SIST-TP CEN/TR 1591-6:2025

SLOVENSKI STANDARD
01-maj-2025
Nadomešča:
SIST CR 13642:2000
Prirobnice in prirobnični spoji - Pravila za konstruiranje prirobničnih spojev,
sestavljenih iz okroglih prirobnic in tesnil - 6. del: Temeljne informacije
Flanges and their joints - Design rules for gasketed circular flange connections - Part 6:
Background information
Flansche und ihre Verbindungen - Regeln für die Auslegung von Flanschverbindungen
mit runden Flanschen und Dichtung - Teil 6: Hintergrund-Informationen
Brides et leurs assemblages - Règles de calcul des assemblages à brides circulaires
avec joint - Partie 6: Document de référence
Ta slovenski standard je istoveten z: CEN/TR 1591-6:2025
ICS:
23.040.60 Prirobnice, oglavki in spojni Flanges, couplings and joints
elementi
23.040.80 Tesnila za cevne zveze Seals for pipe and hose
assemblies
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

CEN/TR 1591-6
TECHNICAL REPORT
RAPPORT TECHNIQUE
January 2025
TECHNISCHER REPORT
ICS 23.040.60 Supersedes CR 13642:1999
English Version
Flanges and their joints - Design rules for gasketed circular
flange connections - Part 6: Background information
Brides et leurs assemblages - Règles de calcul des Flansche und ihre Verbindungen - Regeln für die
assemblages à brides circulaires avec joint - Partie 6: Auslegung von Flanschverbindungen mit runden
Document de référence Flanschen und Dichtung - Teil 6: Hintergrund-
Informationen
This Technical Report was approved by CEN on 22 December 2024. It has been drawn up by the Technical Committee CEN/TC
74.
CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia,
Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway,
Poland, Portugal, Republic of North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Türkiye and
United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION

EUROPÄISCHES KOMITEE FÜR NORMUNG

CEN-CENELEC Management Centre: Rue de la Science 23, B-1040 Brussels
© 2025 CEN All rights of exploitation in any form and by any means reserved Ref. No. CEN/TR 1591-6:2025 E
worldwide for CEN national Members.

Contents Page
European foreword . 4
1 Scope . 5
2 Normative references. 5
3 Terms and definitions . 5
4 Introduction . 5
5 Forces in gasketed joints . 6
5.1 Definition of active and passive forces . 6
5.2 Coupling of internal forces . 6
5.3 Assembly conditions . 7
6 Gasket characteristics . 8
6.1 Mechanical behavior . 8
6.1.1 General . 8
6.1.2 Unloading modulus (E ) . 8
G
6.1.3 Creep/relaxation . 8
6.1.4 Maximum compressive stress . 9
6.2 Sealing criteria . 9
6.3 Effective width . 10
7 Calculations for gaskets . 11
7.1 Effective width of gaskets . 11
7.1.1 Flat gaskets . 11
7.1.2 Effective width of gaskets with curved surfaces. 13
7.2 Elastic stiffness of gaskets . 14
7.3 Load carrying capability of gaskets . 15
8 Calculations for bolts . 17
8.1 Elastic stiffness of bolts . 17
8.2 Load carrying capacity of bolts . 17
8.3 Bolt tightening by torque-wrench . 17
9 Elasticity of shells . 18
9.1 Conical and spherical shells with uniform wall-thickness . 18
9.2 Conical hub with cylindrical shell . 22
10 Elasticity of complete flange . 25
10.1 Flange ring without shell . 25
10.2 Flange ring connected to shell . 28
11 Limit loads of shells . 30
11.1 Conical and spherical shells with constant wall thickness . 30
11.2 Conical hub with cylindrical shell . 35
12 Limit loads for complete flange . 40
12.1 Flange ring without shell . 40
12.2 Flange ring connected to shell . 41
13 Validity limits . 42
13.1 Diameter relations . 42
13.2 Thickness relations . 43
13.3 Angle of connected shell . 44
13.4 Bolt spacing . 44
(informative) Conical and spherical shells (uniform wall thickness) — Verification
of Formulae (9.33) to (9.35) . 47
(informative) Experimental verification . 52
B.1 General . 52
B.2 Test information . 54
(informative) Limit load formulae . 77
C.1 Limit load formulae for any shell . 77
C.2 Limit load formulae for axisymmetric shell. 78
Bibliography . 81
European foreword
This document (CEN/TR 1591-6:2025) has been prepared by Technical Committee CEN/TC 74 “Flanges
and their joints”, the secretariat of which is held by DIN.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. CEN shall not be held responsible for identifying any or all such patent rights.
This document supersedes CR 13642:1999.
CR 13642:1999:
— Update of 6.1.2, 6.1.3, 6.1.4 and 6.2 regarding evolution within EN 1591-1:2024 related to gasket
parameters.
NOTE This is not an exhaustive list of all modifications.
This document is part of a series that consists of the following parts:
 EN 1591-1, Flanges and their joints — Design rules for gasketed circular flange connections — Part 1:
Calculation
 CEN/TR 1591-2, Flanges and their joints — Design rules for gasketed circular flange connections —
Part 2: Gasket parameters
 CEN/TS 1591-3, Flanges and their joints — Design rules for gasketed circular flange connections —
Part 3: Calculation method for metal-to-metal contact type flanged joint
 EN 1591-4, Flanges and their joints — Part 4: Qualification of personnel competency in the assembly of
the bolted connections of critical service pressurized systems
 CEN/TR 1591-5, Flanges and their joints — Design rules for gasketed circular flange connections —
Part 5: Calculation method for full face gasketed joints
 CEN/TR 1591-6, Flanges and their joints — Design rules for gasketed circular flange connections —
Background information
Any feedback and questions on this document should be directed to the users’ national standards body.
A complete listing of these bodies can be found on the CEN website.
1 Scope
This document gives background information for guidance to be used in conjunction with the calculation
method for design rules for gasketed circular flange connections as specified in EN 1591-1:2024.
NOTE References to formulae numbered in this document have a decimal format whilst those in EN 1591-1:2024
are indicated by whole numbers.
2 Normative references
There are no normative references in this document.
3 Terms and definitions
No terms and definitions are listed in this document.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
 ISO Online browsing platform: available at https://www.iso.org/obp/
 IEC Electropedia: available at https://www.electropedia.org/
4 Introduction
Strength assessments in design calculations generally involve a 'proof of load' in the form:
actual loads ≤ allowable loads (4.1)
and similarly, when determining wall thicknesses, etc. The classical basis is linear elasticity theory, for
which proofs are often written:
actual stresses ≤ allowable stresses (4.2)
However, elasticity theory can lead to illogical results such as stress increasing (strength decreasing) with
increasing wall thickness. On the other hand, plasticity theory avoids such inconsistencies and Limit Load
Analysis gives reliable results. Therefore, in EN 1591-1:2024 Limit Load Analysis is used as the basis for
proof of load.
To ensure adequate leak-tightness, it is important that gasket compressive stress Q does not fall below a
certain value. For example, in the ASME Boiler and Pressure Vessel Code, Section VIII ('ASME' hereafter)
the proof for leak-tightness is:
Q ≥ m ⋅ P (4.3)
where m is a 'gasket factor' and P is fluid pressure.
NOTE The general form of a leak-tightness proof is equivalent to a load proof of the form:
actual load ≥ required load (4.4)
It follows that both upper and lower limits are imposed on gasket and bolt loads, as in:
required load ≤ actual load ≤ allowable load (4.5)
Various commonly used design codes (e.g. ASME and related codes) for gasketed joints do not apply this
condition but assume:
required load = actual load (4.6)
and neglect interactions between assembly and subsequent test or service conditions, or additional
assumptions are introduced (e.g. 'gasket force is constant'). This is a poor model of real joints and leakage
problems result.
A code which properly treats all load conditions (assembly, test, and all service conditions) is
TGL 32903/13 (1983), a National Standard of the former German Democratic Republic. Variants of this
have been in use since 1973 and it has been applied to the design of thousands of gasketed joints without
leakage problems. Therefore, when CEN/TC/74/WG 10 "Flanges and their joints - Calculation methods"
was requested to produce a design procedure for gasketed joints, the TGL method was chosen as the basis.
At the request of CEN/TC 54/WG 53 "Unfired pressure vessels – Design methods" the scope of the TGL
version was extended, with basic principles unchanged. The established validity was unaffected by this
extension but behaviour in the new domain has yet to be verified. Examples of validation tests for the
original domain are given in Annex B.
5 Forces in gasketed joints
5.1 Definition of active and passive forces
Active forces (and moments) are those which can, in principle, cause unlimited deformation. Examples in
joints are the axial fluid-pressure force F and an external dead-weight force F . If limited plastic
Q A
deformation is possible without loss of function (as in dished heads) only active forces are required in the
proof of load.
Passive forces (and moments) are due to limited elastic deformation and can only cause limited
deformation (they do affect fatigue). An example is a force due to differential thermal expansion. If limited
plastic deformations can cause loss of function, passive forces are also included in the proof of load.
NOTE This is partially true in the case of leak-tightness. Differential thermal expansion ∆U can cause plastic
I
deformation, with loss of bolt and gasket force, and hence of leak-tightness. Therefore ∆U is included in the
I
calculation of F , F and in the proof of load. On the other hand, scatter of assembly bolt-load is not considered in
GI BI
Formula (118). This is because if assembly bolt-load exceeds the minimum (to ensure subsequent leak-tightness)
and limited plastic deformation occurs in a subsequent condition, the forces can fall to the minimum required
without loss of leak-tightness.
5.2 Coupling of internal forces
Under both assembly conditions and subsequent load conditions the component parts of a gasketed joint
are coupled by internal forces. Therefore, the following geometric relation exists between displacements
of parts:
� � � �
�𝛩𝛩 ⋅ℎ +𝛩𝛩 ⋅ℎ +𝛩𝛩 ⋅ℎ +𝛩𝛩 ⋅ℎ +𝑈𝑈 +𝑈𝑈� =
𝐹𝐹 𝐺𝐺 𝐹𝐹 𝐺𝐺 𝐿𝐿 𝐿𝐿 𝐿𝐿 𝐿𝐿 𝐵𝐵 𝐺𝐺
(𝐼𝐼=0)
� � � �
�Θ ⋅ℎ +Θ ⋅ℎ +Θ ⋅ℎ +Θ ⋅ℎ +𝑈𝑈 +𝑈𝑈 +Δ𝑈𝑈� (5.1)
F 𝐺𝐺 𝐹𝐹 𝐺𝐺 𝐿𝐿 𝐿𝐿 𝐿𝐿 𝐿𝐿 𝐵𝐵 𝐺𝐺 𝐼𝐼
(𝐼𝐼≥1)
Substituting θ , θ (see Formulae (C.1), (C.2) and U , U (Formulae (8.1, 7.27, 7.29)) and the equilibrium
F L B G
condition.
𝐹𝐹 =𝐹𝐹 +𝐹𝐹 +𝐹𝐹 , for all I (5.2)
𝐵𝐵 𝐺𝐺 𝑄𝑄 𝑅𝑅
gives:
F ⋅ Y + F ⋅ Y + F ⋅ Y = F ⋅ Y + F ⋅ Y + F ⋅ Y +ΔU (5.3)
G0 G0 Q0 Q0 R0 R0 GI GI QI QI RI RI I
This is the fundamental formula relating force changes in a joint, subject to F = 0 (P = 0). The flexibility
Q0 I=0
parameters Y , Y , Y , Y are given in Formulae (97) to (100).
B G Q R
If the required gasket force (Formula 102) is known for subsequent conditions (e.g. from pressure P and
I
gasket factor m ), then from Formula (8.3) the assembly force to ensure leak-tightness is:
I
𝐹𝐹 ≥�F ⋅ Y + F ⋅ Y + F ⋅ Y − F ⋅ Y +ΔU�/𝑌𝑌 (5.4)
𝐺𝐺0 GI GI QI QI RI RI R0 R0 I 𝐺𝐺0
which is the basis of Formula (103).
NOTE In Formula (103) of EN 1591-1:2024, an additional term (P ) is introduced to take the gasket creep
QR
phenomena into account (see 6.1.3).
If there is more than one subsequent condition, the largest assembly force is selected in order to be
sufficient for all of them (Formula (105)). In the other subsequent conditions (not associated to the largest
assembly bolt force) the gasket force is greater than required. Gaskets, flanges and bolts ability to
withstand this additional force are checked (Formula (120)).
5.3 Assembly conditions
The bolt-tightening method is selected to produce a bolt load not less than the required minimum; thus,
due to scatter, the target bolt load is greater than this minimum. These effects are considered in
EN 1591-1:2024, subclause 4.4.2. The scatter parameter can be defined in various ways. In
EN 1591-1:2024 the following 'linear definition' is used (as in VDI 2230):
� � � �
𝐹𝐹 =𝐹𝐹 ⋅ (1−𝜀𝜀);                    𝐹𝐹 =𝐹𝐹 ⋅ (1 +𝜀𝜀) (5.5)
𝐵𝐵0 𝐵𝐵0 𝐵𝐵0 𝐵𝐵0
(� � )
𝐹𝐹 +𝐹𝐹
𝐵𝐵0 𝐵𝐵0
�
𝐹𝐹 = ;                       0 <𝜀𝜀 < 1 (5.6)
𝐵𝐵0
An alternative possibility is the "geometric definition":
� � � �
𝐹𝐹 =𝐹𝐹 /(1 +𝜀𝜀);                          𝐹𝐹 =𝐹𝐹 ⋅ (1 +𝜀𝜀) (5.7)
𝐵𝐵0 𝐵𝐵0 𝐵𝐵0 𝐵𝐵0
� � �
𝐹𝐹 =�𝐹𝐹 ⋅𝐹𝐹 ;                      0 <𝜀𝜀 <∞ (5.8)
𝐵𝐵0 𝐵𝐵0 𝐵𝐵0
These definitions are nearly the same for 0 < ε < 0,1 … 0,2, but the alternative definitions account for
disagreements between published data. In EN 1591-1:2024, examples of ε are given based on the linear
definition.
6 Gasket characteristics
6.1 Mechanical behavior
6.1.1 General
Figure 1 — Gasket mechanical behaviour
When structural deformation was not considered, the details of gasket behaviour were of less concern,
but with the present more comprehensive approach they need further consideration. As indicated
schematically in Figure 1, the relationship between the load (Q) and the gasket deflection in compression
(∆e ) is very non-linear for most gasket types.
G
6.1.2 Unloading modulus (E )
G
The initial-loading line (see right side of Figure 1) is strongly non-linear. The unloading-lines (see left side
of Figure 1) were initially considered approximately linear with slope increasing with assembly stress.
In reality, the unloading line is not strictly linear. Its slope reduces at lower stresses. Therefore, to
measure E , a detailed definition is given in EN 13555:2021. The corresponding values of unloading
G
modulus E , are plotted on the left side of the diagram.
G
In EN 1591-1:2024, these values were considered varying almost linearly with assembly stress:
𝐸𝐸 ≈𝐸𝐸 +𝐾𝐾 ⋅𝑄𝑄 (6.1)
𝐺𝐺 0 1 0
In the following revisions of EN 1591-1:2024, based on EN 13555:2021 test results, it has been decided
to use directly tabulated values of E (Q ) without using the linear model given in (6.1).
G 0
6.1.3 Creep/relaxation
When a gasket is subjected to compressive stress an immediate elastic (or elastic-plastic) deformation U
G
occurs followed by creep, increasing deformation with time at constant load. The Figure 2 shows this
diagrammatically.
Figure 2 — Creep/relaxation
In the case of a bolted connection, the load applied on the gasket will not be held constant. The gasket
deflection due to creep will lead to the reduction of the internal forces in the assembly. The amount of
force reduction on the gasket will depend on the compliance of the bolted connection components.
Therefore, the gasket creep/relaxation behaviour is handled through a creep/relaxation factor called P
QR
defined as the ratio of the residual and initial surface pressures. EN 13555:2021 details the test procedure
enabling to measure this parameter.
6.1.4 Maximum compressive stress
The maximum surface pressure (Q ) that can be safely imposed upon the gasket at the service
smax
temperature without damage is determined through EN 13555:2021 tests. In EN 1591-1:2024, the
measured value of Q was modified by a factor c , (Formula (7.43) and Formula (13.24)) that makes
smax G
allowance for the gasket width in relation to thickness. For recording purposes, the explanation for this
correction is maintained in this document even if it is no longer used in the actual version of EN 1591-1:2024.
6.2 Sealing criteria
Q is the minimum level of gasket surface pressure required for tightness class L at assembly (on the
min (L)
effective gasket area). Q is the minimum level of surface pressure required for leakage rate class L
smin(L)
after off-loading. Q is the associated required gasket surface pressure at assembly prior to unloading.
A
Q , Q and Q are variables which are determined in a leakage test according to EN 13555:2021
min (L) A Smin(L)
and which are linked to each other. The lowest acceptable value of Q is equal to Q , in this case
A min(L),I
Q = Q =Q . The higher Q can be chosen, the lower Q can get.
A min(L),I Smin(L),I A Smin(L),I
Key
1 measurement point
2 loading
3 unloading
4 Q (L)
min
5 Qsmin(L)
X effective gasket surface pressure [in MPa]
Y leakage rate [in mg/(m s)]
Figure 3 — Leakage rate as a function of gasket surface pressure (for one specific internal
pressure level of the fluid)
6.3 Effective width
The effective width of a gasket varies with flange rotation, which also causes a radial variation of compressive
stress. Strictly, an iterative calculation is needed to reconcile the changing width, gasket stresses and bolt
load. However, the approach adopted in EN 1591-1:2024 is to calculate gasket-width for the assembly
condition and then assumed this to be unchanged for subsequent conditions. This simplifying assumption
is strictly correct only if gasket-force F and flange rotations do not change. However, the assumption is
G
conservative if the effective width for subsequent conditions is actually smaller than in the assembly
condition, which is often the case.
7 Calculations for gaskets
7.1 Effective width of gaskets
7.1.1 Flat gaskets
Key
b = contact width
Ga
b = effective width
Ge
b = calculated width
Gc
Q(x) = compressive stress
Figure 4 — Flat gaskets
Elastic rotation of flanges (see Formula (10.36)):
� �
� �
Θ +Θ =𝐹𝐹 ⋅�𝑍𝑍 ⋅ℎ /𝐸𝐸 +𝑍𝑍 ⋅ℎ /𝐸𝐸� (7.1)
𝐹𝐹 𝐹𝐹 𝐺𝐺 𝐹𝐹 𝐺𝐺 𝐹𝐹 𝐹𝐹 𝐺𝐺 𝐹𝐹
Elastic deformation of gasket (for 0≤𝑥𝑥≤𝑏𝑏 ):
𝐺𝐺𝐺𝐺
�
𝜀𝜀 =�Θ +Θ �⋅𝑥𝑥/𝑒𝑒 =𝑘𝑘⋅𝑥𝑥 (7.2)
𝐹𝐹 𝐹𝐹 𝜀𝜀
d𝑄𝑄
𝐸𝐸 =𝐸𝐸 +𝐾𝐾 ⋅𝑄𝑄 = (7.3)
𝐺𝐺 0 1
d𝜀𝜀
𝐸𝐸 1
𝑄𝑄 = ⋅ (exp(𝐾𝐾 ⋅𝜀𝜀)− 1)≈𝐸𝐸 ⋅𝜀𝜀⋅�1 + ⋅𝐾𝐾 ⋅𝜀𝜀� (7.4)
1 0 1
𝐾𝐾 2
The resultant gasket force F is:
G
𝑏𝑏
𝐺𝐺𝐺𝐺
( )
𝐹𝐹 =π⋅𝑑𝑑 ⋅∫ 𝑄𝑄𝑥𝑥 ⋅ d𝑥𝑥 (7.5)
𝐺𝐺 𝐺𝐺𝐺𝐺
acting at x = c, given by:
𝑏𝑏 𝑏𝑏
𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺
𝑐𝑐⋅ 𝑄𝑄(𝑥𝑥)⋅ d𝑥𝑥 = 𝑄𝑄(𝑥𝑥)⋅𝑥𝑥⋅ d𝑥𝑥 (7.6)
∫ ∫
0 0
From this follow, step by step:
1 1
𝐹𝐹 =π⋅𝑑𝑑 ⋅𝐸𝐸 ⋅𝑘𝑘⋅ ⋅𝑏𝑏 ⋅�1 + ⋅𝐾𝐾 ⋅𝑘𝑘⋅𝑏𝑏 � (7.7)
𝐺𝐺 𝐺𝐺𝐺𝐺 0 𝐺𝐺𝐺𝐺 1 𝐺𝐺𝐺𝐺
2 3
𝐹𝐹 ⋅𝐺𝐺 ⋅2
𝐺𝐺 𝐺𝐺
𝑏𝑏 = (7.8)
𝐺𝐺𝐺𝐺 1
�
�
π⋅𝑑𝑑 ⋅𝐸𝐸⋅�Θ +Θ �⋅�1+⋅𝐾𝐾⋅𝑘𝑘⋅𝑏𝑏 �
𝐺𝐺𝐺𝐺 0 𝐹𝐹 𝐹𝐹 1 𝐺𝐺𝐺𝐺
1+⋅𝐾𝐾⋅𝑘𝑘⋅𝑏𝑏
1 𝐺𝐺𝐺𝐺
𝑐𝑐 = ⋅𝑏𝑏 ⋅ (7.9)
𝐺𝐺𝐺𝐺 1
1+⋅𝐾𝐾⋅𝑘𝑘⋅𝑏𝑏
1 𝐺𝐺𝐺𝐺
1+⋅𝐾𝐾⋅𝑘𝑘⋅𝑏𝑏
1 𝐺𝐺𝐺𝐺
𝑏𝑏 = 2⋅ (𝑏𝑏 −𝑐𝑐) = ⋅𝑏𝑏 ⋅ (7.10)
𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺 1
1+⋅𝐾𝐾⋅𝑘𝑘⋅𝑏𝑏
1 𝐺𝐺𝐺𝐺
8 1
𝐹𝐹 ⋅𝐺𝐺 ⋅ �1+⋅𝐾𝐾⋅𝑘𝑘⋅𝑏𝑏 �
1 𝐺𝐺𝐺𝐺
𝐺𝐺 𝐺𝐺
9 4
𝑏𝑏 =� ⋅ (7.11)
𝐺𝐺𝐺𝐺 3
� 1
π⋅𝑑𝑑 ⋅�Θ +Θ �⋅𝐸𝐸
𝐺𝐺𝐺𝐺 𝐹𝐹 𝐹𝐹 0
�1+⋅𝐾𝐾⋅𝑘𝑘⋅𝑏𝑏 �
1 𝐺𝐺𝐺𝐺
The remaining elimination of 𝑘𝑘⋅𝑏𝑏 is simplified by assuming 𝐾𝐾 ⋅𝑘𝑘⋅𝑏𝑏 ⋅ ≪ 1, then Formula (7.11)
𝐺𝐺𝐺𝐺 1 𝐺𝐺𝐺𝐺
becomes:
𝐹𝐹 ⋅𝐺𝐺 ⋅
𝐺𝐺 𝐺𝐺
𝑏𝑏 =� (7.12)
𝐺𝐺𝐺𝐺 1
�
π⋅𝑑𝑑 ⋅Θ +Θ �⋅𝐸𝐸⋅�1+⋅𝐾𝐾⋅𝑘𝑘⋅𝑏𝑏 �
�
𝐺𝐺𝐺𝐺 𝐹𝐹 𝐹𝐹 0 1 𝐺𝐺𝐺𝐺
[This approximation gives an error on 𝑏𝑏 ≤ 5 % for 𝐾𝐾 ⋅𝑘𝑘⋅𝑏𝑏 ≤ 10, almost always true!]
𝐺𝐺𝐺𝐺 1 𝐺𝐺𝐺𝐺
𝐹𝐹 will be at least:
𝑄𝑄
� �
where is an average value,
𝐹𝐹 =𝜋𝜋⋅𝑑𝑑 ⋅𝑏𝑏 ⋅𝑄𝑄 𝑄𝑄 (7.13)
𝑄𝑄 𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺
and with Formulae (7.7) and (7.10) this gives:
1+⋅𝐾𝐾⋅𝑘𝑘⋅𝑏𝑏
2⋅𝑏𝑏 4⋅𝑏𝑏 1 𝐺𝐺𝐺𝐺
𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺
𝐸𝐸 ⋅𝑘𝑘⋅𝑏𝑏 =𝑄𝑄⋅ =𝑄𝑄⋅ ⋅ (7.14)
0 𝐺𝐺𝐺𝐺 1 2
3⋅𝑏𝑏
𝑏𝑏 ⋅�1+⋅𝐾𝐾⋅𝑘𝑘⋅𝑏𝑏 � 𝐺𝐺𝐺𝐺
𝐺𝐺𝐺𝐺 1 𝐺𝐺𝐺𝐺 �1+⋅𝐾𝐾⋅𝑘𝑘⋅𝑏𝑏 �
3 1 𝐺𝐺𝐺𝐺
With the approximation: 𝐸𝐸 ⋅𝑘𝑘⋅𝑏𝑏 ≈𝑄𝑄 (7.15)
0 𝐺𝐺𝐺𝐺
and omitting the factor 8/9, Formula (7.12) becomes:
𝐹𝐹 ⋅𝐺𝐺
𝐺𝐺 𝐺𝐺
𝑏𝑏 = =𝑏𝑏 (7.16)
𝐺𝐺𝐺𝐺 � 𝑄𝑄 𝐺𝐺𝐺𝐺(𝐺𝐺𝑒𝑒)
�
π⋅𝑑𝑑 ⋅�Θ +Θ �⋅�𝐸𝐸+𝐾𝐾⋅ �
𝐺𝐺𝐺𝐺 𝐹𝐹 𝐹𝐹 0 1
for elastic behaviour of a gasket
For plastic behaviour:
𝐹𝐹
𝐺𝐺
𝑏𝑏 = (7.17)
𝐺𝐺𝐺𝐺(𝑝𝑝𝑒𝑒)
π⋅𝑑𝑑 ⋅𝑄𝑄
𝐺𝐺𝐺𝐺 smax
True elasto-plastic deformation gives an effective-width greater than for pure-elastic and pure-plastic
deformation, approximately:
2 2
𝑏𝑏 ≈ 𝑏𝑏 +𝑏𝑏 (7.18)
�
𝐺𝐺𝐺𝐺 ( ) ( )
𝐺𝐺𝐺𝐺𝐺𝐺𝑒𝑒 𝐺𝐺𝐺𝐺𝑝𝑝𝑒𝑒
From Formulae (7.1), (7.16) and (7.17):
𝐺𝐺 𝐹𝐹
𝐺𝐺 𝐺𝐺
𝑏𝑏 = +� � (7.19)
𝐺𝐺𝐺𝐺 �
�
𝑍𝑍 𝑍𝑍 𝑄𝑄
𝐹𝐹 𝐹𝐹 π⋅𝑑𝑑 ⋅𝑄𝑄
� 𝐺𝐺𝐺𝐺 smax
π⋅𝑑𝑑 ⋅�ℎ ⋅ +ℎ ⋅ �⋅�𝐸𝐸+𝐾𝐾⋅ �
𝐺𝐺𝐺𝐺 𝐺𝐺 𝐺𝐺 0 1
�
𝐸𝐸 𝐸𝐸 2
𝐹𝐹 𝐹𝐹
This is the formula in EN 1591-1:2024, Table 1 ('Type 1'). The auxiliary notation 𝑏𝑏 is introduced because
𝐺𝐺𝐺𝐺
Formula (7.19) can give 𝑏𝑏 >𝑏𝑏 , but it is necessary that 𝑏𝑏 ≤𝑏𝑏 . (Later the denomination 𝑏𝑏 was
𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺
changed into 𝑏𝑏 , because 𝑏𝑏 and 𝑏𝑏 are too similar.)
𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺
NOTE As introduced in 6.1.2, the model proposed in Formula (6.1) for EG is no longer used in
𝑄𝑄
EN 1591-1:2024 and the expression “𝐸𝐸 +𝐾𝐾 ⋅ “ in Formula (7.19) has been replaced by the parameter “EGm” in
0 1
Formula (63) of EN 1591-1:2024. Nevertheless, it can be observed that the model proposed in Formula (6.1) is still
used (see Formula (7.3)) to obtain Formula (7.19). In EN 1591-1:2024, for metallic gaskets where the value of EG is
considered nearly independent of the applied load (K1*Q << E0), EN 1591-1:2024, Table 1 gives EGm=EG0 ~ E0. For
non-metallic gaskets, for which the EG values are considered to depend highly on the applied load, EN 1591-1:2024,
Table 1 gives the approximation of EGm=0,5*EG0.
7.1.2 Effective width of gaskets with curved surfaces

Key
b = contact width
Ga
b = effective width
Ge
b = calculated width
Gc
Figure 5 — Effective width of gaskets with curved surfaces
For elastic deformation of the gasket (Hertzian contact) the contact width is:
32 𝐹𝐹 ⋅𝑟𝑟⋅�1−𝜈𝜈 �
𝐺𝐺 2
𝑏𝑏 = ⋅ (7.19)
𝐺𝐺𝐺𝐺
π 𝐸𝐸 ⋅π⋅𝑑𝑑
𝐺𝐺 𝐺𝐺𝐺𝐺
and the maximum contact pressure, assumed equal to the mean pressure in 𝑏𝑏 , is:
𝐺𝐺𝐺𝐺
4 𝐹𝐹
𝐺𝐺
𝑄𝑄 = ⋅ (7.20)
smax
π π⋅𝑑𝑑 ⋅𝑏𝑏
𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺
and Q = Q where:
smax
𝐹𝐹
𝐺𝐺
𝑄𝑄 = (7.21)
π⋅𝑑𝑑 ⋅𝑏𝑏
𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺
therefore
π
𝑏𝑏 =𝑏𝑏 ⋅ (7.22)
𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺
i.e.
𝐹𝐹 ⋅𝑟𝑟⋅2⋅π⋅(1−𝜈𝜈 )
𝐺𝐺 2
𝑏𝑏 =� (7.23)
𝐺𝐺𝐺𝐺
𝐸𝐸 ⋅π⋅𝑑𝑑
𝐺𝐺 𝐺𝐺𝐺𝐺
If the contacting surfaces are inclined at angle 𝜑𝜑 to the reference plane as drawn, the normal forces on
𝐺𝐺
the curved surfaces are 𝐹𝐹 / cos𝜑𝜑 (instead of 𝐹𝐹 ), the real contact width increases with �1/ cos𝜑𝜑 , but the
𝐺𝐺 𝐺𝐺 𝐺𝐺 𝐺𝐺
calculated width is the projection on the reference plane (projection factor cos𝜑𝜑 ). Therefore:
𝐺𝐺
𝐹𝐹 ⋅cos𝜑𝜑 ⋅𝑟𝑟⋅2⋅π⋅(1−𝜈𝜈 )
𝐺𝐺 𝐺𝐺 2
𝑏𝑏 =� =𝑏𝑏 (7.24)
𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺(𝐺𝐺𝑒𝑒)
𝐸𝐸 ⋅π⋅𝑑𝑑
𝐺𝐺 𝐺𝐺𝐺𝐺
The plastic width 𝑏𝑏 , and approximation for elasto-plastic behaviour, correspond to Formulae (7.17)
𝐺𝐺𝐺𝐺(𝑝𝑝𝑒𝑒)
( )
and (7.18). FpEN 1591-1:2024, Table 2 is simplified by assuming 2⋅π⋅ 1−𝜈𝜈 ≈ 6 (similar to 8/9≈ 1
for flat gaskets).
Double contact:
Each surface sees half the force hence the calculated width is double that for one surface with half the
force, which is the basis of FpEN 1591-1:2024, Table 2.
7.2 Elastic stiffness of gaskets

Figure 6 — Elastic stiffness of gaskets
A simple model gives:
ΔF ⋅𝐺𝐺
G 𝐺𝐺
�
Δ𝑈𝑈 = ;𝑏𝑏 ≤𝑏𝑏 ≤𝑏𝑏 (7.25)
𝐺𝐺 𝐺𝐺𝐺𝐺 𝐺𝐺 𝐺𝐺𝐺𝐺
�
𝐸𝐸 ⋅𝜋𝜋⋅𝑑𝑑 ⋅𝑏𝑏
𝐺𝐺 𝐺𝐺𝐺𝐺 𝐺𝐺
The active (deformed) width assumed is indicated by the dashed line in the diagram (defined by the angle
π/4) which gives a maximum 'active width' (mid-plane):
𝐺𝐺
𝐺𝐺
( )
𝑏𝑏 =𝑏𝑏 + 2⋅ ,        𝑏𝑏 ≤𝑏𝑏 (7.26)
𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺
and average active width:
𝐺𝐺
𝐺𝐺
� �
𝑏𝑏 =𝑏𝑏 + ,            �𝑏𝑏 ≤𝑏𝑏 � (7.27)
𝐺𝐺 𝐺𝐺𝐺𝐺 𝐺𝐺 𝐺𝐺𝐺𝐺
( )
The following simple formula gives the values for flat gaskets 𝑒𝑒 ≪𝑏𝑏 ,𝑏𝑏 as well as for thick gaskets
𝐺𝐺 𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺
( )
𝑒𝑒 ≫𝑏𝑏 ,𝑏𝑏 :
𝐺𝐺 𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺
𝑏𝑏 +𝐺𝐺 /2
𝐺𝐺𝐺𝐺 𝐺𝐺
�
𝑏𝑏 = ⋅𝑏𝑏 (7.28)
𝐺𝐺 𝐺𝐺𝐺𝐺
𝑏𝑏 +𝐺𝐺 /2
𝐺𝐺𝐺𝐺 𝐺𝐺
writing
Δ𝑈𝑈 =Δ𝐹𝐹 ⋅𝑋𝑋 /𝐸𝐸 (7.29)
𝐺𝐺 𝐺𝐺 𝐺𝐺 𝐺𝐺
gives
𝐺𝐺 𝑏𝑏 +𝐺𝐺 /2
𝐺𝐺 𝐺𝐺𝐺𝐺 𝐺𝐺
𝑋𝑋 = ⋅ ⋅𝑏𝑏 (7.30)
𝐺𝐺 𝐺𝐺𝐺𝐺
𝐴𝐴 𝑏𝑏 +𝐺𝐺 /2
𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺 𝐺𝐺
7.3 Load carrying capability of gaskets
Assumptions:
The gasket is a plastic deformable strip (yield or design stress S ) between two rigid planes. Stresses S
G xx
= S and S = -Q are functions of x only, independent of y and z.
yy zz
Figure 7 — Load carrying capability of gaskets
Limit load formula: Ψ = 1− (𝑛𝑛 +𝑝𝑝) ≥ 0 (7.31)
𝑥𝑥𝑥𝑥
where
𝑆𝑆 𝑄𝑄
𝑥𝑥𝑥𝑥
𝑛𝑛 = =𝑛𝑛 ;𝑝𝑝 = ≤ 1−𝑛𝑛 (7.32)
𝑥𝑥𝑥𝑥 𝑦𝑦𝑦𝑦 𝑥𝑥𝑥𝑥
𝑆𝑆 𝑆𝑆
𝐺𝐺 𝐺𝐺
Equilibrium and Coulomb friction give:
𝑑𝑑𝑆𝑆
𝑥𝑥𝑥𝑥
𝑒𝑒 ⋅ + 2𝑇𝑇 = 0 (7.33)
𝐺𝐺
𝑑𝑑𝑥𝑥
𝑇𝑇≤𝜇𝜇 ⋅𝑄𝑄 (7.34)
𝐺𝐺
From Formulae (7.31) to (7.34) it follows that:
𝑑𝑑𝑛𝑛 2⋅𝜇𝜇
𝑥𝑥𝑥𝑥 𝐺𝐺
+ ⋅ (1−𝑛𝑛 ) = 0 (7.35)
𝑥𝑥𝑥𝑥
𝑑𝑑𝑥𝑥 𝐺𝐺
𝐺𝐺
with boundary condition 𝑛𝑛 = 0 this solution is:
𝑥𝑥𝑥𝑥(𝑥𝑥=0)
𝑥𝑥
( )
𝑛𝑛 = 1− exp�2⋅𝜇𝜇 ⋅ � ;𝑝𝑝 = exp 2⋅𝜇𝜇 ⋅𝑥𝑥/𝑒𝑒 (7.36)
𝑥𝑥𝑥𝑥 𝐺𝐺 𝐺𝐺 𝐺𝐺
𝐺𝐺
𝐺𝐺
Validity of Formulae (7.34) to (7.36) extends to x = x with shear limit:
𝑇𝑇 =𝜇𝜇 ⋅𝑄𝑄≤𝑆𝑆 /√3 (7.37)
𝐺𝐺 𝐺𝐺
1 𝑥𝑥 𝑥𝑥 1 1
1 1
𝑝𝑝 = = exp�2⋅𝜇𝜇 ⋅ � ; = ⋅ ln� � (7.38)
𝐺𝐺
𝜇𝜇 3 𝐺𝐺 𝐺𝐺 2⋅𝜇𝜇 𝜇𝜇 3
√ √
𝐺𝐺 𝐺𝐺 𝐺𝐺 𝐺𝐺 𝐺𝐺
For x>x , Formulae (7.35), (7.36) are replaced by:
𝑑𝑑𝑛𝑛 2
𝑥𝑥𝑥𝑥
+ = 0 (7.39)
𝑑𝑑𝑥𝑥 𝐺𝐺 ⋅ 3
√
𝐺𝐺
( )
2⋅ 𝑥𝑥−𝑥𝑥
𝑛𝑛 =𝑛𝑛 − = 1−𝑝𝑝
𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥1
𝑒𝑒 ⋅ 3
√
𝐺𝐺
1 1 1 𝑥𝑥
𝑝𝑝 = ⋅� ⋅�1− ln� �� + 2⋅ � (7.40)
√3 𝜇𝜇 𝜇𝜇 √3 𝐺𝐺
𝐺𝐺 𝐺𝐺 𝐺𝐺
From Formula (7.38) the limits are:
µ 0,05 0,10 0,20 0,30 0,40 0,50
G
x /e 24,46 8,77 2,65 1,09 0,46 0,14
1 G
The average pressure is:
2 𝑏𝑏 /2
𝐺𝐺
𝑝𝑝̅= ⋅ 𝑝𝑝⋅𝑑𝑑𝑥𝑥 (7.41)
∫
𝑏𝑏
𝐺𝐺
Some results from Formulae (7.36), (7.40) are shown in Figure 8 a), and corresponding results from
Formula (7.41) in Figure 8 b). The corresponding polynomial series for small 𝜇𝜇 ⋅𝑏𝑏 /𝑒𝑒 is:
𝐺𝐺 𝐺𝐺 𝐺𝐺
1 𝜇𝜇 ⋅𝑏𝑏 1 𝜇𝜇 ⋅𝑏𝑏
𝐺𝐺 𝐺𝐺 𝐺𝐺 𝐺𝐺
𝑝𝑝̅= 1 + ⋅� � + ⋅� � +. (7.42)
2 𝐺𝐺 6 𝐺𝐺
𝐺𝐺 𝐺𝐺
Therefore, the approximation (broken lines in Figure 8 b))
𝜇𝜇 ⋅𝑏𝑏
𝐺𝐺 𝐺𝐺
𝑝𝑝̅= 1 + (= c in EN 1591-1:2024) (7.43)
G
2⋅𝐺𝐺
𝐺𝐺
is conservative whilst 0≤𝜇𝜇 < 1/ 3.
√
𝐺𝐺
a) b)
Figure 8 — Load carrying capability of gaskets
8 Calculations for bolts
8.1 Elastic stiffness of bolts
The axial elongation of the bolts is:
𝑋𝑋
𝐵𝐵
Δ𝑈𝑈 =Δ𝐹𝐹 ⋅ (8.1)
𝐵𝐵 𝐵𝐵
𝐸𝐸
𝐵𝐵
where
4 𝑒𝑒 𝑒𝑒 0,8
𝑠𝑠 𝐺𝐺
𝑋𝑋 = ⋅� + + � (8.2)
𝐵𝐵 2 2
π⋅𝑛𝑛 𝑑𝑑 𝑑𝑑 𝑑𝑑
𝐵𝐵 𝐵𝐵0
𝐵𝐵𝑠𝑠 𝐵𝐵𝐺𝐺
The last term in brackets is an approximation for the elastic deformation of the two nuts (or nut and bolt
head).
8.2 Load carrying capacity of bolts
Maximum permissible tensile force (for sum of n bolts) is:
B
𝐹𝐹 ≤𝑆𝑆 ⋅𝐴𝐴 (8.3)
𝐵𝐵 𝐵𝐵 𝐵𝐵
π
( ( ))
𝐴𝐴 =𝑛𝑛 ⋅ ⋅ min𝑑𝑑 ,𝑑𝑑 (8.4)
𝐵𝐵 𝐵𝐵 𝐵𝐵𝐺𝐺 𝐵𝐵𝐵𝐵
( )
𝑑𝑑 = 𝑑𝑑 +𝑑𝑑 /2 (8.5)
𝐵𝐵𝐺𝐺 𝐵𝐵2 𝐵𝐵3
Formula (8.5) represents ISO effective diameter.
8.3 Bolt tightening by torque-wrench
To obtain bolt load F (for one bolt) using a torque-wrench, given thread pitch p and thread friction
B0 t
coefficient
µ , requires torsional moment:
t
𝑑𝑑 𝑝𝑝 𝜇𝜇
𝐵𝐵2 𝐺𝐺 𝐺𝐺
𝑀𝑀 =𝐹𝐹 ⋅ ⋅� + � (8.6)
𝑇𝑇0 𝐵𝐵0
2 π⋅𝑑𝑑 cos𝛼𝛼
𝐵𝐵2
(α = thread angle, commonly 60° or 55°)
Assuming d ≤ d (see Formula (8.4)) the limit load formula is:
Be B
2 2
𝐹𝐹 𝑀𝑀
𝐵𝐵0 𝑇𝑇0 2
� � + 3⋅� � ≤𝑆𝑆 (8.7)
2 2 𝐵𝐵
π/4⋅𝑑𝑑 π/12⋅𝑑𝑑
𝐵𝐵𝐺𝐺 𝐵𝐵𝐺𝐺
π 3⋅𝑑𝑑 𝑝𝑝 𝜇𝜇
2 𝐵𝐵2 𝐺𝐺 𝐺𝐺
�
𝐹𝐹 ≤𝑆𝑆 ⋅ ⋅𝑑𝑑 / 1 + 3⋅� ⋅� + �� (8.8)
𝐵𝐵0 𝐵𝐵 𝐵𝐵𝐺𝐺
4 2⋅𝑑𝑑 π⋅𝑑𝑑 cos𝛼𝛼
𝐵𝐵𝐺𝐺 𝐵𝐵2
( )
Bolt threads commonly 𝑝𝑝 /𝜋𝜋⋅𝑑𝑑 ≈ 0,05 … 0,03 have and 𝜇𝜇 ≈ 0,1 … 0,2, hence:
𝐺𝐺 𝐵𝐵2 𝐺𝐺
π
𝐹𝐹 ≤𝑆𝑆 ⋅ ⋅𝑑𝑑 /(1,1 … 1,2) (8.9)
𝐵𝐵0 𝐵𝐵 𝐵𝐵𝐺𝐺
This scatter is less than that of most tightening methods (EN 1591-1:2024, Annex C), further the torsional
moment M is active only while the wrench is moving. After tightening, the torsional moment M is
T0 T0
passive and can relax by passive deformation or creep. It is therefore neglected in this document, this is
established practice in Germany.
9 Elasticity of shells
9.1 Conical and spherical shells with uniform wall-thickness
Equilibrium conditions are:
′
(𝑟𝑟⋅𝑁𝑁 ) + sin𝜑𝜑⋅𝑁𝑁 + ⋅ (𝑟𝑟⋅𝑄𝑄 ) = 0 (9.1)
𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 𝑢𝑢
𝑟𝑟
𝑘𝑘
′
( ) ( )
𝑟𝑟⋅𝑄𝑄 − ⋅ 𝑟𝑟⋅𝑁𝑁 − cos𝜑𝜑⋅𝑁𝑁 +𝑟𝑟⋅𝑃𝑃 = 0 (9.2)
𝑢𝑢 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣
𝑟𝑟
𝑘𝑘
′
(𝑟𝑟⋅𝑀𝑀 ) + sin𝜑𝜑⋅𝑀𝑀 − (𝑟𝑟⋅𝑄𝑄 ) = 0 (9.3)
𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 𝑢𝑢
(N , M correspond to the circumferential direction v)
vv vv
Elastic relations are:
𝐸𝐸⋅𝐺𝐺 sin𝜑𝜑 1 cos𝜑𝜑
′
𝑁𝑁 = ⋅ 𝑈𝑈 −𝜈𝜈⋅ ⋅𝑈𝑈 +� + �⋅𝑊𝑊 (9.4)
� �
𝑢𝑢𝑢𝑢
1−𝜈𝜈 𝑟𝑟 𝑟𝑟 𝑟𝑟
𝑘𝑘
𝐸𝐸⋅𝐺𝐺 sin𝜑𝜑 1 cos𝜑𝜑
′
𝑁𝑁 = ⋅ 𝜈𝜈⋅𝑈𝑈 − ⋅𝑈𝑈 +� + �⋅𝑊𝑊 (9.5)
� �
𝑣𝑣𝑣𝑣
1−𝜈𝜈 𝑟𝑟 𝑟𝑟 𝑟𝑟
𝑘𝑘
𝐸𝐸⋅𝐺𝐺 sin𝜑𝜑
′′
𝑀𝑀 = ⋅�−𝑊𝑊 +𝜈𝜈⋅ ⋅𝑊𝑊′� (9.6)
𝑢𝑢𝑢𝑢
12⋅(1−𝜈𝜈 ) 𝑟𝑟
𝐸𝐸⋅𝐺𝐺 sin𝜑𝜑
′′
𝑀𝑀 = ⋅ −𝜈𝜈⋅𝑊𝑊 + ⋅𝑊𝑊′ (9.7)
� �
𝑣𝑣𝑣𝑣
12⋅(1−𝜈𝜈 ) 𝑟𝑟
Key
u, w: coordinates, mm
U, W: displacements, mm
N , Q : forces/ length, N/mm
uu u
N , Q : same forces at u=0, N/mm
s s
M : moment/length, Nmm/mm
uu
M : same moment at u=0, Nmm/mm
s
Θ = W' inclination
Figure 9 — Conical and spherical shells with uniform wall-thickness
𝑑𝑑( ) 1 𝑑𝑑( )
′ ′ ′
( )
= = ⋅ ;   𝑟𝑟 =− sin𝜑𝜑 ;      𝑧𝑧 = + cos𝜑𝜑
𝑑𝑑𝑑𝑑 𝑟𝑟 𝑑𝑑𝜑𝜑
𝑘𝑘
r = const
k
From Formulae (9.3), (9.6), (9.7) the shear force is:
3 2
𝐸𝐸⋅𝐺𝐺 sin𝜑𝜑 cos𝜑𝜑 sin 𝜑𝜑
′′′ ′′
𝑄𝑄 = ⋅ −𝑊𝑊 +𝜈𝜈⋅ ⋅𝑊𝑊 +�𝜈𝜈⋅ + �⋅𝑊𝑊′ (9.8)
� �
𝑢𝑢
2 2
12⋅(1−𝜈𝜈 ) 𝑟𝑟 𝑟𝑟 ⋅𝑟𝑟 𝑟𝑟
𝑘𝑘
Transformed and simplified equilibrium conditions:
′
(𝑟𝑟⋅𝑁𝑁 ) + sin𝜑𝜑⋅𝑁𝑁 ≈ 0,          ( 𝑟𝑟⋅𝑄𝑄 /𝑟𝑟 relatively small) (9.9)
𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 𝑢𝑢 𝑘𝑘
′′ ′
( ) ( ) ( )
−𝑟𝑟⋅𝑀𝑀 − sin𝜑𝜑⋅𝑀𝑀 + ⋅ 𝑟𝑟⋅𝑁𝑁 + cos𝜑𝜑⋅𝑁𝑁 =𝑃𝑃⋅𝑟𝑟 (9.10)
𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣
𝑟𝑟
𝑘𝑘
( )
Substitution of Formulae (9.4) to (9.7) in Formulae (9.9), (9.10) and division by 𝐸𝐸⋅𝑒𝑒⋅𝑟𝑟/ 1−𝜈𝜈 gives
two differential formulae:
sin𝜑𝜑 1 𝜈𝜈⋅cos𝜑𝜑
′′ ′ ′
𝑈𝑈 − ⋅𝑈𝑈 +. . . +� + �⋅𝑊𝑊 +. . . = 0  (9.11)
𝑟𝑟 𝑟𝑟 𝑟𝑟
𝑘𝑘
2 2 2
1 𝜈𝜈⋅cos𝜑𝜑 𝐺𝐺 2⋅sin𝜑𝜑 1 2⋅𝜈𝜈⋅cos𝜑𝜑 cos 𝜑𝜑 𝑃𝑃⋅�1−𝜈𝜈 �
′ ′′′′ ′′′
� + �⋅𝑈𝑈 +. . . + ⋅ 𝑊𝑊 − ⋅𝑊𝑊 +. . . +� + + �⋅𝑊𝑊 = (9.12)
� �
𝑟𝑟 𝑟𝑟 12 𝑟𝑟 𝑟𝑟 𝑟𝑟 ⋅𝑟𝑟 𝑟𝑟 𝐸𝐸⋅𝐺𝐺
𝑘𝑘 𝑘𝑘 𝑘𝑘
lower derivatives are neglected (indicated by '…')
Approximate solution for P = 0:
Putting  𝑈𝑈 =𝐴𝐴⋅ exp(𝜒𝜒.𝑑𝑑) and 𝑊𝑊 =𝐶𝐶⋅ exp(𝜒𝜒.𝑑𝑑) gives:
𝑠𝑠𝑠𝑠𝑛𝑛𝜑𝜑 1 𝜈𝜈.𝑐𝑐𝑐𝑐𝑠𝑠𝜑𝜑 1 𝜈𝜈.𝑐𝑐𝑐𝑐𝑠𝑠𝜑𝜑
�𝜒𝜒 − .𝑥𝑥 +⋯� .𝐴𝐴 +�� + � .𝜒𝜒 +⋯� .�� + � .𝜒𝜒 +⋯� .𝐴𝐴
𝑟𝑟 𝑟𝑟 𝑟𝑟 𝑟𝑟 𝑟𝑟
𝑘𝑘 𝑘𝑘
2 2
𝑒𝑒 2⋅ sin𝜑𝜑 1 2.𝜈𝜈.𝑐𝑐𝑐𝑐𝑠𝑠𝜑𝜑 𝑐𝑐𝑐𝑐𝑠𝑠 𝜑𝜑
4 3
+� ⋅�𝜒𝜒 − ⋅𝜒𝜒 +. . .� +� + + ��
2 2
12 𝑟𝑟 𝑟𝑟 𝑟𝑟 .𝑟𝑟 𝑟𝑟
𝑘𝑘 𝑘𝑘
The determinant of those two formulae is zero if the following relation is fulfilled.
2 2
𝐺𝐺 3⋅sin𝜑𝜑 cos 𝜑𝜑
6 5 2 2
⋅�𝜒𝜒 − ⋅𝜒𝜒 +. . .� +𝜒𝜒 ⋅ (1−𝜈𝜈 )⋅ +. . . = 0 (9.13)
12 𝑟𝑟 𝑟𝑟
The roots of this are 𝜒𝜒 = 0
2 2
12⋅�1−𝜈𝜈 �⋅cos 𝜑𝜑 3⋅sin𝜑𝜑
4 3
and 𝜒𝜒 =− +𝜒𝜒 ⋅
2 2
𝐺𝐺 ⋅𝑟𝑟 𝑟𝑟
A first approximation for the latter is:
2 2
( )
12⋅ 1−𝜈𝜈 ⋅ cos 𝜑𝜑
�
𝜒𝜒 = ±2⋅ i⋅ ;
2 2
⋅𝑟𝑟
4⋅𝑒𝑒
2 2
( )
12⋅ 1−𝜈𝜈 ⋅ cos 𝜑𝜑
�
𝜒𝜒 = (±1 ± i)⋅
2 2
⋅𝑟𝑟
4⋅𝑒𝑒
where i =−1.
A second approximation is:
2 2 2 2
12⋅ (1−𝜈𝜈 )⋅ cos 𝜑𝜑 3⋅ sin𝜑𝜑 4⋅𝑒𝑒 ⋅𝑟𝑟
� ( ) �
𝜒𝜒 = ±2⋅ i⋅ ⋅�1 + ±1 ± i ⋅ ⋅ �
2 2 2 2
4⋅𝑒𝑒 ⋅𝑟𝑟 2⋅𝑟𝑟 12⋅ (1−𝜈𝜈 )⋅ cos 𝜑𝜑
and the ratio of this to the first approximation is:
𝜒𝜒 𝑒𝑒
𝐵𝐵𝐺𝐺𝐺𝐺𝑠𝑠𝑛𝑛𝑑𝑑
= 1 + (1 ± i)⋅ 3⋅ sin𝜑𝜑 ;          𝑑𝑑 = 2𝑟𝑟
�
�
𝜒𝜒 2
𝑓𝑓𝐺𝐺𝑟𝑟𝐵𝐵𝐺𝐺 𝑑𝑑⋅ cos𝜑𝜑⋅�12⋅ (1−𝜈𝜈 )
It can be assumed that the first approximation is sufficient, if
𝑒𝑒
3⋅ |sin𝜑𝜑|⋅ =Δ≤ 0,15 … 0,25,
�
𝑑𝑑⋅ cos𝜑𝜑⋅�12⋅ (1−𝜈𝜈 )
i.e.
Δ 𝑑𝑑
�
|sin𝜑𝜑|≤ ⋅ ⋅ cos𝜑𝜑⋅�12⋅ (1−𝜈𝜈 ),
3 𝑒𝑒
which can be simplified to:
cos𝜑𝜑 ≥ 1/[1 + (0,005 … 0,015)⋅𝑑𝑑 /𝑒𝑒 ] (9.14)
𝐵𝐵 𝐵𝐵 𝐵𝐵
Hence the first approximation is valid for ϕ ≥ 60° with d /e ≥ 100 …. 200. Numerical justification is
S S S
given in Annex A.
Solution of Formulae (9.11), (9.12) as first approximation (in (9.11) U´ and in (9.12) W´´´ neglected):
𝑢𝑢 𝑢𝑢 𝑢𝑢 𝑢𝑢
𝑈𝑈 =𝐴𝐴 +𝐴𝐴 ⋅ + exp�− ��𝐴𝐴 ⋅ cos +𝐴𝐴 ⋅ sin � (9.15)
0 1 2 3
𝑒𝑒 𝑒𝑒 𝑒𝑒 𝑒𝑒
𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑠𝑠
𝑢𝑢 𝑢𝑢 𝑢𝑢 𝑢𝑢
𝑊𝑊 =𝐶𝐶 +𝐶𝐶 ⋅ + exp�− ��𝐶𝐶 ⋅ cos +𝐶𝐶 ⋅ sin � (9.16)
0 1 2 3
𝑒𝑒 𝑒𝑒 𝑒𝑒 𝑒𝑒
𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑠𝑠
2 2
4⋅𝐺𝐺 ⋅𝑟𝑟
𝑙𝑙 =� ,   (𝑟𝑟≈𝑟𝑟 ,𝜑𝜑≈𝜑𝜑 ) (9.17)
𝐵𝐵 𝐵𝐵 𝐵𝐵
2 2
( )
12⋅1−𝜈𝜈 ⋅cos 𝜑𝜑
Substitution in (9.11), (9.12) gets:
2 2
1 1 𝜈𝜈⋅cos𝜑𝜑 1 2⋅𝜈𝜈⋅cos𝜑𝜑 cos 𝜑𝜑 𝑃𝑃⋅�1−𝜈𝜈 �
𝐶𝐶 = 0; 𝐴𝐴 ⋅ ⋅� + � +𝐶𝐶 ⋅� + + � = + (9.18)
1 1 0 2 2
𝑒𝑒 𝑟𝑟 𝑟𝑟 𝑟𝑟 𝑟𝑟 ⋅𝑟𝑟 𝑟𝑟 𝐸𝐸⋅𝐺𝐺
𝑠𝑠 𝑘𝑘 𝑘𝑘
𝑘𝑘
𝑒𝑒 1 𝜈𝜈⋅cos𝜑𝜑 𝑒𝑒 1 𝜈𝜈⋅cos𝜑𝜑
𝑠𝑠 𝑠𝑠
𝐴𝐴 = ⋅� + �⋅ (𝐶𝐶 +𝐶𝐶 );     𝐴𝐴 = ⋅� + �⋅ (𝐶𝐶 −𝐶𝐶 ) (9.19)
2 3 2 3 3 2
2 𝑟𝑟 𝑟𝑟 2 𝑟𝑟 𝑟𝑟
𝑘𝑘 𝑘𝑘
Boundary conditions at u = 0:
𝑊𝑊 =𝑊𝑊 =𝐶𝐶 +𝐶𝐶   →    𝐶𝐶 =𝑊𝑊−𝐶𝐶 (9.20)
(0) 𝐵𝐵 0 2 2 𝐵𝐵 0
′
( ) ( )
𝑊𝑊 =Θ = ⋅ 𝐶𝐶 −𝐶𝐶 +𝐶𝐶      →     𝐶𝐶 =Θ ⋅𝑙𝑙 + 𝑊𝑊−𝐶𝐶 (9.21)
(0) 𝐵𝐵 1 2 3 3 𝐵𝐵 𝐵𝐵 𝐵𝐵 0
𝑒𝑒
𝑠𝑠
𝑃𝑃⋅𝑟𝑟 𝐹𝐹
𝑠𝑠 𝑅𝑅
𝑁𝑁 ⋅ cos𝜑𝜑 +𝑄𝑄 ⋅ sin𝜑𝜑≡𝑁𝑁 ⋅ cos𝜑𝜑 −𝑄𝑄 ⋅ sin𝜑𝜑 = + (9.22)
𝑢𝑢𝑢𝑢(0) 𝑢𝑢(0) 𝐵𝐵 𝐵𝐵 𝐵𝐵 𝐵𝐵
2 2π𝑟𝑟
𝑠𝑠
𝐸𝐸⋅𝐺𝐺 1 𝜈𝜈⋅cos𝜑𝜑 𝐸𝐸⋅𝐺𝐺 1 1 𝜈𝜈⋅cos𝜑𝜑
′
𝑁𝑁 ≈ ⋅�𝑈𝑈 +� + � .𝑊𝑊� = ⋅�𝐴𝐴 ⋅ +𝐶𝐶 ⋅� + �� (9.23)
𝑢𝑢𝑢𝑢(0) 1 0
2 2
1−𝜈𝜈 𝑟𝑟 𝑟𝑟 1−𝜈𝜈 𝑒𝑒 𝑟𝑟 𝑟𝑟
𝑘𝑘 𝑠𝑠 𝑘𝑘
(0)
𝐸𝐸⋅𝐺𝐺 𝐸𝐸⋅𝐺𝐺 2
′′′
𝑄𝑄 ≈− ⋅𝑊𝑊 =− ⋅ ⋅ [2⋅ (𝑊𝑊−𝐶𝐶 ) +Θ ⋅𝑙𝑙 ] (9.24)
𝑢𝑢(0) 2 (0) 2 3 𝐵𝐵 0 𝐵𝐵 𝐵𝐵
( ) ( )
12⋅1−𝜈𝜈 12⋅1−𝜈𝜈 𝑒𝑒
𝑠𝑠
𝐸𝐸⋅𝐺𝐺 𝐸𝐸⋅𝐺𝐺 2
3 ′′
[ ( ) ]
𝑀𝑀 ≈− ⋅𝑊𝑊 = + ⋅ ⋅ 1⋅ 𝑊𝑊−𝐶𝐶 +𝛩𝛩 ⋅𝑙𝑙 (9.25)
𝑢𝑢𝑢𝑢(0) ( ) 2 𝐵𝐵 0 𝐵𝐵 𝐵𝐵
2 0 2
12⋅(1−𝜈𝜈 ) 12⋅(1−𝜈𝜈 ) 𝑒𝑒
𝑠𝑠
Formulae (9.23) and (9.24) show Q << N ; therefore (9.22) gives:
U UU
1 1 v⋅cos φ r F 1−v
S R
A ⋅ + C ⋅� + �≈ ⋅�P + �⋅ (9.26)
1 0 2
l r r 2⋅cos φ π⋅r E⋅e
S k S
S
Formulae (9.26) and (9.18) determine A and C :
1 0
𝑟𝑟 1 𝑟𝑟 𝐹𝐹
S R
S
𝐶𝐶 = ⋅�𝑃𝑃− ⋅� +𝑣𝑣�⋅�𝑃𝑃 + �� (9.27)
2 2
𝐸𝐸⋅𝐺𝐺⋅cos 𝜑𝜑 2 𝑟𝑟 ⋅cos 𝜑𝜑 𝜋𝜋⋅𝑟𝑟
S K S
S
A and A need not be calculated explicitly.
1 0
Finally (with E = E , e = e , r = r = d / 2 ) the results for the elastic case are as follows:
S S S S
𝐸𝐸 ⋅𝐺𝐺 ⋅2⋅cos 𝜑𝜑
S S S 2
( ) ( )
−𝑄𝑄 =𝑄𝑄 = ⋅�𝑊𝑊 −𝑊𝑊 ⋅𝑘𝑘 ⋅ 2⋅𝑙𝑙 + Θ −Θ ⋅𝑘𝑘 ⋅𝑙𝑙� (9.28)
u(0) S 2 S 0 1 S S 0 2 S
𝑑𝑑
S
𝐸𝐸 ⋅𝐺𝐺 ⋅2⋅cos 𝜑𝜑
S S S 2 3
( ) ( )
+𝑀𝑀 =𝑀𝑀 = ⋅�𝑊𝑊 −𝑊𝑊 ⋅𝑘𝑘 ⋅𝑙𝑙 + Θ −Θ ⋅𝑘𝑘 ⋅𝑙𝑙� (9.29)
uu(0) S 2 S 0 2 S S 0 3 S
𝑑𝑑
S
1 1 𝜋𝜋
+𝑁𝑁 =𝑁𝑁 = ⋅ 𝑄𝑄 ⋅ sin 𝜑𝜑 + ⋅�𝑃𝑃⋅ ⋅𝑑𝑑 +𝐹𝐹� (9.30)
� �
uu(0) S S S S R
cos𝜑𝜑 𝜋𝜋⋅𝑑𝑑 4
S S
𝑃𝑃⋅𝑑𝑑 ⋅𝑘𝑘 𝐹𝐹 ⋅𝑘𝑘 𝛼𝛼 ⋅Δ𝑇𝑇 ⋅𝑑𝑑
4 R 6 S S S
S
𝐶𝐶 =𝑊𝑊 = + + (9.31)
0 0
2 2
𝐸𝐸 ⋅𝐺𝐺 ⋅4⋅cos 𝜑𝜑 𝜋𝜋⋅𝐸𝐸 ⋅𝐺𝐺 ⋅cos 𝜑𝜑 2⋅cos 𝜑𝜑
S S S S S S S
𝑃𝑃⋅𝑑𝑑 ⋅𝑘𝑘 𝐹𝐹 ⋅𝑘𝑘
5 R 7
S
Θ = + (9.32)
2 2
𝐸𝐸 ⋅𝐺𝐺 ⋅𝑒𝑒 ⋅4⋅cos 𝜑𝜑 𝜋𝜋⋅𝐸𝐸 ⋅𝐺𝐺 ⋅𝑒𝑒 ⋅cos 𝜑𝜑
S S S S S S S S
Additional coefficients k to k and Θ are included to facilitate numerical comparison with the analytical
1 7 0
solution, for which:
k = k = k = 1 ; k = k = 0 ; and/or Θ = 0 (9.33)
1 2 3 5 7 0
1 𝑟𝑟 1 𝑟𝑟
S S
𝑘𝑘 = 1− ⋅� +𝑣𝑣� ; 𝑘𝑘 =− ⋅� +𝑣𝑣� (9.34)
4 6
2 𝑟𝑟 ⋅cos 𝜑𝜑 2 𝑟𝑟 ⋅cos 𝜑𝜑
k S k S
subject to:
cos 𝜑𝜑 ≥ 1/[1 + (0,005 … 0,015)⋅𝑑𝑑 /𝑒𝑒 ] (9.35)
S S S
9.2 Conical hub with cylindrical shell
The elastic stiffness of the system in Figure 10 has been calculated numerically (using computer program
'ROSCHA', TU Dresden).
For simplicity it was assumed that the system could be represented by an equivalent cylindrical shell as
follows:
𝑒𝑒 ≤𝑒𝑒 ≤𝑒𝑒 ; 𝑟𝑟 =𝑟𝑟 +𝑒𝑒 /2; 𝑟𝑟 =𝑟𝑟 +𝑒𝑒 /2
1 E 2 1 0 1 E 0 E
From Formulae (9.28) and (9.29):
𝐸𝐸 ⋅𝐺𝐺 𝐸𝐸 ⋅𝐺𝐺
S 1 S E
2 2
[ ]
𝑄𝑄 = ⋅ 𝑊𝑊 ⋅ 2⋅𝑙𝑙 ⋅𝑘𝑘 +Θ ⋅𝑙𝑙 ⋅𝑘𝑘 = ⋅�𝑊𝑊 ⋅ 2⋅𝑙𝑙 +Θ ⋅𝑙𝑙 � (9.36)
2 2 2 1 1 2 1 2 2 2 E 2 E
2⋅𝑟𝑟 2⋅𝑟𝑟
1 E
𝐸𝐸 ⋅𝐺𝐺 𝐸𝐸 ⋅𝐺𝐺
S 1 S E
2 3 2 3
[ ]
𝑀𝑀 = ⋅ 𝑊𝑊 ⋅𝑙𝑙 ⋅𝑘𝑘 +Θ ⋅𝑙𝑙 ⋅𝑘𝑘 = ⋅�𝑊𝑊 ⋅𝑙𝑙 +Θ ⋅𝑙𝑙 � (9.37)
2 2 2 1 3 2 1 3 2 2 E 2 E
2⋅𝑟𝑟 2⋅𝑟𝑟
1 E
2⋅𝑟𝑟⋅𝐺𝐺 2⋅𝑟𝑟 ⋅𝐺𝐺
1 1 E E
𝑙𝑙 = 𝑙𝑙 = (9.38)
� �
1 E
2 2
�12⋅(1−𝑣𝑣 ) �12⋅(1−𝑣𝑣 )
Calculations were performed for the following value:
𝑑𝑑 /𝑒𝑒 = 10 … 1000; 𝛽𝛽 =𝑒𝑒 /𝑒𝑒 = 1,5; 2; 3; 4; 6
1 1 2 1
𝑣𝑣 = 0,30; 𝜒𝜒 =𝑙𝑙 / 2⋅𝑟𝑟 ⋅𝑒𝑒 = 0,55; 1,10; 2,20; 4,40
�
H 1 1
Figure 10 — Conical hub with cylindrical shell
Values of factors k , k , k were then obtained by comparison of coefficients in Formulae (9.36) and (9.37)
1 2 3
2/5
1/2
𝑒𝑒 𝑟𝑟 𝑒𝑒 𝑟𝑟 𝑒𝑒 𝑟𝑟
E E 2/3 E E E E
= ⋅𝑘𝑘 ; =� ⋅𝑘𝑘� ; =� ⋅𝑘𝑘�
�
2 3
𝑒𝑒 𝑟𝑟 𝑒𝑒 𝑟𝑟 𝑒𝑒 𝑟𝑟
1 1 1 1 1 1
(9.39)
Calculated results are plotted in Figure 11, vertical bars indicate scatter, and the fitted curves are given
by:
𝑒𝑒 (𝛽𝛽− 1)⋅𝜒𝜒
E
≈ 1 +
𝑒𝑒 𝛽𝛽/3 +𝜒𝜒
(9.40)
Figure 11 — Calculated results
Effective diameters d for different cases are defined in Figure 12.
E
a) b) c)
Figure 12 — Effective diameters d for different cases
E
Maximum and minimum values are:
𝑑𝑑 = min(𝑑𝑑 −𝑒𝑒 +𝑒𝑒 ;𝑑𝑑 +𝑒𝑒 −𝑒𝑒 ) (9.41)
Emax 1 1 E 2 2 E
𝑑𝑑 = min (𝑑𝑑 +𝑒𝑒 −𝑒𝑒 ;𝑑𝑑 −𝑒𝑒 +𝑒𝑒 ) (9.42)
Emin 1 1 E 2 2 E
If the outer surface is cylindrical (Figure 12 a)):
𝑑𝑑 =𝑑𝑑 +𝑒𝑒 −𝑒𝑒 =𝑑𝑑 =𝑑𝑑 +𝑒𝑒 −𝑒𝑒 (9.43)
Emax 2 2 E Emin 1 1 E
and if the inner surface is cylindrical (Figure 12 c)):
𝑑𝑑 =𝑑𝑑 −𝑒𝑒 +𝑒𝑒 =𝑑𝑑 =𝑑𝑑 −𝑒𝑒 +𝑒𝑒  (9.44)
Emax 1 1 2 Emin 2 2 E
these two extreme cases are exact.
For intermediate cases assume:
𝑑𝑑 = (𝑑𝑑 +𝑑𝑑 )/2 (9.45)
E E𝐺𝐺𝐺𝐺𝑥𝑥 Emin
with this assumption the symmetric case (Figure 12 b)) is also exact:
1+𝜀𝜀
( )
Putting 𝑒𝑒 = ⋅ 𝑒𝑒 +𝑒𝑒 (9.46)
E 1 2
and d = d F
...