Effects of Analysis Method, Bolt Pre-Stress, and Cover Plate Thickness on
the Behavior of Bolted Flanges of Different Sizes
by
Christopher Michael Wowk
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING
Major Subject: MECHANICAL ENGINEERING
Approved:
_________________________________________
Norberto Lemcoff, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
May 2015
© Copyright 2015
by
Christopher Wowk
All Rights Reserved
i
CONTENTS
LIST OF TABLES ............................................................................................................................... iv
LIST OF FIGURES .............................................................................................................................. v
ACKNOWLEDGMENT...................................................................................................................... vi
ABSTRACT...................................................................................................................................... vii
NOMENCLATURE.......................................................................................................................... viii
1. Introduction.............................................................................................................................. 1
1.1
Background .................................................................................................................... 1
1.2
Problem Description ...................................................................................................... 2
2. Theory/Methodology ............................................................................................................... 4
2.1
2.2
2.3
Theory ............................................................................................................................ 4
2.1.1
Joint Separation Behavior ................................................................................. 4
2.1.2
Stress in the Center of the Cover ...................................................................... 8
Methodology.................................................................................................................. 9
2.2.1
Solid Element Finite Element Model ................................................................ 9
2.2.2
Shell and Beam Element Finite Element Model ............................................. 13
Evaluation vs Analysis Type ......................................................................................... 14
3. Results and Discussion ........................................................................................................... 17
3.1
3.2
3.3
Joint Separation Behavior ............................................................................................ 17
3.1.1
Joint Separation Behavior Due to Differing Bolt Pre-Stress and
Analysis Type................................................................................................... 17
3.1.2
Maximum Joint Separation Due to Cover Plate Thickness ............................. 21
3.1.3
Maximum Joint Separation Due to Normalized Bolt Pre-Stress and
Nominal Pipe Size............................................................................................ 22
Location of Contact Outside Bolt Circle ....................................................................... 23
3.2.1
Location of Contact Due to Differing Bolt Pre-Stress...................................... 23
3.2.2
Normalized Location of Contact Due to Nominal Pipe Size ............................ 24
Stress in the Center of the Cover Plate ........................................................................ 25
ii
3.3.1
Stress in the Center of the Cover Plate Due to Differing Bolt Preload ........... 26
3.3.2
Stress in the Center of the Cover Plate Due to Cover Plate Thickness ........... 28
4. Conclusions............................................................................................................................. 30
5. References .............................................................................................................................. 32
Appendix A ...................................................................................................................................... 1
Appendix B ...................................................................................................................................... 1
iii
LIST OF TABLES
Table 1 – Flange Component Geometric Parameters..................................................................... 3
Table 2 - Material Summary............................................................................................................ 3
Table 3 – Analyses Performed ...................................................................................................... 16
iv
LIST OF FIGURES
Figure 1 – Typical Class 3, Category 1 Appendix Y Joint (Half Section) ............................................ 2
Figure 2 – Free Body Diagram of Class 3, Category 1 Flange (Waters and Schneider 1969) ........... 5
Figure 3 – Force and Moment Diagram for Annular Ring Portion of Cover Plate & Flange ............ 6
Figure 4 – Meshed Solid Element ABAQUS Model of Class 3, Category 1 Flange Pair .................. 10
Figure 5 – Solid Element Model Boundary Conditions and Loads ................................................. 11
Figure 6 – COPEN Extraction Path.................................................................................................. 12
Figure 7 – Symbol Plot of CNORMF for 80% Yield Bolt Pre-Stress Case ........................................ 12
Figure 8 – Meshed Shell/Beam Element ABAQUS Model of Class 3, Category 1 Flange Pair ....... 14
Figure 9 – Maximum Joint Separation at Sealing Element vs. Bolt Pre-Stress .............................. 18
Figure 10 – Joint Separation vs. Distance from the Outside Diameter of Flange – Radial
Beam Theory Analysis Method .................................................................................... 20
Figure 11 – Joint Separation vs. Distance from the Outside Diameter of Flange – Solid
Element Finite Element Model .................................................................................... 20
Figure 12 – Joint Separation vs. Distance from the Outside Diameter of Flange – Shell/Beam
Element Finite Element Model .................................................................................... 21
Figure 13 – Maximum Joint Separation at Sealing Element vs. Bolt Pre-Stress (Different
Cover Plate Thicknesses) ............................................................................................. 22
Figure 14 – Maximum Joint Separation at Sealing Element vs. Bolt Pre-Stress (4 and 16 NPS
Sizes) ............................................................................................................................ 23
Figure 15 – Normalized Contact Distance from Bolt Circle vs Bolt Pre-Stress............................... 24
Figure 16 – Normalized Contact Distance from Bolt Circle vs Bolt Pre-Stress (16 NPS Flange
Pair).............................................................................................................................. 25
Figure 17 – Radial Stress (S11) in Cover Plate vs Bolt Pre-Stress ................................................... 27
Figure 18 – Tangential Stress (S22) in Cover Plate vs Bolt Pre-Stress............................................ 27
Figure 19 – Radial Stress (S11) in Cover Plate vs Cover Plate Thickness ....................................... 29
Figure 20 – Tangential Stress (S22) in Cover Plate vs Cover Plate Thickness ................................ 29
v
ACKNOWLEDGMENT
I would like to thank my project advisor, Professor Norberto Lemcoff, for his support and understanding during the process of completing this project. His advice and patience were invaluable
to this project.
I would also like to thank the many professors I have had over the course of my education at
Rensselaer Polytechnic Institute. The knowledge and insight in the field of mechanical engineering
they have transferred to me during my time here were also invaluable to the completion of this
project.
Finally, I would like to thank my wife, Lara, and my parents for all of their love and support, without which completion of this project and degree program would not have been possible.
vi
ABSTRACT
Guidelines for sizing the flange geometry and bolting requirements for flanges with contact outside the bolting circle are provided in Non-Mandatory Appendix Y of the American Society of
Mechanical Engineers Boiler and Pressure Vessel Code. However, Appendix Y does not specify a
method for determining the separation of the flange components when subjected to internal
pressure loading, an important parameter when selecting a sealing element. This paper investigates the behavior of the flanged joint for Class 3, Category 1 Appendix Y flange pairs using the
radial beam theory method, a finite element analysis using solid elements, and a finite element
analysis using shell and beam elements. This investigation identified similarities between the results of the different analysis methods and also potential shortcomings and limitations for the
analysis methods.
All analysis types predicted the same relationships between bolt pre-stress and cover plate thickness on joint behavior; joint separation decreases with increasing bolt pre-stress and cover plate
thickness. Although the relationships were the same, slightly different results were predicted by
each analysis method. Shell/beam element finite element models predicted smaller joint separations and lower stresses in the center of the cover plate when compared to the other methods.
Radial beam theory and solid element finite element models were found to be in generally good
agreement for joint separation and location of contact at high bolt pre-stresses, however the radial beam theory method was unable to predict complete separation of the joint at low bolt prestress.
vii
NOMENCLATURE
a
Width of the Outermost Region of the Radial Beam
b
Distance from Bolt Circle to Location of Contact
bmax
Distance from Bolt Circle to Outside Diameter of Flange
B1
Diameter of the Sealing Element
C
Width of the Innermost Region of the Radial Beam
CNORMF
Normal Force due to Contact Acting on Each Node
COORD1
Radial Location of Each Node
COPEN
Distance of Each Node to Contact Surface
f
Bolt Hole Flexibility Constant
f’
Bolt Hole Flexibility Constant
f”
Bolt Hole Flexibility Constant
F
Axial Force Acting On Cover Plate/Flange Due To Pressure
I
Moment Of Inertia
ℓ
Distance from the Bolt Circle to the Seal
L
Length of the Flange from Point of Contact to Seal
M
Bending Moment Acting on Flange/Cover Plate
M1
Total Bending Moment Acting on Cover Plate
MSII
Total Bending Moment Acting on Cover Plate
Nbolts
Number of Bolts in Bolt Circle
P
Internal Pressure of the Joint
Rm
Radius of the Sealing Element
S11
Radial Stress Predicted By ABAQUS
S22
Tangential Stress Predicted By ABAQUS
SRIIBC
Radial Stress in the Center of the Cover Plate Predicted By BPVC
viii
SRIIWS
Radial Stress in the Center of the Cover Plate Predicted By RBT
STIIBC
Tangential Stress in the Center of the Cover Plate Predicted By BPVC
STIIWS
Radial Stress in the Center of the Cover Plate Predicted By RBT
t
Thickness of the Cover Plate or Flange
tII
Thickness of the Cover Plate
v(x)
Separation of the Joint as a Function of x
x
Distance from Outside Diameter
Greek Letters
θ
Rotation of the Radial Beam
θi
Rotation of the Radial Beam at Outside Diameter
θreq_sol
Required Angular Section for Solid Element FEA
θreq_shell
Required Angular Section for Shell/Beam Element FEA
ν
Poisson’s ratio
Subscripts
A
Referring to the outermost portion of the radial beam
B
Referring to the innermost portion of the radial beam
ix
ACRONYMS
ASME
American Society of Mechanical Engineers
BPVC
Boiler and Pressure Vessel Code
FEA
Finite Element Analysis
RBT
Radial Beam Theory
x
1. Introduction
1.1 Background
Bolted pressure vessel flanges are typical for fluid power applications when disassembly of the
joint is required for maintenance or access to the internals of the system. Bolted flanges can be
broken into two general categories: flanges with no contact beyond the bolting circle, and flanges
with contact beyond the bolting circle. Flanges without contact beyond the bolting circle are
stressed during bolt up. Flanges with contact between the mating flanges outside the bolting circle are not stressed until pressurization. The behavior of these flanges are dependent on the prestress carried in the bolts and the interaction between the flanges in contact. This project will
focus on the behavior of the latter type of flanges.
Standards shapes and sizes exist for flanges with contact outside the bolting circle; however, if a
custom design is needed, it is the responsibility of the designer to ensure a leak-free joint is maintained at the design pressure, and that the components of the assembly do not fail in service.
Guidelines for sizing the flange geometry and bolting requirements are provided in Non-Mandatory Appendix Y of the American Society of Mechanical Engineers (ASME) Boiler and Pressure
Vessel Code (BPVC) (ASME 2013). The basis of the analysis method utilized by Appendix Y involves
considering each flange as a collection of discrete, radial beams whose behavior can be described
through beam theory (Schneider 1968). The analytical procedures presented in Appendix Y cover
many different configurations of flanged joints with contact outside the bolt circle. Past work has
been performed on the agreement between the behavior of symmetric flange pairs predicted by
the radial beam theory, classical plate theory, and finite element analysis (Galai and Bouzid 2010).
1
1.2 Problem Description
This project will focus on the behavior of a typical flat faced flange with a flat cover plate, as
predicted by three different analysis methods: a) radial beam theory (RBT), b) finite element analysis (FEA) using solid elements, and c) finite element analysis using shell elements for the
components of the flange and beam elements for the bolts, for differing cover plate thicknesses,
bolt preload, and flange sizes. In Appendix Y nomenclature, this joint configuration is designated
as a Class 3, Category 1 Appendix Y flange pair. The flange pair is made up of a cover plate,
flange/hub, and fastener hardware. A typical joint of this type is shown in Figure 1.
Figure 1 – Typical Class 3, Category 1 Appendix Y Joint (Half Section)
Joint separation behavior is an important characteristic in determining the leak behavior of the
joint, as it quantifies separation of the flanges at the location of the sealing element. In addition
to the investigation into the behavior of the joint under pressure loading, additional analysis will
be performed to characterize the stress predictions for different analysis methods in the center
of the cover plate.
2
The geometry of the flanged joints evaluated by this project are based on standard sizes given in
ASME B16.5, Pipe Flanges and Flanged Fittings for Class 1500 flanges, and are shown in Table 1.
The ASME B16.5 class designation dictates the operating pressure rating for the flange components, with Class 1500 being rated for a 1,500 psi operating pressure. Materials allowed by ASME
B16.5 were used for all joint components. The cover and flange/hub are made from ASTM A515
Grade 60 carbon steel, and the bolts are made of ASTM A193 Grade B7 stainless steel. Material
properties are shown in Table 2.
Table 1 – Flange Component Geometric Parameters
NPS Size
4
16
Outside Diameter of Flange and Cover (in)
12.25
32.50
Flange and Cover Plate Thickness (in)
2.12
5.75
Bolt Circle Diameter (in)
9.50
27.75
Bolt Hole Diameter (in)
1.38
2.63
8
16
Nominal Bolt Diameter (in)
1.25
2.50
Bolt Tensile Diameter (in)
1.11
2.26
Bolt Tensile Area (in)
0.969
4.00
Pipe Bore (in)
4.60
16.19
Hub Outside Diameter (in)
6.38
21.75
Flange/Hub Length (in)
3.56
10.25
Center of Seal Groove Diameter (in)
5.49
18.97
Number of Bolts
Table 2 - Material Summary
Flange
Component
Material
Specification
Young’s
Modulus
Yield
Strength
Poisson’s
Ratio
Cover Plate &
Flange/Hub
ASTM A515
Grade 60
30,000 ksi
32 ksi
0.3
Bolts
ASTM A193
Grade B7
30,000 ksi
105 ksi
0.3
3
2. Theory/Methodology
2.1 Theory
This project will investigate the joint separation behavior and cover plate stresses for Class 3, Category 1 Appendix Y flanges with varying cover plate thickness, bolt preload, and nominal pipe size.
Bolted joint behavior will be determined using the RBT developed by Waters and Schneider
(1969), equations given in Appendix Y, finite element models consisting of solid elements, and
finite element models consisting of more computationally efficient shell and beam elements.
2.1.1
Joint Separation Behavior
Appendix Y of the BPVC provides a method for sizing Class 3, Category 1 flange pairs. Included in
this method are equations to predict the rotation of the cover plate and flange at the location of
the sealing element, stresses in the flange components, and the pre-stress and operating stress
of the bolts to ensure contact outside the bolt circle at a location specified by the designer. Although useful in confirming a design will satisfy the stress requirements of the BPVC, the equations
in Appendix Y do not readily support determination of the joint separation behavior, or allow for
analysis of cases of low bolt pre-stress where the relative rotations of the cover and flange at the
outside diameter are non-zero. In order to evaluate cases with low bolt pre-stress, additional investigation and manipulation of the method used to create the Appendix Y method is required.
This method was first proposed by Schneider (1968), and considers the bending behavior of the
flange components as a series of radial beams. Schneider (1968) initially applied this method to
identical flange pairs where symmetry could be exploited to simplify the analysis, but later modified the identical flange pair method for use in analysis with non-identical flange pairs (Waters
and Schneider 1969). Waters and Schneider’s (1969) method for evaluating non-identical flange
pairs is presented below.
4
A free body diagram of a Class 3, Category 1 Appendix Y flange configuration is shown in Figure 2.
Fluid pressure acts on the area of the cover and flange within the diameter of the seal, which is
assumed to be located at the mid-thickness of the hub, Rm. The internal pressure not only acts to
separate the joint between the cover and flange, but also creates a large overturning moment
which creates a counter-clockwise rotation of the outer portion of the flange.
Figure 2 – Free Body Diagram of Class 3, Category 1 Flange (Waters and
Schneider 1969)
Contact is assumed to occur as a uniform line load outside the bolt circle between the flange and
cover, at a distance b, to react the prying loads acting on the bolts. The distance between the bolt
circle and the location of contact, b, is determined by equating the moments acting on the flange
and cover, respectively, about the location of contact, R. The true location of R is dependent on
the preload of the bolts and the separation of the flanges. In the method presented by Waters
and Schneider (1969), it is more computationally efficient to select a location of R and then solve
for the required bolt preload to create contact at the selected R, than to specify the preload and
solve for R.
5
For non-identical flanges, Waters and Schneider’s method assumes that the two sets of moments
act on the flanges. A system of balanced forces and moments acts equally on the flange and
cover. This system of forces and moments produces the same flange separation behavior as the
identical flange pair method (Schneider 1968). The remaining unbalanced system of forces and
moments causes rigid body rotation of the flanges as a pair, and does not act to further separate
the joint.
The joint must be considered as four separate pieces as shown in Figure 2. The outer rings of the
cover plate and flange must be analyzed as systems of discrete, radial beams in order to use beam
theory to determine displacements. The free body and moment diagram for the annular portion
of the cover plate and flange considered as a beam is shown in Figure 3.
Figure 3 – Force and Moment Diagram for Annular Ring Portion of Cover Plate &
Flange
6
The moment along the length of the radial beam can be expressed as:
𝑀𝐴 =
𝑓" 𝑀+𝑓"𝐹ℓ
𝑥
𝑏
𝑀𝐵 = (𝑓𝑀 + 𝑓 ′ 𝐹ℓ) − 𝑓 ′ 𝐹(𝑥 − 𝑏)
0≤x<b
(1)
b≤x≤L
(2)
Note that in the above equations, factors f, f’, and f” are included to account for the increased
flexibility allowed by the bolt holes. The curvature of the beam can be determined by integrating
the expression M/EI. To account for the decrease in width of the beam as x increases (moves
towards the center), the beam will be considered as made up of two regions, the outer region
with width a, and the inner with width c. To maintain continuity in the circumferential direction,
the term (1-ν2) is applied to the moment of inertia equations.
𝑎𝑡 3
𝐼𝐴 = 12∗(1−𝜈2 )
0≤x<b
(3)
b≤x≤L
(4)
𝑏 𝑀𝐴
𝑑𝑥
𝐸𝐼𝐴
0≤x<b
(5)
𝐿 𝑀𝐵
𝑑𝑥
𝐸𝐼𝐵
b≤x≤L
(6)
0≤𝑥<𝑏
𝑏≤𝑥≤𝐿
(7)
𝑐𝑡 3
𝐼𝐵 = 12∗(1−𝜈2 )
The curvature of the beam can be determined as:
𝜃𝐴 (𝑥) = ∫0
𝜃𝐵 (𝑥) = ∫𝑏
𝜃(𝑥) = {
𝜃𝐴 (𝑥)
𝜃𝐵 (𝑥)
𝑓𝑜𝑟
with boundary conditions θA(0) = θi and θA(b)= θB(b). When b<bmax, θi is equal to zero, since the
flange and cover are in contact over the length from b to bmax. When contact occurs at the outside
diameter, b=bmax, a non-zero rotation can occur at the point of contact.
The deflection of the beam can then be determined as:
𝑏
𝑣𝐴 (𝑥) = ∫0 𝜃𝐴 (𝑥) 𝑑𝑥
0≤x<b
7
(8)
𝐿
𝑣𝐵 (𝑥) = ∫𝑏 𝜃𝐵 (𝑥) 𝑑𝑥
𝑣 (𝑥)
𝑣(𝑥) = { 𝐴
𝑣𝐵 (𝑥)
𝑓𝑜𝑟
b≤x≤L
(9)
0≤𝑥<𝑏
𝑏≤𝑥≤𝐿
(10)
with boundary conditions vA(0) = 0 and vA(b)= vB(b). The expression for v(x), too cumbersome to
be presented in this report, is dependent on the reaction forces and moments that must be determined by solving the system of equations for the entire flange assembly. The equations
necessary to determine the forces and moments were presented by Waters and Schneider (1969),
and are shown in the Appendix A Maple worksheet. To solve for the forces, a value of b and/or θi
must be assumed in order to calculate the resulting bolt pre-stress and joint separation. Trial and
error was used to determine the values of b and θi for the bolt preloads investigated in this project. The joint separation between the cover plate and flange was then calculated based on the
resulting values of b and θi. In addition to the values of b determined by RBT, for higher bolt prestresses, equations in Appendix Y were used to predict the location of contact as function of bolt
pre-stress (ASME 2013).
2.1.2
Stress in the Center of the Cover
Stress in the center of the cover was determined using equations presented in Appendix Y of the
BPVC (ASME 2103) and by Waters and Schneider (1969). Appendix Y of the BPVC predicts the
radial and tangential stresses at the center of the cover plate to be equal. The stresses can be
calculated using equation 38 of Appendix Y:
𝑆𝑅𝐼𝐼𝐵𝐶 = 𝑆𝑇𝐼𝐼𝐵𝐶 =
0.3094∗𝑃∗𝐵12
2
𝑡𝐼𝐼
6∗𝑀𝑆𝐼𝐼
2
1 ∗𝑡𝐼𝐼
− 𝜋∗𝐵
(11)
where P is the operating pressure, B1 is the diameter of the sealing element, tII is the thickness of
the cover, and MSII is the total flange moment for the cover plate at the sealing element diameter,
which can be calculated using the equations in Appendix Y of BPVC.
8
Waters and Schneider (1969) offer an equation to predict the radial and tangential stresses at the
center of the cover also. As in Appendix Y, both the radial and tangential stresses at the center of
the cover plate are predicted to be equal. The magnitude of the stresses are given by equation 48
of Waters and Schneider (1969):
3
𝑃∗𝑅𝑚2 ∗(3+𝑣)
−
8
𝑆𝑅𝐼𝐼𝑆𝑊𝑆 = 𝑆𝑇𝐼𝐼𝑊𝑆 = 𝑡 2 ∗ [
𝐼𝐼
2 ∗ 𝑀1 ]
(12)
where P is the operating pressure, Rm is the radius of the sealing element, tII is the thickness of
the cover plate, ν is the Poisson’s ratio of the cover material, and M1 is the total flange moment
for the cover plate at the location of the sealing element, which can be calculated as described in
Section 2.1.1.
2.2 Methodology
Maple worksheets, provided in Appendix A, were created to perform the RBT analysis. Microsoft
Excel spreadsheets, provided in Appendix B, were developed to perform the Appendix Y calculations. In addition to the analytical modeling performed by the Maple worksheets and Microsoft
Excel spreadsheets, joint behavior was investigated using ABAQUS finite element models. Models
using solid elements, and more computationally efficient shell and beam elements, were created
and used for evaluating the behavior of the bolted joints.
2.2.1
Solid Element Finite Element Model
Cylindrical symmetry of the flanged joint was exploited to reduce the number of elements needed
to properly characterize the contact between the cover plate, flange, and bolts. An angular segment of the flange pair from the center of one bolt to the midpoint of the cover/flange between
the initial bolt hole and subsequent bolt hole is all that is required to fully define the behavior of
the joint. The angular segment is dependent on the number of bolts in the bolt circle, and is determined by:
9
180
𝜃𝑟𝑒𝑞−𝑠𝑜𝑙𝑖𝑑 = 𝑛
(13)
𝑏𝑜𝑙𝑡𝑠
So, for a flange with eight bolts in the bolt pattern, a 22.5° segment is required, and for a flange
with 16 bolts in the bolt pattern, an 11.25° segment is required.
Solid models of the joint segment were created using ABAQUS CAE with the dimensions shown in
Table 1. The parts were partitioned to force nodes of the contact surfaces between the bolts,
cover plate, and flange to align, to allow for improved contact recognition. All parts were meshed
using C3D8 elements (8 noded linear brick elements). The meshed assembly is shown in Figure 4.
Figure 4 – Meshed Solid Element ABAQUS Model of Class 3, Category 1 Flange
Pair
To account for the effects of symmetry, a cylindrical coordinate system was created and symmetrical boundary conditions were applied to the appropriate faces of the parts. Axial displacement
of the assembly was restrained by fixing the axial displacement of the free face of the hub to zero.
Contact was defined throughout the model with the use of the “General Contact” interaction.
The analysis was performed in two steps. In the first step, the pre-stress was applied to the bolt.
This was done using a “Bolt Preload” load applied to the middle of the bolt. Since only half of the
10
bolt appeared in the model, only half of the preload was required to be applied in order to generate the required bolt pre-stress. In the second step, a pressure load of 1,500 psi was applied to
the cover from the center up to the location of the hypothetical sealing element at Rm, the bore
of flange/hub, and also from the face of the flange up to the location of the hypothetical sealing
element. Boundary conditions and loads applied to the assembly are shown in Figure 5.
Figure 5 – Solid Element Model Boundary Conditions and Loads
To determine the joint separation behavior of the solid element model, the Contact Opening
(COPEN) field output was extracted from the model after the pressure load was applied along a
path at the edge of the flange between bolt circles. This field output tracks the distance between
two faces in close proximity as part of ABAQUS’s contact algorithm. The extraction path for COPEN
is shown in Figure 6.
11
Figure 6 – COPEN Extraction Path
Since the solid element FEA model allowed for contact between the flange and cover at multiple
radii outside the bolt circle, the location of contact predicted by the solid element FEA model was
approximated by calculating the total moment acting on the inside face of the cover due to contact with the flange, and dividing it by the total contact force acting on the inside face of the cover.
To accomplish this, the Contact Normal (CNORMF) and radial node location (COORD1) field outputs were extracted for every node on the inside surface of the cover plate. The location of
contact and distance from the bolt circle could then be calculated as shown in Equations (14)
through (17).
Figure 7 – Symbol Plot of CNORMF for 80% Yield Bolt Pre-Stress Case
12
𝑇𝑜𝑡𝑎𝑙 𝑀𝑜𝑚𝑒𝑛𝑡 = ∑𝑛𝑖=0 𝐶𝑂𝑂𝑅𝐷1𝑛 ∗ 𝐶𝑁𝑂𝑅𝑀𝐹𝑛
(14)
𝑇𝑜𝑡𝑎𝑙 𝐹𝑜𝑟𝑐𝑒 = ∑𝑛𝑖=0 𝐶𝑁𝑂𝑅𝑀𝐹𝑛
(15)
𝐿𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝐶𝑜𝑛𝑡𝑎𝑐𝑡 =
𝑏=
𝐷𝐵𝑜𝑙𝑡𝐶𝑖𝑟𝑐𝑙𝑒
2
𝑇𝑜𝑡𝑎𝑙 𝑀𝑜𝑚𝑒𝑛𝑡
𝑇𝑜𝑡𝑎𝑙 𝐹𝑜𝑟𝑐𝑒
(16)
− 𝐿𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝐶𝑜𝑛𝑡𝑎𝑐𝑡
(17)
Radial and tangential stresses in the center of the cover were directly extracted from the FEA
model. The analysis results were transformed into cylindrical coordinates, and the radial and tangential stress were extracted as S11 and S22, respectively.
2.2.2
Shell and Beam Element Finite Element Model
Cylindrical symmetry of the flanged joint was once again exploited to reduce the number of elements needed to properly characterize the contact between the cover plate, flange, and bolts. An
angular segment of the flange pair from subsequent midpoints between bolt holes was required
to fully define the behavior of the joint, since it is not feasible to model half the bolt with a beam
element. The angular segment required was determined by the number of bolts in the bolt circle,
and is calculated from:
360
𝜃𝑟𝑒𝑞−𝑠ℎ𝑒𝑙𝑙 = 𝑛
(18)
𝑏𝑜𝑙𝑡𝑠
So, for a flange with eight bolts in the bolt pattern, a 45° segment is required, and for a flange
with 16 bolts in the bolt pattern, a 22.5° segment is required.
Shell models of the flange segment were created using ABAQUS CAE with the dimensions shown
in Table 1. The bolt was represented in the analysis as a B33 element (2 node cubic beam in space)
with a circular cross section equal to the bolt’s stress area. Each end of the beam was attached to
the cover and flange, with a kinematic coupling constraint to represent the bolt head and nut. The
influence radius for the kinematic couplings were set equal to the diameter of the head of the
13
bolt. The parts were partitioned to force nodes of the contact surfaces between the cover plate
and flange to align to allow for improved contact recognition. The cover plate and flange were
meshed using S4 elements (4 node doubly curved general-purpose shell, finite membrane strain
elements). Shell offsets were set so that the inner faces of the cover plate and flange/hub had
zero separation. The meshed assembly is shown in Figure 8.
Figure 8 – Meshed Shell/Beam Element ABAQUS Model of Class 3, Category 1
Flange Pair
Boundary conditions, load steps, and contact interactions were applied to the shell model in the
same manner as they were applied to the solid element model with the following exception: the
full bolt preload was applied since the full cross-section of the bolt was included in the analysis.
The joint separation behavior, radial location of contact between the cover plate and flange, and
stress in the center of the cover plate were extracted from the shell/beam model in the same
manner as they were extracted from the solid element model, as described in Section 2.2.1.
2.3 Evaluation vs Analysis Type
In order to reduce the number of analyses required to compare the effects of the multiple independent variables investigated by this project, not all combinations of nominal pipe size, bolt
14
preload, and cover plate thickness were evaluated with each analysis method. Instead, the influences of each independent variable were evaluated using some of the analysis methods
presented in Sections 2.1 and 2.2 in order to allow for general statements regarding their effect
to be made. The analysis type performed for each perturbation of independent variable is shown
in Table 3.
15
Table 3 – Analyses Performed
4"
NPS Size
Analysis
Method
Cover Thickness
Bolt Pre-Stress
Joint
Separation
Along Length
t1
0
Pressure
Load
0.5*t1
25%
Yield
80%
Yield
0
Pressure
Load
Radial Beam Theory
Solid Element FEA
Shell Element FEA
Radial Beam Theory
Radial
Location of
Contact
Appendix Y Equations
Solid Element FEA
Shell Element FEA
Radial Beam Theory
Stress In
Center of
Cover
Appendix Y Equations
Solid Element FEA
Shell Element FEA
Maximum
Joint
Separation
16"
Radial Beam Theory
Solid Element FEA
Shell Element FEA
16
25%
Yield
2*t1
80%
Yield
0
Pressure
Load
t2
25%
Yield
80%
Yield
0
Pressure
Load
25%
Yield
80%
Yield
3. Results and Discussion
3.1 Joint Separation Behavior
The separation of the joint between the cover plate and flange with differing analysis types, bolt
preload, cover plate thickness, and nominal pipe size, under a 1,500 psi internal pressure load,
was investigated. The separation of the joint from the outside diameter of the flanges inward to
the location of the sealing element was evaluated for a standard sized 4 NPS Class 3, Category 1
Appendix Y flange pair with differing bolt preloads using different analysis methods. The maximum separation of the joint at the sealing element was also determined for the same joint with
different cover plate thicknesses using RBT, solid element finite element models, and shell/beam
element finite element models. RBT and solid element finite element models were also used to
compare the joint separation of standard sized 4 NPS and 16 NPS Class 3, Category 1 Appendix Y
flange pairs with differing preloads.
3.1.1
Joint Separation Behavior Due to Differing Bolt Pre-Stress and Analysis
Type
The maximum joint separation at the sealing element as a function of the pre-stress in the bolts
is shown in Figure 9. It can be seen that the three analysis methods show a similar relationship,
and the joint separation decreases as the bolt pre-stress increases. This relationship is important,
as a larger separation of the cover plate and flanges indicates a potential for the joint to leak in
service, because the sealing element can only expand a finite amount. Typically, bolts in pressure
containing applications are torqued to a pre-stress equal to at least the load they will see in service, and in this way reduce cyclic stresses caused by pressurization and depressurization cycles.
The results of this analysis indicate that a preload greater than the operating load may be required
for Class 3, Category 1 Appendix Y flange pairs in order to minimize joint separation at the sealing
element. The magnitude of the preload is dependent on the capabilities of the sealing element,
17
however the results of this project indicate that a preload at least 6x the operating load can significantly reduce joint separation at the seal.
Additionally, at higher pre-stresses for each analysis method, the total joint separation appears
to approach a non-zero limit. It is suspected that these limits are the result of flexure of the cover
plate and flange/hub. This non-zero lower limit at the sealing element indicates that the capabilities of the sealing element must be matched to the expected joint separation, even when high
bolt pre-stresses are utilized in order to ensure a leak-free joint.
Figure 9 – Maximum Joint Separation at Sealing Element vs. Bolt Pre-Stress
The total joint separation as a function of the distance from the outside diameter of the flange
using RBT is shown in Figure 10, using solid element FEA models in Figure 11, and using shell/beam
element FEA models in Figure 12. The general magnitudes of separation from RBT and solid element FEA are in good agreement for low bolt pre-stresses. The shell/beam element FEA models
predicts much smaller joint separations for all preloads. Upon further evaluation of the deflection
curves, the separation along the length of the joint appears to be linear for the RBT and solid
18
element FEA results, whereas the separation appears to be exponential for the shell/beam element FEA results. This suggests that the elongation of the bolts has a larger influence in the joint
separation when RBT and solid element FEA are used, and the flexibility of the cover plate and
flange has a larger influence when the shell/beam element FEA analysis is used. This difference in
results is suspected to be caused by the artificial stiffening of the bolts caused by the kinematic
coupling constraint used to model the bolt heads in the shell/beam element FEA models.
For the RBT analysis, there seems to be a limit where there is a much smaller variation in the
behavior of the joint at higher bolt pre-stresses, as not much difference is seen between the 25%
and 80% Yield curves. This limit is not seen when the joint is analyzed using either FEA methods.
Both FEA methods predict complete separation of the cover plate and flange at the outside diameter when the bolts are not preloaded. This complete separation was not observed in the RBT
analysis. It must be noted that complete separation of the flanges is not supported by the RBT
analysis method, as contact is assumed to occur at least at the outside diameter. This limitation
in the RBT analysis indicates a potential shortcoming in the RBT analysis at lower bolt pre-stresses.
19
Figure 10 – Joint Separation vs. Distance from the Outside Diameter of Flange –
Radial Beam Theory Analysis Method
Figure 11 – Joint Separation vs. Distance from the Outside Diameter of Flange –
Solid Element Finite Element Model
20
Figure 12 – Joint Separation vs. Distance from the Outside Diameter of Flange –
Shell/Beam Element Finite Element Model
3.1.2
Maximum Joint Separation Due to Cover Plate Thickness
The maximum joint separation at the sealing element as a function of the pre-stress in the bolts
and cover plate thickness is shown in Figure 13. The maximum joint separation was calculated
using RBT and solid element FEA models. It can be seen that both analysis methods show similar
behavior: joint separation decreases as bolt pre-stress and cover plate thickness increase. For the
thinner cover plate models, the results of the RBT and solid element FEA diverge as bolt preload
increases. For the standard size and double thickness models, both analysis methods showed
good agreement, however smaller joint separations were predicted at higher preloads by the solid
element FEA models. The results from this analysis indicate that joint separation can be decreased
by increasing the bolt pre-stress or cover plate thickness.
21
Figure 13 – Maximum Joint Separation at Sealing Element vs. Bolt Pre-Stress
(Different Cover Plate Thicknesses)
3.1.3
Maximum Joint Separation Due to Normalized Bolt Pre-Stress and Nominal
Pipe Size
The maximum joint separation at the sealing element as a function of the pre-stress in the bolts
for standard sized 4 and 16 NPS flange pairs is shown in Figure 14. The maximum joint separation
for the 16 NPS flange pair was calculated using RBT and solid element FEA models, whereas only
the results of the RBT analysis are shown for the 4 NPS flange pair. The RBT and solid element FEA
results are in good agreement for maximum separation of the flanges at the sealing element. It
can be seen that greater joint separation is predicted for the larger size flange pair when the bolt
pre-stress is low. As bolt pre-stress increases, the joint separation of the larger flange pair converges to the predicted separation for the smaller flange pair.
22
Figure 14 – Maximum Joint Separation at Sealing Element vs. Bolt Pre-Stress (4
and 16 NPS Sizes)
3.2 Location of Contact Outside Bolt Circle
The location of contact outside the bolt circle under a 1,500 psi internal pressure with differing
analysis types, bolt pre-stress, and nominal pipe sizes was also investigated. The location of contact was calculated using RBT, Appendix Y equations, solid element FEA models, and shell/beam
FEA models, for a 4 NPS Class 3, Category 1 Appendix Y flange pair with varying bolt pre-stress.
The radial location of contact for a 16 NPS flange pair was also calculated using RBT, Appendix Y
equations, and solid element FEA models, and the results were normalized and compared to the
results for the 4 NPS flange pair.
3.2.1
Location of Contact Due to Differing Bolt Pre-Stress
The normalized contact distance from the bolt circle due to bolt pre-stress predicted by the different analysis methods is shown in Figure 15. All of the analysis methods are in good agreement
at bolt pre-stresses greater than 20% of the bolt yield strength. Agreement between the analysis
23
methods is expected because the location of contact is the result of a force and moment balance
about the bolt circle. However, slight variations do exist between the analytical and FEA methods
at lower preloads. These variations are caused by the lack of capability of the analytical methods
to predict complete separation of the joint at low bolt pre-stresses. For the RBT results, contact is
assumed to occur at the outside diameter of the joint, bmax, until the bolt pre-stress is great
enough to force the slope of the cover plate and flange/hub to be zero at the outside diameter of
the joint. A similar assumption is carried in the Appendix Y equations, where the minimum bolt
pre-stress that is able to be calculated corresponds to contact at the outside diameter and a zero
slope between the flanges.
Figure 15 – Normalized Contact Distance from Bolt Circle vs Bolt Pre-Stress
3.2.2
Normalized Location of Contact Due to Nominal Pipe Size
The normalized contact distance from the bolt circle, due to the bolt pre-stress predicted by the
different analysis methods for the 16 NPS flange pair, is shown in Figure 16. The contact distance
was calculated using RBT, Appendix Y equations, and solid element FEA models. All three analysis
24
methods show the expected relationship: the distance of contact from the bolt circle decreases
as bolt pre-stress increases. There appears to be more variation in the contact location results
between the different analyses for the 16 NPS flange pair than for the 4 NPS flange pair. The cause
of the observed variation is unclear, as the location of contact is the result of a force and moment
balance around the bolt circle.
Figure 16 – Normalized Contact Distance from Bolt Circle vs Bolt Pre-Stress
(16 NPS Flange Pair)
3.3 Stress in the Center of the Cover Plate
The radial and tangential stresses in the center of the cover plate, under the internal pressure
load, were investigated using differing analysis types, bolt pre-stresses, and cover plate thicknesses. The stresses were calculated using RBT, Appendix Y equations, solid element FEA models,
and shell/beam FEA models for a 4 NPS Class 3, Category 1 Appendix Y flange pair with varying
bolt pre-stress. The effect of cover plate thickness on these stresses was also investigated.
25
3.3.1
Stress in the Center of the Cover Plate Due to Differing Bolt Preload
Radial (S11) and tangential (S22) stresses in the center of the cover plate for a standard sized 4
NPS Class 3, Category 1 Appendix Y flange pair as a function of bolt pre-stress are shown in Figure
17 and Figure 18, respectively. The calculated stresses vary greatly between the four analysis
types for both radial and tangential stress. RBT calculated values for stress appear to approach a
limit as the bolt pre-stress increases. Appendix Y results show a fairly constant stress over the
entire bolt pre-stress range studied in the present analysis. Both FEA methods calculated stresses
that decreased as the pre-stress approached 25% of the bolt yield strength, but increased as the
bolt pre-stress was increased further. Additional mesh refinement work, that is outside the purview of this project, is required in order to determine if the cover plate stress increase with
increasing bolt pre-stress is due to element distortion near the center of the cover plate. Although
it is unclear what is the cause for the differences in stress results for the four analysis types, it is
suspected that the shell/beam element FEA model yields incorrect stress results because of the
large differences between them and the results of the other analysis types, and the previously
mentioned increased rigidity the kinematic coupling representing the bolt head adds to the
model.
26
Figure 17 – Radial Stress (S11) in Cover Plate vs Bolt Pre-Stress
Figure 18 – Tangential Stress (S22) in Cover Plate vs Bolt Pre-Stress
27
3.3.2
Stress in the Center of the Cover Plate Due to Cover Plate Thickness
Radial (S11) and tangential (S22) stresses in the center of the cover plate for a 4 NPS Class 3,
Category 1 Appendix Y flange pair with different cover plate thicknesses are shown in Figure 19
and Figure 20, respectively. The thickness of the cover plate is normalized by the standard thickness specified in ASME B16.5. The stresses were calculated using RBT, Appendix Y equations, and
solid element FEA models. Due to the large stress variations noted in Section 3.3.1 regarding the
shell/beam element FEA results, the shell/beam FEA analysis method was not included when investigating the effect of cover plate thickness on stress.
Figure 19 and Figure 20 show that the stress in the cover plate decreases as cover plate thickness
increases. This relationship is expected. Since the pressure area is the same regardless of cover
plate thickness, the forces and moments acting on the cover plate are expected to be independent
of cover plate thickness. Although the forces and moments are independent of thickness, the
stress in a plate is inversely proportional to its thickness squared.
For the thinner cover plate, there were differences in stresses calculated by each analysis method.
As the cover plate thickness increases, the results from the different analysis methods converge.
The results of the different analysis methods were shown to be in good agreement for the standard and double-thickness cover plates.
28
Figure 19 – Radial Stress (S11) in Cover Plate vs Cover Plate Thickness
Figure 20 – Tangential Stress (S22) in Cover Plate vs Cover Plate Thickness
29
4. Conclusions
The joint separation behavior and cover plate stresses for Class 3, Category 1 Appendix Y flanges
with varying cover plate thickness, bolt preload, and nominal pipe size were investigated in this
project. Bolted joint behavior was determined using the RBT analysis method developed by Waters and Schneider (1969), equations given in Appendix Y of ASME BPVC, finite element models
consisting of solid elements, and finite element models consisting of more computationally efficient shell and beam elements. The following conclusions were able to be drawn from the
analyses performed by this project:
Differences exist in the joint behaviors predicted by RBT, solid element FEA models, and
shell/beam FEA models. Shell/beam FEA models predicted joint separations much smaller than
the other analysis types. The suspected cause of this difference is the increased rigidity added to
the model by the kinematic coupling constraint that represented the interaction between the
heads of the fastener and cover plate and flange. The magnitude of joint separation predicted by
RBT and solid element FEA models were generally in good agreement. However, it was determined that RBT lacks the capability to predict complete separation of the cover plate and flange
faces at low bolt pre-stresses. Additionally, a limit was found to exist for the RBT analysis, where
joint separation does not decrease any longer as the bolt pre-stress increases.
The thickness of the cover plate and the overall size of the flange pair was also found to have an
influence on joint separation. Joint separation decreased as thickness of the cover plate increased.
For the 16 NPS flange pair, predicted joint separations for lower bolt pre-stresses were greater
than those predicted for the 4 NPS flange pair, but the separations appeared to converge as the
bolt pre-stress was increased.
30
Predictions for the radial location of contact between the cover plate and flange for the four analysis types were found to be in good agreement for the 4 NPS flange pair. Comparison of the
normalized location of contact between flange pair sizes were found not to show any direct relationship.
There appeared to be differences in predicted radial and tangential stresses for the four different
analysis types; however, all showed that the stresses decrease as cover plate thickness increases.
The shell/beam element FEA results were generally much lower than the results of the other analysis methods. It is possible that the method of modeling bolt head contact artificially stiffened the
joint, leading to lower cover plate deflections and stresses. Additionally, further mesh refinement
analysis must be performed to determine whether the increase in stress between bolt prestresses of 25% and 80% yield strength of the bolt were caused by excessive element distortion
at the center of the cover plate for the solid element FEA model.
31
5. References
ASME Boiler and Pressure Vessel Code, Section VIII – Rules for Construction of Pressure Vessels, 2013 Edition, Appendix Y.
Galai, H and Bouzid, A.H, “Analytical Modeling of Flat Face Flanges with Metal-to-Metal Contact Beyond the Bolt Circle,” Journal of Pressure Vessel Technology, Vol 132, 2010, 061207061207-8.
Schneider, R.W, “Flat Face Flanges with Metal-to-Metal Contact Beyond the Bolt Circle,”
Transactions of the ASME, Vol 90, 1968, pgs 82-88.
Waters, E.O and Schneider, R.W, “Axisymmetric, Nonidentical, Flat Face Flanges with Metalto-Metal Contact Beyond the Bolt Circle,” Journal of Engineering for Industry, Vol 91, 1969,
pgs 615-622.
32
Appendix A
Sample MAPLE Worksheet to Determine Joint Behavior using
Radial Beam Theory
Included in this appendix is the MAPLE worksheet used to determine the flange behavior using RBT
for the standard sized 4 NPS flange with no bolt pre-stress. The behavior of different flange sizes,
cover plate thicknesses, and bolt preload can be determined by changing the necessary values in
the “Case Constants” and “Calculations” sections.
A1
A2
A3
A4
A5
A6
Appendix B
Sample EXCEL Worksheet to Determine Bolt Pre-Stress with
Varying Locations of Contact Using Appendix Y Formulas
Included in this appendix is the Excel worksheet used to determine the required bolt pre-stress to
engage contact at differing distances using Appendix Y equations for the standard sized 4 NPS
flange. The behavior of different flange sizes and cover plate thicknesses can be determined by
changing the necessary values in the “Flange Properties” section.
Material Properties
E
30000000
nu
0.3
psi
P
Flange Properties
t1
1500
psi
2.12
in
t2
Bolt Hole
Diameter
Flange Outside
Diameter
2.12
in
1.38
in
12.25
in
Pipe Bore
4.6
in
Diameter of Seal
Bolt Circle
Diameter
Pipe Outside
Diameter
5.49
in
9.5
in
6.38
in
g0
0.89
g1
0.89
re1
Bolt Properties
Tensile Diameter
Stress Area
Number of Bolts
Bolt Length
1
1.11
in
0.969
in^2
8
5.35
in
B1
Calculated Values
Estar1
Estar2
X
AR
rB
a
β
F
V
Fprime
H
HD
HT
h0
hc or "b"
JS
JP
MP
C1
C2
C3
C4
θrb1
θrb2
Ms1
Ms2
Mu1
Mu2
Mb1
Mb2
θB1
θB2
HC
Wm1
Sigb
Si
% YS
Fbi
SR2_c
ST2_c
285843840
285843840
0.5
0.369909594
0.039464491
1.980874317
1.365209472
0.9082
0.550103
5.685849602
35507.96878
24928.53771
10579.43107
2.023363536
0.103
0.668475969
0.400964295
73547.40081
-0.204855533
-7474.244217
-0.12541143
-12579.73298
-1.52528E-05
1.52528E-05
-12032.94804
-8367.399809
-1,833
1,833
-10,200
-10,200
0.00012
0.00015
615,022
650,530
83,918
83,916
80%
81,315
3,760
3,760
hD
hT
0.25
0.681993
0.414482
73547.4
-0.202947
-8089.674
-0.122282
-12929.27
-1.45E-05
1.45E-05
-12421.71
-8932.051
-1,745
1,745
-10,677
-10,677
0.00013
0.00016
251,482
286,990
37,021
37,013
35%
35,866
3,804
3,804
B2
0.36
0.692108
0.424596
73547.4
-0.201543
-8542.753
-0.119964
-13188.21
-1.4E-05
1.4E-05
-12708.72
-9348.314
-1,680
1,680
-11,029
-11,029
0.00013
0.00016
173,664
209,172
26,983
26,966
26%
26,130
3,836
3,836
2.005
2.2275
1
0.750959
0.483447
73547.4
-0.19374
-11059.3
-0.10686
-14652
-1.1E-05
1.1E-05
-14315.61
-11669.12
-1,323
1,323
-12,992
-12,992
0.00015
0.00017
60,555
96,063
12,392
12,265
12%
11,885
4,016
4,016
1.375
0.7854415
0.5179298
73547.401
-0.189443
-12445.25
-0.099474
-15477.14
-9.39E-06
9.389E-06
-15210.16
-12953.7
-1,128
1,128
-14,082
-14,082
0.00015
0.00017
43,248
78,756
10,159
9,924
9%
9,617
4,115
4,115