EN 14025:2003
(Main)Tanks for the transport of dangerous goods - Metallic pressure tanks - Design and construction
Tanks for the transport of dangerous goods - Metallic pressure tanks - Design and construction
This standard deals with the design and construction of metallic pressure tanks for the transport of dangerous goods, including their openings, closures and structures but excluding service equipment.
Tanks für die Beförderung gefährlicher Güter - Drucktanks aus Metall - Auslegung und Bau
Citernes destinées au transport de matières dangereuses - Citernes métalliques sous pression - Conception et fabrication
La présente Norme européenne spécifie les exigences minimales en matière de conception et de fabrication des citernes métalliques sous pression ayant une pression maximale de service supérieure à 50 kPa (0,5 bar) utilisées pour le transport de matières dangereuses par route et par chemin de fer. La présente norme inclut les exigences relatives à leurs ouvertures, fermetures et structures, à l'exclusion toutefois des exigences relatives aux équipements de service. Pour les citernes utilisées pour le transport de GPL, voir l'EN 12493, pour les citernes utilisées pour le transport de liquides cryogéniques, voir les EN 13530-1 et EN 13530-2.
Posode za prevoz nevarnih snovi – Kovinske tlačne cisterne – Konstruiranje in izdelava
General Information
Relations
Standards Content (Sample)
SLOVENSKI STANDARD
SIST EN 14025:2004
01-junij-2004
3RVRGH]DSUHYR]QHYDUQLKVQRYL±.RYLQVNHWODþQHFLVWHUQH±.RQVWUXLUDQMHLQ
L]GHODYD
Tanks for the transport of dangerous goods - Metallic pressure tanks - Design and
construction
Tanks für die Beförderung gefährlicher Güter - Drucktanks aus Metall - Auslegung und
Bau
Citernes destinées au transport de matieres dangereuses - Citernes métalliques sous
pression - Conception et fabrication
Ta slovenski standard je istoveten z: EN 14025:2003
ICS:
13.300 Varstvo pred nevarnimi Protection against dangerous
izdelki goods
23.020.20 Posode in vsebniki, montirani Vessels and containers
na vozila mounted on vehicles
SIST EN 14025:2004 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
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EUROPEAN STANDARD
EN 14025
NORME EUROPÉENNE
EUROPÄISCHE NORM
November 2003
ICS 13.300; 23.020.20
English version
Tanks for the transport of dangerous goods - Metallic pressure
tanks - Design and construction
Citernes destinées au transport de matières dangereuses - Tanks für die Beförderung gefährlicher Güter - Drucktanks
Citernes métalliques sous pression - Conception et aus Metall - Auslegung und Bau
fabrication
This European Standard was approved by CEN on 25 September 2003.
CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European
Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such national
standards may be obtained on application to the Management Centre or to any CEN member.
This European Standard exists in three official versions (English, French, German). A version in any other language made by translation
under the responsibility of a CEN member into its own language and notified to the Management Centre has the same status as the official
versions.
CEN members are the national standards bodies of Austria, Belgium, Czech Republic, Denmark, Finland, France, Germany, Greece,
Hungary, Iceland, Ireland, Italy, Luxembourg, Malta, Netherlands, Norway, Portugal, Slovakia, Spain, Sweden, Switzerland and United
Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION
EUROPÄISCHES KOMITEE FÜR NORMUNG
Management Centre: rue de Stassart, 36 B-1050 Brussels
© 2003 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN 14025:2003 E
worldwide for CEN national Members.
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EN 14025:2003 (E)
Contents
page
Foreword.3
1 Scope .4
2 Normative references .4
3 Definitions and Symbols .5
3.1 Definition.5
3.2 Symbols .5
4 Materials.6
4.1 General.6
4.2 Compatibility .6
5 Design .6
5.1 General.6
5.2 Reduction of shell thickness .7
5.3 Protection of the shell .7
6 Calculation.7
6.1 General.7
6.2 Design criteria .8
6.3 Calculation for internal pressure.9
6.4 Calculation for external pressure.25
6.5 Tank to frame connection/interface .26
7 Construction and Manufacturing .26
7.1 General Requirements.26
7.2 Cutting.27
7.3 Forming.27
7.4 Welding .28
7.5 Manufacturing tolerances .30
8 Repairs.31
8.1 General.31
8.2 Repair of surface defects in the parent metal.31
8.3 Repair of weld defects.31
Annex A (informative) RID/ADR references (edition 2003).32
A.1 Introduction .32
A.2 Tank-vehicles (Extract from chapter 6.8 ADR) , Tank-containers (Extract from chapter 6.8
RID/ADR) and Tank-wagons (Extract from chapter 6.8 RID) .32
Annex B (informative) Sample calculation.41
B.1 Introduction .41
B.2 Dimensions, characteristics of materials, operating and testing conditions.41
B.3 Calculation according to branch A .42
B.4 Calculation according to branch B .42
B.5 Calculation according to branch C .42
B.6 Calculation according to branch D .44
B.7 Results .50
Bibliography .52
2
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EN 14025:2003 (E)
Foreword
This document (EN 14025:2003) has been prepared by Technical Committee CEN/TC 296 “Tanks for the transport
of dangerous goods”, the secretariat of which is held by AFNOR.
This European Standard shall be given the status of a national standard, either by publication of an identical text or
by endorsement, at the latest by April 2004, and conflicting national standards shall be withdrawn at the latest by
April 2004.
This European Standard has been prepared under a mandate given to CEN by the European Commission and the
European Free Trade Association, and supports the objectives of the framework Directives on Transport of
Dangerous goods.
The standard is submitted for reference into the RID and/or in the technical annexes of the ADR.
Therefore the standards listed in the normative references and covering basic requirements of the RID/ADR not
addressed within the present standard are normative only when the standards themselves are referred to in the
RID and/or in the technical annexes of the ADR.
Annexes A and B are informative.
According to the CEN/CENELEC Internal Regulations, the national standards organizations of the following
countries are bound to implement this European Standard: Austria, Belgium, Czech Republic, Denmark, Finland,
France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Luxembourg, Malta, Netherlands, Norway, Portugal,
Slovakia, Spain, Sweden, Switzerland and the United Kingdom.
3
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EN 14025:2003 (E)
1 Scope
This European Standard specifies the minimum requirements for the design and construction of metallic pressure
tanks having a maximum working pressure exceeding 50 kPa (0,5 bar) for the transport of dangerous goods by road
and rail. This standard includes requirements for openings, closures and structural equipment; it does not cover
requirements of service equipment. For road tankers for the transport of LPG see EN 12493. For tanks for the
transport of cryogenic liquids see EN 13530-1 and -2.
2 Normative references
This European Standard incorporates by dated or undated reference, provisions from other publications. These
normative references are cited at the appropriate places in the text and the publications are listed hereafter. For
dated references, subsequent amendments to or revisions of any of these publications apply to this European
Standard only when incorporated in it by amendment or revision. For undated references the latest edition of the
publication referred to applies (including amendments).
EN 287–1, Approval testing of welders - Fusion welding - Part 1: Steels.
EN 287–2, Approval testing of welders - Fusion welding - Part 2: Aluminium and aluminium alloys.
EN 288, Specification and qualification of welding procedures for metallic materials — Parts 1 - 9
EN 473, Non destructive testing — Qualification and certification of NDT personnel — General principles.
EN 729-1, Quality requirements for welding – Fusion welding of metallic materials – Part 1: Guidelines for selection
and use.
EN 729-2, Quality requirements for welding – Fusion welding of metallic materials–- Part 2: Comprehensive quality
requirements.
EN 729-3, Quality requirements for welding – Fusion welding of metallic materials–- Part 3: Standard quality
requirements.
EN 970, Non-destructive examination of fusion welds — Visual examination.
EN 1435, Non-destructive examination of welds — Radiographic examination of welded joints.
EN 1591-1, Flanges and their joints — Design rules for gasketed circular flange connections — Part 1: Calculation
method.
EN 1708–1, Welding — Basic weld joint details in steel — Part 1: Pressurized components.
EN 1714, Non destructive examination of welds — Ultrasonic examination of welded joints.
EN 12285-1 : 2003, Workshop fabricated steel tanks - Part 1: Horizontal cylindrical single skin and double skin
tanks for the underground storage of flammable and non-flammable water polluting liquids
EN 12493, Welded steel tanks for liquefied petroleum gas (LPG) — Road tankers — Design and manufacture.
EN 12561–6, Railway applications – Tank wagons – Part 6: Manholes.
prEN 13094 : 2000, Tanks for the transport of dangerous goods — Low-pressure metallic tanks — Design and
construction.
EN 13445-2 : 2002, Unfired pressure vessels — Part 2: Materials.
EN 13445-3 : 2002, Unfired pressure vessels — Part 3: Design.
EN 13445-4, Unfired pressure vessels — Part 4: Fabrication.
4
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EN 14025:2003 (E)
EN 25817, Arc-welded joints in steel - Guidance on quality levels for imperfections (ISO 5817:1992).
EN 30042, Arc-welded joints in aluminium and its weldable alloys — Guidance on quality levels for imperfections
(ISO 10042:1992).
ISO 1496–3, Series 1 freight containers — Specification and testing — Part 3 : Tank containers for liquids, gases
and pressurized dry bulk.
ISO 7005–1, Metallic flanges — Part 1 : Steel flanges.
3 Definitions and Symbols
3.1 Definition
For the purpose of this standard the term "pressure-tank" means a tank as defined in the international regulations
for the transport of dangerous goods by road or rail having a maximum working pressure or a test pressure
exceeding 50 kPa (0,5 bar).
3.2 Symbols
The following general symbols are used throughout the text; they are listed in alphabetical order. Special symbols are
explained with the relevant formulae.
A, A
p ; fm ; fb ; fp cross sectional areas reached for the calculation of nozzles reinforcement
(see 6.3.5.2 and Figure 7)
d inside diameter of an opening
i
D mean diameter of the cylindrical part of the tank at the junction of a cone
C
D outside diameter of the cylindrical part of the tank or the straight flange of the dished end
e
D inside diameter of the cylindrical part of the tank or the straight flange of dished end
i
e minimum required wall thickness (in mm) of the tank
e wall thickness of a conical part of a tank
k
e wall thickness of a hemispherical end
R
E Young's modulus
f nominal design stress (allowable stress)
d
h inside height of an ellipsoidal dished end
K shape factor of ellipsoidal ends
MWP maximum working pressure, in MPa
p design pressure, in MPa
p
dyn equivalent dynamic pressure
p test pressure, in MPa
test
p vapour pressure at 50 °C or at the design temperature, whichever is the higher; to be taken as the
vap
numerical value of the absolute pressure
5
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EN 14025:2003 (E)
r inner knuckle radius, in mm
R inside spherical radius of the central part of a torispherical end
R guaranteed (upper) minimum yield strength, in N/mm² or guaranteed minimum 0,2 % proof strength in
e
N/mm² (for austenitic steel the 1 % proof strength may be chosen)
R guaranteed minimum tensile strength, in N/mm²
m
lwelding coefficient
S
4 Materials
4.1 General
The tank shell shall be fabricated from metallic materials which shall be resistant to brittle fracture and of adequate
impact strength within the design temperature range. The material shall be suitable for shaping. EN 13445-2 applies,
but see also A.2.7.
Welded shells shall be fabricated from a material which has been shown to have acceptable welding characteristics.
4.2 Compatibility
Shells, fittings, and pipework shall be constructed from materials which are:
a) Substantially immune to attack by the substance(s) intended to be transported; or
b) Properly passivated or neutralised by chemical reaction; or
c) Lined with corrosion-resistant material directly bonded to the shell or attached by equivalent means.
Gaskets shall be made of materials not subject to attack by the substances intended to be transported. The
materials of the tank, including any devices, gaskets, linings and accessories, shall not adversely affect the
substances intended to be transported in the tank.
Guidelines on material specifications in relation to the substances to be transported may be taken from annex A of
EN 12285-1 : 2003.
5 Design
5.1 General
Tanks shall be designed to withstand without loss of contents the
1) Operating conditions including static and dynamic forces in normal conditions of carriage;
2) Test conditions;
3) Explosion pressure proof conditions (if required).
under consideration of clause 6.
NOTE The requirements coming from the relevant regulations (i. e. RID and ADR) are given in annex A for information.
6
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EN 14025:2003 (E)
5.2 Reduction of shell thickness
The minimum wall thickness of shells (see A.2.12) are allowed to be reduced by a maximum of 2 mm in the case of
mild steel or of an equivalent thickness of another metal, if protection of the shell against damage through lateral
impact or overturning is provided (see 5.3 and A.2.13).
For shells of rail tank wagons no reduction of the minimum wall thickness due to protection is allowed.
5.3 Protection of the shell
Shells of tank containers are protected against damage if one of the following measures is provided:
a structure, in which the shell is supported by a complete skeleton including longitudinal and transverse structural
members. This structure shall conform to the requirements of ISO 1496-3;
a double wall construction, where the aggregate thickness of the outer metal wall and the shell wall itself is not
less than the minimum wall thickness prescribed in A.2.12 for reduced wall thickness;
a "sandwich" construction, which means shells made with double walls having an intermediate layer of rigid solid
materials (e. g. foam, at least 50 mm thick), where the outer wall has a thickness of at least 0,5 mm of steel,
0,8 mm of aluminium or 2 mm of a plastics material reinforced with glass fibre. For other layer materials (e.g.
mineral wool, at least 100 mm thick), the outer wall has a thickness of at least 0,8 mm of austenitic steel. Other
combinations of materials used to provide protection against damage shall be shown to have equivalent strength
to the minimum thickness required in accordance with A.2.12. One method of comparing the strength of sheets of
materials is given in annex C of prEN 13094:2000.
For shells of road tankers see also 5.3.2 of prEN 13094:2000.
6 Calculation
6.1 General
The calculation scheme given in Figure 1 shows how to determine the wall thickness of a shell to meet the
requirements of this standard and the relevant regulations (i. e. RID and ADR).
7
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EN 14025:2003 (E)
Metallic pressure tanks
Design Data
RID/ ADR EN 14025
Cylindrical wall, ends and All tank parts
covers
AB C D
Minimum wall Equivalent wall Test conditions Operating conditions
thickness thickness
formula(e) formula(e)
Determining Determining Determining factors: Determining
factors: factors:
factors:
P /D /s /lT i
e / R / A
P D/ s / lo m1 1
T/
p D /f /li d s
P /D /sC
With/ without
additional
protection
Largest value Largest value wall thickness
e or e
1
other parts
Largest value cylindrical
wall, ends
and covers
Required minimum Required
thickness dimensions
Figure 1 — Calculation scheme for the wall thickness of metallic pressure tanks for the transport of
dangerous goods
The wall thickness shall not be less than the maximum value given by the following equations applied for the
operating and for the test conditions.
6.2 Design criteria
Design criteria (loads, allowable stresses, design temperature etc.) to be applied shall be taken from Table 1.
8
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EN 14025:2003 (E)
Table 1 — design criteria
Operating conditions Test conditions
MWP
p b
p
test
but not less than
a
p + p [if applicable]
vap dyn
f for ferritic steels and min { R / 1,5; R / 2,4}
d e m
aluminium alloys
f for austenitic steels R / 1,5
d e
b
min { 0,75 R ; 0,5 R }
e m
with 30 % £ A £ 35 %
f for austenitic steels max { R / 1,5; min (R / 1,2 ; R / 3)}
d e e m
with A > 35 %
20 °C provided that the operating
temperature of the tank is within the range
Temperature at the pressure test
– 20 °C to + 50 °C. When the operating
Design temperature
(normally +20 °C).
temperature is outside this range then the
design temperature shall be taken as the
extreme value of the operating temperature.
a
The dynamic forces shall be taken into account. This may be done by introducing an equivalent dynamic pressure p determined on the
dyn
basis of the dynamic forces specified in the relevant regulations (see annex A) but not less than 35 kPa ( 0,35 bar). The largest compartment
shall be taken into account.
b
To be taken from the relevant regulation, (see also annex A).
6.3 Calculation for internal pressure
6.3.1 General
The thickness of the cylindrical section of the shell shall be determined in accordance with 6.3.2. The thickness of the
ends (including partitions) of the shell shall be determined in accordance with 6.3.3; for partitions and surge plates see
also A.2.16. The thickness of a conical section and the reinforcement of the cone to cylinder junction shall be
determined in accordance with 6.3.4. For flanges, joints and their bolts see 6.3.6.
Openings in the shell shall be designed in accordance with 6.3.5.
6.3.2 Wall thickness of the cylindrical section
The wall thickness shall not be less than the value given by equation (1).
p·D
i
e = (1)
2 f·l-p
d S
6.3.3 Wall thickness of ends
6.3.3.1 General
The thickness of the ends, subject to the limitations in 6.3.3.2, shall not be less than that given by 6.3.3.3 and 6.3.3.4,
6.3.3.5 or 6.3.3.6 as applicable.
6.3.3.2 Design limitations
The following design limitations shall apply to the tank ends (compare Figure 2):
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EN 14025:2003 (E)
(a) Hemispherical ends 0,001 D£e£ 0,16 D
e e
(b) Torispherical ends 0,001 D£e£0,08 D
e e
0,06 D £r£0,2 D
i i
r‡2 e
R£D
e
(c) Ellipsoidal ends 0,001D£e£ 0,08 D
i i
3,4£ D /h £4,4
i
The four relationships in (b) and the two relationships in (c) shall be simultaneously fulfilled.
NOTE Kloepper and Korbbogen-type ends are particular cases of torispherical ends:
kloepper type
torispherical end for which R/D = 1,0 and r/D = 0,1
e e
korbbogen type
torispherical end for which R/D = 0,8 and r/D = 0,154
e e
Figure 2 — Geometry of ends
6.3.3.3 Thickness of the flange of the end
The wall thickness of the cylindrical or straight flange of the end shall not be less than the thickness as determined in
accordance with 6.3.2 for a cylindrical section having the same inside diameter D .
i
6.3.3.4 Thickness of hemispherical ends
The thickness of a hemispherical end shall be not less than that determined by equation (2).
p·D
i
e = (2)
4 f·l-p
d S
6.3.3.5 Thickness of torispherical ends
6.3.3.5.1 General
The minimum thickness of torispherical ends shall be the greatest of the values of e e or e as determined by
y, s b
equations (3) to (5).
10
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EN 14025:2003 (E)
p (0,75·R + 0,2·D )
i
e =b(3)
y e
f
d
where is calculated from formulae (6) to (14).
e
p·R
e = (4)
s
2 f·l-0,5·p
d S
2
( )
0,825 3
p D
i
e = (0,75·R + 0,2·D ) (5)
b i
111 f r
dŁł
Łł
If ends are manufactured from several elements then the welding coefficient l may be taken equal to 1,0 (for
S
thickness calculations only) if the weld crosses the crown area 0,6 D (see Figure 3).
e
l = 0,8 resp. 1,0 l = 1,0 l = 1,0l = 0,8 resp.1,0
S S S S
according to 7.4.3 according to 7.4.3
a) weld outside 0,6 D b) weld inside 0,6 D c) composed ends
e e
Figure 3 — Position of welds in ends
6.3.3.5.2 Formulae for calculation of b
e
Y = min (e / R ; 0,04) (6)
Z = log (1/ Y ) (7)
X = r / D (8)
i
1
N = 1,006-(9)
4
6,2 + (90 Y )
For X = 0,06:
3 2
b= N (-0,3635 Z + 2,2124 Z-3,2937 Z +1,8873) (10)
e 0,06
For 0,06 < X < 0,1:
b= 25 [(0,1-X )b+ (X-0,06)b] (11)
e e 0,06 e 0,1
11
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EN 14025:2003 (E)
For X = 0,1:
3 2
b= N (-0,1833 Z +1,0383 Z-1,2943 Z + 0,837) (12)
e 0,1
For 0,1 < X < 0,2:
b= 10 [(0,2-X )b+ (X-0,1)b] (13)
e e 0,1 e 0,2
For X = 0,2:
2
b= max { 0,95 (0,56-1,94 Y-82,5 Y ) ; 0,5 } (14)
e 0,2
6.3.3.5.3 Openings within the knuckle area of Kloepper- and Korbbogen-type ends
6.3.3.5.3.1 In this clause rules are given for increasing the thickness of a dished end to compensate for branches
within the knuckle area.
The rules are limited in application to Kloepper- and Korbbogen-type ends for which
a) 6£d / D 0, (15)
i e
d
i
b) £6,7 (16)
e·D
e
The increased thickness required by this clause applies to the whole knuckle area. Welded-on compensation is not
permitted.
6.3.3.5.3.2 Determine b from the procedure in Table 2
k
Replace p by (b p) in equation (3) to arrive at the required thickness. Equations (4) and (5) continue and apply
k
without modification.
Table 2 — Procedure for calculation of weakening factor bb for openings in the knuckle region (design)
k
Step Kloepper type end Korbbogen type end
p p
1
=
V log 1000 V = log 1000
f
f
dŁłŁdł
2 3
2 2
A = max {0,5 ; 0,264 + 0,938 V-0,592 V + 0,14 V }
A = 0,54 + 0,41V + 0,0441V
3 2 2
B = min {4,2 ; 4,91-2,165 V + 0,151V } B = 7,77-4,53 V + 0,7441V
4
d d d d
i i i i
b= max A + B ;1+ 0,3 Bb= max A + B ; 1+ 0,5 B
k k
D D D D
e e e eŁłŁł
6.3.3.6 Thickness of ellipsoidal ends
An ellipsoidal end is an end that is made on a former that has a true ellipsoidal shape. Ellipsoidal ends shall have a
shape factor K with a value between 1,7 and 2,2.
D
i
K = (17)
2 h
12
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EN 14025:2003 (E)
Ellipsoidal ends shall be designed as torispherical ends in accordance with 6.3.3.5 with values of r and R as given
by the following:
0,5
r = -0,08 D (18)
i
KŁł
R = (0,44 K + 0,02) D (19)
i
6.3.4 Wall thickness of conical sections
6.3.4.1 General
The conical section of the tank shell shall conform to the design limitations in 6.3.4.2. The wall thickness shall be
determined in accordance with 6.3.4.3. For more information compare also EN 13445-3.
6.3.4.2 Design limitations of the conical section
This clause gives rules for right circular cones and cone/cylinder intersections where the cone and the cylinder are of
the same axis of rotation. It applies only to cones with an apex angle not greater than 120 ° and cones for which
e·cosa/ D > 0,001
(20)
c
For offset cones between two cylinders the cylinders shall have parallel centrelines offset from each other by a
distance not greater than the difference of the radii. A required thickness shall be calculated in accordance with
6.3.4.3.3 for the junction at the large end and in accordance with 6.3.4.3.4 for the junction at the small end. Since the
thickness calculated above is the minimum allowable at that point along the case, it is permissible to build a cone from
plates of different thickness provided that at every point the minimum is achieved.
Figure 4 — Junction between cylinder and cone; angle aa
6.3.4.3 Determination of the wall thickness of the conical section
6.3.4.3.1 The minimum thickness of the wall of the conical section e of the shell shall be not less than the
k
lowest value given by equation (21), unless otherwise required by the subclauses of 6.3.4.3.
p·D 1
i
e = (21)
k
2 f·l-p cosaŁłd SŁł
6.3.4.3.2 Wall thickness at the junction of cylindrical and conical sections of the shell.
The conditions and design calculations for the large end of the cone are given in 6.3.4.3.3 and for the small end of
the cone in 6.3.4.3.4.
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EN 14025:2003 (E)
6.3.4.3.3 Large end of the cone without a knuckle at the joint
The joint shall be made with a butt weld and there shall be smooth transition across the joint from cylindrical section
to the conical section.
The joint shall be subjected to 100 % radiographic or ultrasonic inspection, unless the actual thickness is 40 %
greater than the minimum thickness determined in accordance with this subclause.
The minimum thickness of the wall of the shell at the junction of the conical and cylindrical sections shall be
determined from the following two equations by iteration.
p·D·bc c
e = (22)
1
2 f
d
1 D tanac
b=-0,15 (23)
c
3 e
1+1/ cosa1ŁłŁł
This thickness shall be maintained for a distance of at least 1,4 , from the junction along the cylinder. The
1
required thickness e of the cone adjacent to the junction is the greater of e and e . This thickness shall be
2 k
maintained for a distance of at least 1,4 from the junction along the cone, see Figure 4. The distance and
2 1
are determined as follows:
k
= D·e (24)
1 c 1
D e·c 2
= (25)
2
cosa
...
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