IEC 60865-1:2011

Short-circuit currents - Calculation of effects - Part 1: Definitions and calculation methods

IEC 60865-1:2011 is applicable to the mechanical and thermal effects of short-circuit currents. It contains procedures for the calculation of: the electromagnetic effect on rigid conductors and flexible conductors, the thermal effect on bare conductors. For cables and insulated conductors, reference is made, for example, to IEC 60949 and IEC 60986. For the electromagnetic and thermal effects in d.c. auxiliary installations of power plants and substations reference is made to IEC 61660-2. Only a.c. systems are dealt with in this standard. This third edition cancels and replaces the second edition published in 1993 and constitutes a technical revision. The main changes with respect to the previous edition are:
- The determinations for automatic reclosure together with rigid conductors have been revised.
- The influence of mid-span droppers to the span has been included. For vertical cable-connection the displacement and the tensile force onto the lower fixing point may now be calculated. Additional recommendations for foundation loads due to tensile forces have been added. The subclause for determination of the thermal equivalent short-circuits current has been deleted (it is now part of IEC 60909-0). The regulations for thermal effects of electrical equipment have been deleted. The standard has been reorganized and some of the symbols have been changed to follow the conceptual characteristic of international standards.

Courants de court-circuit - Calcul des effects - Partie 1: Définitions et méthodes de calcul

La CEI 60865-1:2011 s'applique aux effets mécaniques et thermiques des courants de court-circuit. Elle contient les modes opératoires de calcul de: l'effet électromagnétique sur les conducteurs rigides et souples; l'effet thermique sur les conducteurs nus. Pour les câbles et les conducteurs isolés, se référer par exemple à la CEI 60949 et à la CEI 60986. Pour les effets électromagnétiques et thermiques dans les installations auxiliaires alimentées en courant continu dans les centrales et les postes, il est fait référence à la CEI 61660-2. Seuls les systèmes alimentés en courant alternatif sont abordés dans la présente norme. Cette troisième édition annule et remplace la deuxième édition parue en 1993 et constitue une révision technique. Les modifications principales par rapport à l'édition précédente sont les suivantes:
- Les déterminations liées à la refermeture automatique ainsi qu'aux conducteurs rigides ont été révisées.
- L'influence des billettes de support en milieu de portée sur la portée a été incluse. Pour le raccordement de câbles verticaux, le déplacement et la force de traction sur le point de fixation le plus bas peuvent à présent être calculés. Des recommandations supplémentaires liées aux charges pour les fondations dues aux forces de traction ont été ajoutées. Le paragraphe relatif à la détermination du courant de court-circuit thermique équivalent a été supprimé (il fait à présent partie de la CEI 60909-0). Les régulations concernant les effets thermiques du matériel électrique ont été supprimées. La norme a été restructurée et certains symboles ont été modifiés afin de mieux correspondre aux caractéristiques conceptuelles des normes internationales.

General Information

• Status: Published
• Publication Date: 23-Oct-2011
• ICS: 17.220.01 - Electricity. Magnetism. General aspects
• ICS: 29.240.20 - Power transmission and distribution lines
• Technical Committee: TC 73 - Short-circuit currents
• Drafting Committee: MT 2 - TC 73/MT 2
• Current Stage: PPUB - Publication issued
• Start Date: 24-Oct-2011
• Completion Date: 31-Jan-2012

Relations

• Revises: IEC 60865-1:1993 - Short-circuit currents - Calculation of effects - Part 1: Definitions and calculation methods
• Effective Date: 05-Sep-2023

Overview

IEC 60865-1:2011 is an international standard published by the International Electrotechnical Commission (IEC) that details methods for calculating the mechanical and thermal effects of short-circuit currents in alternating current (AC) electrical systems. This third edition updates and replaces the 1993 version, providing revised calculation procedures, enhanced considerations for conductor arrangements, and clarifications on mechanical stresses and thermal impacts.

The standard is essential for engineers and professionals involved in power system design, specifying how to assess electromagnetic forces on conductors and thermal effects on bare conductors during short-circuits. It also addresses the influence of components such as mid-span droppers and automatic reclosing on system behavior. IEC 60865-1 references other related standards for insulated cables and DC auxiliary systems, ensuring comprehensive coverage of various power installation scenarios.

Key Topics

• Short-Circuit Current Effects
  The standard focuses on calculating both electromagnetic and thermal effects caused by short-circuit currents in AC systems. It comprehensively covers rigid and flexible conductor arrangements and their behavior during fault conditions.

• Mechanical Stresses on Conductors
  Methods for determining peak electromagnetic forces acting between main and sub-conductors are provided. It includes formulas to calculate stresses, section modulus, and permissible conductor stress under short-circuit faults.

• Flexible Conductor Behavior
  Special attention is given to calculating tensile forces during short-circuits, accounting for factors like conductor swing, elongation, and sag. The role of mid-span droppers and vertical conductors (droppers) is included to predict accurate tension and displacement effects.

• Thermal Effects Calculation
  The standard explains how to compute the thermal equivalent short-circuit current and the associated rise in conductor temperature. It provides guidance on short-time withstand current density, conductor temperature limits during faults, and the impact of short-circuit duration on thermal strength.

• Automatic Reclosing Considerations
  Revised procedures on how automatic reclosing influences force calculations on rigid conductors ensure that transient effects and repeated fault scenarios are properly evaluated.

• Load Calculations on Structures and Foundations
  The document offers recommendations for calculating loads on electrical structures, insulators, connectors, and foundations due to conductor forces during short-circuits, improving safety and structural integrity assessment.

• Symbol and Terminology Harmonization
  The standard aligns its symbols and conceptual framework with other international IEC standards to maintain consistency across electrotechnical documentation.

Applications

• Power Plant and Substation Design
  IEC 60865-1 is critical for electrical engineers designing and analyzing busbar systems, overhead lines, and cable connections in substations to ensure they can withstand mechanical and thermal stresses during faults.

• Electrical Equipment Testing
  Provides manufacturers and testers with precise calculation methods to verify mechanical strength and thermal endurance of conductors under short-circuit conditions.

• Structural Engineering for Electrical Installations
  Enables calculation of conductor-induced stresses on poles, towers, insulators, and foundations, facilitating proper structural design and safety compliance.

• Reliability Analysis in Power Systems
  Helps operators and planners model the impact of short-circuits, including reclosing strategies, to optimize system robustness and fault handling.

• Safety Standards Compliance
  Ensures adherence to international safety and performance standards by providing standardized calculation procedures for mechanical and thermal effects of fault currents.

Related Standards

• IEC 60949 – Calculation of the thermal rating of cables for short-circuits.
• IEC 60986 – Additional methods related to cables and insulated conductors under short-circuit conditions.
• IEC 60909-0 – Calculation of short-circuit currents in three-phase AC systems, including the thermal equivalent short-circuit current procedure now removed from IEC 60865-1.
• IEC 61660-2 – Guidelines for electromagnetic and thermal effects in DC auxiliary installations of power plants and substations.

---

Keywords: IEC 60865-1, short-circuit currents, calculation methods, mechanical effects, thermal effects, conductor stresses, rigid conductors, flexible conductors, automatic reclosing, power system design, electrical standards, conductor tensile forces, thermal short-circuit strength, AC power systems.

Standards Content (Sample)

IEC 60865-1 ®
Edition 3.0 2011-10
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
Short-circuit currents – Calculation of effects –
Part 1: Definitions and calculation methods

Courants de court-circuit – Calcul des effects –
Partie 1: Définitions et méthodes de calcul

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IEC 60865-1 ®
Edition 3.0 2011-10
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
Short-circuit currents – Calculation of effects –
Part 1: Definitions and calculation methods

Courants de court-circuit – Calcul des effects –
Partie 1: Définitions et méthodes de calcul

INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
COMMISSION
ELECTROTECHNIQUE
PRICE CODE
INTERNATIONALE
CODE PRIX XA
ISBN 978-2-88912-771-9
ICS 17.220.01; 29.240.20
– 2 – 60865-1  IEC:2011
CONTENTS
FOREWORD . 4
1 Scope . 6
2 Normative references . 6
3 Terms, definitions, symbols and units . 7
3.1 Terms and definitions . 7
3.2 Symbols and units . 9
4 General . 12
5 Rigid conductor arrangements . 13
5.1 General . 13
5.2 Calculation of electromagnetic forces . 13
5.2.1 Calculation of peak force between the main conductors during a
three-phase short-circuit . 13
5.2.2 Calculation of peak force between the main conductors during a line-
to-line short-circuit . 13
5.2.3 Calculation of peak value of force between coplanar sub-conductors . 14
5.3 Effective distance between main conductors and between sub-conductors . 14
5.4 Calculation of stresses in rigid conductors . 16
5.4.1 Calculation of stresses . 16
5.4.2 Section modulus and factor q of main conductor composed of sub-
conductors . 17
5.4.3 Permitted conductor stress . 20
5.5 Structure loads due to rigid conductors . 21
5.6 Consideration of automatic reclosing . 21
5.7 Calculation with special regard to conductor oscillation . 22
5.7.1 General . 22
5.7.2 Determination of relevant natural frequency . 23
5.7.3 The factors V , V , V , V and V . 23
F σm σs rm rs
6 Flexible conductor arrangements . 26
6.1 General . 26
6.2 Effects on horizontal main conductors . 27
6.2.1 General . 27
6.2.2 Characteristic dimensions and parameter . 27
6.2.3 Tensile force F during short-circuit caused by swing out (short-
t,d
circuit tensile force) without dropper in midspan . 30
6.2.4 Dynamic change of sag due to elongation of conductor and change of
shape of the conductor curve . 31
6.2.5 Tensile force F during short-circuit caused by swing out (short-
t,d
circuit tensile force) with dropper in the middle of the span . 32
6.2.6 Tensile force F after short-circuit caused by drop (drop force) . 33
f,d
6.2.7 Horizontal span displacement b and minimum air clearance a . 33
h min
6.3 Effects on vertical main conductors (droppers) . 34
6.4 Effects on bundled conductors . 35
6.4.1 Characteristic dimensions and parameter . 35
6.4.2 Tensile force F in the case of clashing sub-conductors . 38
pi,d
6.4.3 Tensile force F in the case of non-clashing sub-conductors . 38
pi,d
6.5 Structure loads due to flexible conductors . 41
6.5.1 Design load for post insulators, their supports and connectors . 41

60865-1  IEC:2011 – 3 –
6.5.2 Design load for structures, insulators and connectors with tensile
forces transmitted by insulator chains . 41
6.5.3 Design load for foundations . 42
7 The thermal effect on bare conductors . 42
7.1 General . 42
7.2 Calculation of thermal equivalent short-circuit current . 42
7.3 Calculation of temperature rise and rated short-time withstand current
density for conductors . 43
7.4 Calculation of thermal short-time strength for different durations of the short-
circuit . 44
Annex A (normative) Equations for calculation of diagrams . 46
Bibliography . 51

Figure 1 – Factor k for calculating the effective conductor distance . 15
1s
Figure 2 – Loading direction and bending axis for multiple conductor arrangements . 18
Figure 3 – Factor e for the influence of connecting pieces in Equation (17) . 24
Figure 4 – Factors V , V and V to be used with the three-phase and line-to-line
F σm σs
short-circuits . 25
Figure 5 – Factors V and V to be used with three-phase automatic reclosing . 26
rm rs
Figure 6 – Maximum swing out angle δ for a given maximum short-circuit duration T . 30
max k1
Figure 7 – Factor ψ for tensile force in flexible conductors . 31
Figure 8 – Geometry of a dropper . 33
Figure 9 – ν as a function of ν . 37
2 1
180°
Figure 10 – ν ·sin as a function of a /d . 37
3 s
n
Figure 11 – ξ as a function of j and ε . 38
st
Figure 12 – η as a function of j and ε . 41
st
Figure 13 – Relation between rated short-circuit withstand current density (T = 1 s)
kr
and conductor temperature . 44

Table 1 – Effective distance a between sub-conductors for rectangular cross-section
s
dimensions . 16
Table 2 – Maximum possible values of V V , V V , V V . 19
σm rm σs rs F rm
Table 3 – Factors α, β, γ for different busbar support arrangements . 20
Table 4 – Factor q . 22
Table 5 – Section moduli W of main conductors with two or more stiffening elements
m
between two adjacent supports. The stiffening elements are black. . 22
Table 6 – Recommended highest temperatures for mechanically stressed conductors
during a short-circuit . 43

– 4 – 60865-1  IEC:2011
INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
SHORT-CIRCUIT CURRENTS –
CALCULATION OF EFFECTS –
Part 1: Definitions and calculation methods

FOREWORD
1) The International Electrotechnical Commission (IEC) is a worldwide organization for standardization comprising
all national electrotechnical committees (IEC National Committees). The object of IEC is to promote
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8) Attention is drawn to the Normative references cited in this publication. Use of the referenced publications is
indispensable for the correct application of this publication.
9) Attention is drawn to the possibility that some of the elements of this IEC Publication may be the subject of
patent rights. IEC shall not be held responsible for identifying any or all such patent rights.
International Standard IEC 60865-1 has been prepared by IEC technical committee 73: Short-
circuit currents.
This third edition cancels and replaces the second edition published in 1993. This edition
constitutes a technical revision.
The main changes with respect to the previous edition are listed below:
• The determinations for automatic reclosure together with rigid conductors have been
revised.
• The influence of mid-span droppers to the span has been included.
• For vertical cable-connection the displacement and the tensile force onto the lower fixing
point may now be calculated.
• Additional recommendations for foundation loads due to tensile forces have been added.

60865-1  IEC:2011 – 5 –
• The subclause for determination of the thermal equivalent short-circuits current has been
deleted (it is now part of IEC 60909-0).
• The regulations for thermal effects of electrical equipment have been deleted.
• The standard has been reorganized and some of the symbols have been changed to
follow the conceptual characteristic of international standards.
The text of this standard is based on the following documents:
CDV Report on voting
73/152/CDV 73/153/RVC
Full information on the voting for the approval of this standard can be found in the report on
voting indicated in the above table.
This publication has been drafted in accordance with the ISO/IEC Directives, Part 2.
A list of all parts of the IEC 60865 series, under the general title, Short-circuit currents –
Calculation of effects can be found on the IEC website.
The committee has decided that the contents of this publication will remain unchanged until
the stability date indicated on the IEC web site under "http://webstore.iec.ch" in the data
related to the specific publication. At this date, the publication will be
• reconfirmed,
• withdrawn,
• replaced by a revised edition, or
• amended.
– 6 – 60865-1  IEC:2011
SHORT-CIRCUIT CURRENTS –
CALCULATION OF EFFECTS –
Part 1: Definitions and calculation methods

1 Scope
This part of IEC 60865 is applicable to the mechanical and thermal effects of short-circuit
currents. It contains procedures for the calculation of
– the electromagnetic effect on rigid conductors and flexible conductors,
– the thermal effect on bare conductors.
For cables and insulated conductors, reference is made, for example, to IEC 60949 and
IEC 60986. For the electromagnetic and thermal effects in d.c. auxiliary installations of power
plants and substations reference is made to IEC 61660-2.
Only a.c. systems are dealt with in this standard.
The following points should, in particular, be noted:
a) The calculation of short-circuit currents should be based on IEC 60909. For the
determination of the greatest possible short-circuit current, additional information from
other IEC standards may be referred to, e.g. details about the underlying circuitry of the
calculation or details about current-limiting devices, if this leads to a reduction of the
mechanical stress.
b) Short-circuit duration used in this standard depends on the protection concept and should
be considered in that sense.
c) These standardized procedures are adjusted to practical requirements and contain
simplifications which are conservative. Testing or more detailed methods of calculation or
both may be used.
d) In Clause 5 of this standard, for arrangements with rigid conductors, only the stresses
caused by short-circuit currents are calculated. Furthermore, other stresses can exist, e.g.
caused by dead-load, wind, ice, operating forces or earthquakes. The combination of
these loads with the short-circuit loading should be part of an agreement and/or be given
by standards, e.g. erection-codes.
The tensile forces in arrangements with flexible conductors include the effects of dead-
load. With respect to the combination of other loads the considerations given above are
valid.
e) The calculated loads are design loads and should be used as exceptional loads without
any additional partial safety factor according to installation codes of, for example,
IEC 61936-1 [1] .
2 Normative references
The following referenced documents are indispensable for the application of this document.
For dated references, only the edition cited applies. For undated references, the latest edition
of the referenced document (including any amendments) applies.
IEC 60909 (all parts) Short-circuit current calculation in three-phase a.c. systems
—————————
Figures in square brackets refer to the bibliography.

60865-1  IEC:2011 – 7 –
IEC 60909-0, Short-circuit currents in three-phase a.c. systems – Part 0: Calculation of
currents
IEC 60949, Calculation of thermally permissible short-circuit currents, taking into account
non-adiabatic heating effects
IEC 60986, Short-circuit temperature limits of electric cables with rated voltages from 6 kV
(U = 7,2 kV) up to 30 kV (U = 36 kV)
m m
IEC 61660-2, Short-circuit currents in d.c. auxiliary installations in power plants and
substations – Part 2: Calculation of effects
3 Terms, definitions, symbols and units
3.1 Terms and definitions
For the purposes of this document the following terms and definitions apply.
3.1.1
main conductor
conductor or arrangement composed of a number of conductors which carries the total current
in one phase
3.1.2
sub-conductor
single conductor which carries a certain part of the total current in one phase and is a part of
the main conductor
3.1.3
fixed support
support of a rigid conductor in which moments are imposed in the regarded plane
3.1.4
simple support
support of a rigid conductor in which no moments are imposed in the regarded plane
3.1.5
connecting piece
any additional mass within a span which does not belong to the uniform conductor material,
includingamong others, spacers, stiffening elements, bar overlappings, branchings, etc.
3.1.6
spacer
mechanical element between sub-conductors, rigid or flexible, which, at the point of
installation, maintains the clearance between sub-conductors
3.1.7
stiffening element
special spacer intended to reduce the mechanical stress of rigid conductors
3.1.8
relevant natural frequency
f
cm
first natural frequency of the free vibration of a single span beam without damping and natural
frequency of order ν of beams with ν spans without damping

– 8 – 60865-1  IEC:2011
3.1.9
short-circuit tensile force
F
t,d
maximum tensile force (design value) in a flexible main conductor due to swing out reached
during the short-circuit
3.1.10
drop force
F
f,d
maximum tensile force (design value) in a flexible main conductor which occurs when the
span drops down after swing out
3.1.11
pinch force
F
pi,d
maximum tensile force (design value) in a bundled flexible conductor during the short-circuit
due to the attraction of the sub-conductors in the bundle
3.1.12
duration of the first short-circuit current flow
T
k1
time interval between the initiation of the short-circuit and the first breaking of the current
3.1.13
thermal equivalent short-circuit current
I
th
r.m.s. value of current having the same thermal effect and the same duration as the actual
short-circuit current, which can contain d.c. component and can subside in time
3.1.14
thermal equivalent short-circuit current density
S
th
ratio of the thermal equivalent short-circuit current and the cross-section area of the
conductor
3.1.15
rated short-time withstand current density, S , for conductors
thr
r.m.s. value of the current density which a conductor is able to withstand for the rated short
time
3.1.16
duration of short-circuit current
T
k
sum of the time durations of the short-circuit current flow from the initiation of the first short-
circuit to the final breaking of the current in all phases
3.1.17
rated short-time
T
kr
time duration for which a conductor can withstand a current density equal to its rated short-
time withstand current density

60865-1  IEC:2011 – 9 –
3.2 Symbols and units
All equations used in this standard are quantity equations in which quantity symbols represent
physical quantities possessing both numerical values and dimensions.
The symbols used in this standard and the SI-units concerned are given in the following lists.
A Cross-section of one main-conductor m

A Cross-section of one sub-conductor m
s
a Centre-line distance between conductors m
a Effective distance between main conductors m
m
a Minimum air clearance m
min
a Effective distance between sub-conductors m
s
a Centre-line distance between sub-conductor 1 and m
1n
sub-conductor n
a Centre-line distance between sub-conductors m
1s
b Maximum horizontal displacement m
h
b Dimension of a main conductor perpendicular to the direction of m
m
the force
b Dimension of a sub-conductor perpendicular to the direction m
s
of the force
C Dilatation factor   1
D
C Form factor 1
F
c Dimension of a main conductor in the direction of the force m
m
c Dimension of a sub-conductor in the direction of the force m
s
4 2
c Material constant m /(A s)
th
d Outer diameter of a tubular or flexible conductor m
E Young's modulus N/m
E Actual Young's modulus N/m
eff
e Factor for the influence of connecting pieces 1
F Force acting between two parallel long conductors during a short- N
circuit
Characteristic electromagnetic force per unit length on flexible N/m
F′
main conductors
F Force between main conductors during a short-circuit N
m
F Force between main conductors during a line-to-line short-circuit N
m2
F Force on the central main conductor during a balanced three- N
m3
phase short-circuit
F Force on support of rigid conductors (peak value, design value) N
r,d
F Drop force of one main conductor (design value) N
f,d
F Pinch force of one main conductor (design value) N
pi,d
F Force between sub-conductors during a short-circuit N
s
F Static tensile force of one flexible main conductor N
st
F Short-circuit tensile force of one main conductor (design value) N
t,d
– 10 – 60865-1  IEC:2011
F Short-circuit current force between the sub-conductors in a bundle N
ν
f System frequency Hz
f Relevant natural frequency of a main conductor Hz
cm
f Relevant natural frequency of a sub-conductor Hz
cs
f Dynamic conductor sag at midspan m
ed
f Equivalent static conductor sag at midspan m
es
f Static conductor sag at midspan m
st
f Stress corresponding to the yield point N/m
y
g Conventional value of acceleration of gravity m/s
h Height of the dropper m
′′ Initial symmetrical three-phase short-circuit current (r.m.s.) A
I
k
′′ Initial line-to-earth short-circuit current (r.m.s.) A
I
k1
′′ Initial symmetrical line-to-line short-circuit current (r.m.s.) A
I
k2
I Thermal equivalent short-circuit current A
th
i Peak short-circuit current A
p
i Peak short-circuit current in case of a line-to-line short-circuit A
p2
i , i Instantaneous values of the currents in the conductors A
1 2
J Second moment of main conductor area m
m
J Second moment of sub-conductor area m
s
j Parameter determining the bundle configuration during short- 1
circuit current flow
k Number of sets of spacers or stiffening elements 1
k Factor for the effective distance between sub-conductor 1 and 1
1n
sub-conductor, n
k Factor for effective conductor distance 1
1s
l Centre-line distance between supports m
l Cord length of a flexible main conductor in the span m
c
l Length of one insulator chain m
i
l Centre-line distance between connecting pieces or between one m
s
connecting piece and the adjacent support
l Cord length of a dropper m
v
′ Mass per unit length of main conductor kg/m
m
m
′ Mass per unit length of one sub-conductor kg/m
m
s
m Total mass of one set of connecting pieces kg
z
N Stiffness norm of an installation with flexible conductors 1/N
n Number of sub-conductors of a main conductor  1
q Factor of plasticity 1
r The ratio of electromechanic force on a conductor under short- 1
circuit conditions to gravity
S Resultant spring constant of both supports of one span N/m

S Thermal equivalent short-circuit current density A/mm
th
60865-1  IEC:2011 – 11 –
S Rated short-time withstand current density A/mm
thr
T Period of conductor oscillation s
T Duration of short-circuit current s
k
T Duration of short-circuit i at repeating short-circuits s
ki
T Rated short-time s
kr
T Duration of the first short-circuit current flow s
k1
T Resulting period of the conductor oscillation during the short- s
res
circuit current flow
t Wall thickness of tubes m
V Ratio of dynamic and static force on supports 1
F
V Ratio of dynamic stress (forces on the supports, contribution of 1
rm
main conductor bending stress) caused by forces between main
conductors with unsuccessful three-phase automatic reclosing and
dynamic stress with successful three-phase automatic reclosing
V Ratio of contribution of dynamic stress caused by forces between 1
rs
sub-conductors with unsuccessful three-phase automatic reclosing
and contribution of dynamic stress with successful three-phase
automatic reclosing
V Ratio of dynamic and static contribution of main conductor stress 1
σm
V Ratio of dynamic and static contribution of sub-conductor stress 1
σs
W Section modulus of main conductor m
m
W Section modulus of sub-conductor m
s
w Width of dropper m
Factor for force on support 1
α
β Factor for main conductor stress 1
Factor for relevant natural frequency estimation 1
γ
Actual maximum swing-out angle due to the limitation of the swing- degrees
δ
out movement by the dropper
Swing-out angle at the end of the short-circuit current flow degrees
δ
end
Maximum swing-out angle degrees
δ
max
δ Angular direction of the force degrees
Elastic expansion 1
ε
ela
Strain factor of the bundle contraction 1
ε , ε
pi st
Thermal expansion 1
ε
th
Stress factor of the flexible main conductor 1
ζ
Factor for calculating F in the case of non-clashing sub- 1
η
pi,d
conductors
Conductor temperature of the beginning of a short-circuit °C
θ
b
Conductor temperature at the end of a short-circuit °C
θ
e
Factor for the calculation of the peak short-circuit current 1
κ
µ
Magnetic constant, permeability of vacuum H/m
Number of spans of a continuous beam 1
ν
– 12 – 60865-1  IEC:2011
ν , ν , ν , Factors for calculating F
e 1 2 pi,d
ν , ν ,
3 4
Factor for calculating F in the case of clashing sub-conductors 1
ξ
pi,d
Lowest value of cable stress when Young's modulus becomes N/m
σ
fin
constant
σ Bending stress caused by the forces between main conductors N/m
m,d
(design value)
σ Bending stress caused by the forces between sub-conductors N/m
s,d
(design value)
σ Total conductor stress (design value) N/m
tot,d
χ Quantity for the maximum swing-out angle 1
Factors for the tensile force in a flexible conductor 1
ϕ, ψ
4 General
With the calculation methods presented in this standard
• stresses in rigid conductors,
• tensile forces in flexible conductors,
• forces on insulators and substructures, which might expose them to bending, tension
and/or compression,
• span displacements of flexible conductors and
• heating of conductors
can be estimated.
Electromagnetic forces are induced in conductors by the currents flowing through them.
Where such electromagnetic forces interact on parallel conductors, they cause stresses that
have to be taken into account at the substations. For this reason:
• the forces between parallel conductors are set forth in the following clauses;
• the electromagnetic force components set up by conductors with bends and/or cross-overs
may normally be disregarded.
In the case of metal-clad systems, the change of the electromagnetic forces between the
conductors due to magnetic shielding can be taken into account. In addition, however, the
forces acting between each conductor and its enclosure and between the enclosures shall be
considered.
When parallel conductors are long compared to the distance between them, the forces will be
evenly distributed along the conductors and are given by Equation (1)
µ l
F = ii (1)
2π a
where
i and i are the instantaneous values of the currents in the conductors;
1 2
l is the centre-line distance between the supports;
a is the centre-line distance between the conductors.

60865-1  IEC:2011 – 13 –
When the currents in the two conductors have the same direction, the forces are attractive.
When the directions of the currents are opposite, the forces are repulsive.
5 Rigid conductor arrangements
5.1 General
Rigid conductors can be supported in different ways, either fixed or simple or in a combination
of both. Depending on the type of support and the number of supports, the stresses in the
conductors and the forces on the supports will be different for the same short-circuit current.
The equations given also include the elasticity of the supports.
The stresses in the conductors and the forces on the supports also depend on the ratio
between the relevant natural frequency of the mechanical system and the electrical system
frequency. For example, in the case of resonance or near to resonance, the stresses and
forces in the system can be amplified. If f / f < 0,5 the response of the system decreases
cm
and the maximum stresses are in the outer phases.
5.2 Calculation of electromagnetic forces
5.2.1 Calculation of peak force between the main conductors during a three-phase
short-circuit
In a three-phase system with the main conductors arranged with the same centre-line
distances on the same plane, the maximum force acts on the central main conductor during a
three-phase short-circuit and is given by:
µ 3 l
0 2
Fi= (2)
m3 p
22π a
m
where
i is the peak value of the short-circuit current in the case of a balanced three-phase
p
short-circuit. For the calculation, see the IEC 60909 series;
l is the maximum centre-line distance between adjacent supports;
a is the effective distance between main conductors in 5.3.
m
NOTE Equation (2) can also be used for calculating the resulting peak force when conductors with circular cross-
sections are in the corners of an equilateral triangle and where a is the length of the side of the triangle.
m
5.2.2 Calculation of peak force between the main conductors during a line-to-line
short-circuit
The maximum force acting between the conductors carrying the short-circuit current during a
line-to-line short-circuit in a three-phase system or in a two-line single-phase-system is given
by:
µ
l
Fi= (3)
m2 p2
2π a
m
where
i is the peak short-circuit current in the case of a line-to-line short-circuit;
p2
l is the maximum centre-line distance between adjacent supports;
a is the effective distance between main conductors in 5.3.
m
– 14 – 60865-1  IEC:2011
5.2.3 Calculation of peak value of force between coplanar sub-conductors
The maximum force acts on the outer sub-conductors and is between two adjacent connecting
pieces given by:
 i 
µ l
p
0s
F = (4)
 
s
 
2π na
  s
where
n is the number of sub-conductors;
l is the maximum existing centre-line distance between two adjacent connecting pieces;
s
a is the effective distance between sub-conductors;
s
i is equal to i for a three-phase system or to i for a two-line single-phase system.
p p p2
5.3 Effective distance between main conductors and between sub-conductors
The forces between conductors carrying short-circuit currents depend on the geometrical
configuration and the profile of the conductors. For this reason the effective distance a
m
between main conductors has been introduced in 5.2.1 and 5.2.2 and the effective distance a
s
between sub-conductors in 5.2.3. They shall be taken as follows:
Effective distance a between coplanar main conductors with the centre-line distance a:
m
• Main conductors consisting of single circular cross-sections:
a = a (5)
m
• Main conductors consisting of single rectangular cross-sections and main conductors
composed of sub-conductors with rectangular cross-sections:
a
a = (6)
m
k
k shall be taken from Figure 1, with a = a, b = b and c = c .
12 1s s m s m
Effective distance a between the n coplanar sub-conductors of a main conductor:
s
• Sub-conductors with circular cross-sections:
11 1 1 1 1
+ + ++++ (7)
aa a a a a
s 12 13 14 1s 1n
• Sub-conductors with rectangular cross-sections:
Some values for a are given in Table 1. For other distances and sub-conductor
s
dimensions the equation
1 k k k k k
12 13 14 1s 1n
+ + ++ ++ (8)
aa a a a a
s 12 13 14 1s 1n
can be used. The values for k ,.k shall be taken from Figure 1.
12 1n
=
=
60865-1  IEC:2011 – 15 –
IEC  2481/11
Figure 1 – Factor k for calculating the effective conductor distance
1s
For programming, the Equation is given in Clause A.2.

– 16 – 60865-1  IEC:2011
Table 1 – Effective distance a between sub-conductors
s
a
for rectangular cross-section dimensions
b
s
Rectangular cross sections 0,04 0,05 0,06 0,08 0,10 0,12 0,16 0,20
c
s
0,005 0,020 0,024 0,027 0,033 0,040 – – –

0,010 0,028 0,031 0,034 0,041 0,047 0,054 0,067 0,080

0,005 – 0,013 0,015 0,018 0,022 – – –

0,010 0,017 0,019 0,020 0,023 0,027 0,030 0,037 0,043

0,005 – – – – – – – –
0,010 0,014 0,015 0,016 0,018 0,020 0,022 0,026 0,031

0,005 – 0,014 0,015 0,018 0,020 – – –

0,010 0,017 0,018 0,020 0,022 0,025 0,027 0,032 –

a
All dimensions are given in metres.

5.4 Calculation of stresses in rigid conductors
5.4.1 Calculation of stresses
Conductors have to be fixed in a way that axial forces can be disregarded. Under this
assumption the forces acting are bending forces and the general equation for the bending
stress caused by the forces between main conductors is given by:
Fl
m
σβ= VV (9)
m,d σm rm
8W
m
where
F is either the value F of three-phase systems according to Equation (2) or F of two-
m m3 m2
line single-phase systems according to Equation (3);
W is the section modulus of the main conductor and shall be calculated with respect to
m
the direction of forces between main conductors.

60865-1  IEC:2011 – 17 –
The bending stress caused by the forces between sub-conductors is given by:
Fl
ss
σ = VV (10)
s,d σs rs
16W
s
where
F according to Equation (4) shall be used;
s
W is the section modulus of the sub-conductor and shall be calculated with respect to the
s
direction of forces between sub-conductors.
V , V , V and V are factors which take into account the dynamic phenomena, and β is a
σm σs rm rs
factor depending on the type and the number of supports. The maximum possible values of
V V and V V shall be taken from Table 2 and the factorβ shall be taken from Table 3.
σm rm σs rs
NOTE The factor β describes the reduction of the bending stress at the place of its supports, taking into account
the plastic deformation of the conductor (see Table 3).
Non-uniform spans in continuous beams may be treated, with sufficient degree of accuracy by
assuming the maximum span applied throughout. This means that
• the end supports are not subjected to greater stress than the inner ones,
• span lengths less than 20 % of the adjacent ones shall be avoided. If that does not prove
to be possible, the conductors shall be decoupled using flexible joints at the supports. If
there is a flexible joint within a span, the length of this span should be less than 70 % of
the lengths of the adjacent spans.
If it is not evident whether a beam is supported or fixed, the worst case shall be taken into
account.
For further consideration, see 5.7
5.4.2 Section modulus and factor q of main conductor composed of sub-conductors
The bending stress and, consequently, the mechanical withstand of the conductor, depends
on the section modulus.
If the stress occurs in accordance with Figure 2a, the section modulus W is independent of
m
the number of connecting pieces and is equal to the sum of the section moduli W of the sub-
s
conductors (W with respect to the axis x-x). The factor q has then the value 1,5 for
s
rectangular cross-sections and 1,19 for U and I sections.
If the stress occurs in accordance with Figure 2b and in the case there is only one or no
stiffening element within a supported distance, the section modulus W is equal to the sum of
m
of the sub-conductors (W with respect to the axis y-y). The factor q has
the section moduli W
s s
then the value 1,5 for rectangular cross-sections and 1,83 for U and I sections.
When, within a supported distance, there are two or more stiffening elements, higher values
of section moduli may be used:
• for main conductors composed of sub-conductors of rectangular cross-sections with a
space between the bars equal to the bar thickness, the section moduli are given in
Table 5;
• for conductor groups having U and I cross-sections, 50 % of the section moduli with
respect to the axis 0-0 should be used.

– 18 – 60865-1  IEC:2011
The factor q then has a value of 1,5 for rectangular cross-section and 1,83 for U and I
sections.
IEC  2482/11
Figure 2a – Loading along the surface

IEC  2483/11
Figure 2b – Loading perpendicular to the surface
Figure 2 – Loading direction and bending axis
for multiple conductor arrangements

60865-1  IEC:2011 – 19 –
Table 2 – Maximum possible values of V V , V V , V V
σm rm σs rs F rm
System
Type of Without three-
With three-phase With and without three-phase
short- phase automatic
automatic reclosing automatic reclosing
circuit reclosing
V V ,V V V V , V V V V
σm rm σs rs σm rm σs rs F rm
First Second
current current
flow flow
range
σ
tot,d
2,0 for ≤ 0,5 1
0,8 f
y
0,8 f σ
y
t
...