with the uniform
radial ice thickness t
c
. Note that this ice thickness
depends on L, although the uniform radial thickness
does not. This dependence on Lstems from the
shape of these objects changing as ice accretes,
while a round cross section remains round. A
corresponding equation for the ice thickness on a
bar with width Loriented perpendicular to the drop
trajectory can be derived in a similar way.
Table 1 shows the values of n,mand qfor a few
shapes. In the last column of this table is the ice
thickness calculated from (5) that is consistent with a
30 mm ice thickness on a cylinder for L= 0.1 m. In
these conditions, the depth of ice on a cold roof per-
pendicular to the rain drop trajectory is found from (3)
to be 84.8 mm. For the square and angle, for which
p0=d¼2ffiffiffi
2
pis smaller than p, the ice thickness is
greater than the 30 mm on a round cross section. For
the channel and I-beam, for which p0=d¼3ffiffiffi
2
pis
greater than p, the ice thickness is less than on the
round cross section. The nonstructural E-shaped section
is included to illustrate the effect of increasing perim-
eter on the equivalent uniform ice thickness.
Table 1
Shape parameters for five structural sections and an E-shaped section,
and ice thicknesses consistent with a 30-mm radial thickness
Shape nmqt
0
(mm)
Square 4 4 0 32.7
Angle 4 5 1 33.1
Channel 6 6 2 23.4
H section 6 8 4 23.7
E section 8 8 4 17.8
Bar 2 4 0 40.9
Fig. 3. Ratio of ice areas; solid circles are A
c
(t
c
)/A(t
0
) and open circles are A(t
c
)/A(t
0
).
K.F. Jones, A.B. Peabody / Cold Regions Science and Technology 44 (2006) 145–148 147
4. The application of the radial ice thickness to
structural shapes
The calculation in the previous section could be
made more general to allow for sides with different
lengths (e.g. rectangles, angles with unequal legs) and
for specified thicknesses for the leg, web, or flange of
the structural section. However, the purpose of this
paper is not to provide these detailed calculations, but
to show how to use the uniform radial thickness on a
cylinder to obtain consistent ice loads on these other
shapes.
The uniform radial ice thickness that is calculated for
a round cross section is often assumed to apply to other
shapes as well (e.g. ASCE, 2000; ISO, 1998; TIA,
1996). However, as indicated in the examples in the
previous section, the uniform ice thickness actually
varies with the shape of the cross section. The ice
area is the best measure for comparing ice loads on
different shapes. The ice area A(t) for a uniform ice
thickness ton our simple shapes is given by
AtðÞ¼nLt qt2þm
4pt2:ð6Þ
The area of ice with thickness t
c
on a cylinder is
Actc
ðÞ¼pdtcþt2
c
:ð7Þ
We can compare the area of ice (6) on a structural
shape, on which is accreted a thickness of ice t
0
calcu-
lated from (5) that is consistent with the amount of
precipitation resulting in an ice thickness t
c
on a cylin-
der with diameter d¼ffiffiffi
2
pLwith a) the area of ice of
thickness t
c
on the cylinder from (7), and b) the area of
ice (6) on the structural shape assuming an ice thickness
t
c
. The first ratio is A
c
(t
c
)/A(t
0
) and the second ratio is
A(t
c
)/A(t
0
). These ratios are plotted in Fig. 3 for the six
cross sections. Note that the first ratio, using a consis-
tent ice thickness on the structural shapes, is near 1 and
approaches 1 asymptotically as Lincreases. However,
the second ratio, which comes from applying the same
ice thickness on the structural shapes as on the cylinder,
is much greater than or less than 1, depending on the
shape. Furthermore, that ratio does not tend toward 1
with increasing L.
This comparison shows that the design uniform
radial ice thickness t
c
can be used to determine the
design ice load on other structural shapes by applying
the area of ice on the cylinder (7) to those shapes. This
approach gives ice loads on the cylinder and the other
shapes that are consistent with each other. In contrast,
applying the radial ice thickness t
c
directly to structural
shapes results in ice loads that are excessively high or
low, depending on the shape. The radial ice thickness t
c
can still be used in determining the projected area for
specifying the design wind-on-ice load. However, one
should keep in mind the effect of the accreted ice on the
drag coefficient, as well as the wide variety of shapes of
accreted ice that occur in nature. The ice shapes on
structures in a particular storm will depend on the
specific meteorological conditions at that location as
well as on the orientation and shape of the components
and their relative positions in the structure.
This method of applying mapped extreme radial ice
thicknesses to non-round cross sections is specified in
ASCE Standard 7 Minimum design loads for buildings
and other structures beginning with the 2002 revision
(ASCE, 2002).
Acknowledgments
This work was supported by the Corps of Engineers
Engineering Research and Development Center work
package Near Surface Properties and Sensor Signal
Interaction with Terrain, work unit Icing on Structures
(Jones), and Hydro Quebec and the Natural Science and
Engineering Research Council of Canada (Peabody).
References
ASCE, 2000. Minimum Design Loads for Buildings and Other Struc-
tures, ASCE Standard 7-98. American Society of Civil Engineers,
Reston, VA.
ASCE, K.F., 2002. Minimum Design Loads for Buildings and Other
Structures, ASCE Standard 7-02. American Society of Civil
Engineers, Reston, VA.
ISO, 1998, Atmospheric Icing on structures ISO/TC 98/SC 3 (com-
mittee draft, January 1998).
Jones, K.F., 1996. Ice Accretion in Freezing Rain, CRREL Report 96-
2. Cold Regions Research and Engineering Laboratory, Hanover,
NH.
Jones, K.F., 1998. A simple model for freezing rain ice loads. Atmo-
spheric Research 46, 87 – 97.
TIA, 1996. Structural Standards for Steel Antenna Towers and An-
tenna Supporting Structures TIA/EIA-222-F. Telecommunications
Industry Association, Arlington, VA.
K.F. Jones, A.B. Peabody / Cold Regions Science and Technology 44 (2006) 145–148148
