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
CONTENTS
LIST OF TABLES .............................................................................................................................. iii
LIST OF FIGURES............................................................................................................................. iv
ACKNOWLEDGMENT ...................................................................................................................... v
ABSTRACT ...................................................................................................................................... vi
NOMENCLATURE .......................................................................................................................... vii
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 Opening 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 ........................................................................................................... 16
3.1
3.2
3.3
Joint Separation Behavior ............................................................................................ 16
3.1.1
Joint Separation Behavior Due to Differing Bolt Pre-Stress and
Analysis Type .................................................................................................. 16
3.1.2
Maximum Joint Separation Due to Cover Plate Thickness ............................. 20
3.1.3
Maximum Joint Separation Due to Normalized Bolt Pre-Stress and
Nominal Pipe Size ........................................................................................... 21
Location of Contact Outside Bolt Circle ....................................................................... 22
3.2.1
Location of Contact Due to Differing Bolt Pre-Stress ..................................... 22
3.2.2
Normalized Location of Contact Due to Nominal Pipe Size............................ 24
Stress in the Center of the Cover Plate ....................................................................... 25
i
3.3.1
Stress in the Center of the Cover Plate Due to Differing Bolt Preload ........... 25
3.3.2
Stress in the Center of the Cover Plate Due to Cover Plate Thickness ........... 27
4. Conclusions ............................................................................................................................ 29
5. References.............................................................................................................................. 31
ii
LIST OF TABLES
Table 1 – Flange Component Geometric Parameters .................................................................... 3
Table 2 - Material Summary ........................................................................................................... 3
Table 3 – Analyses Performed ...................................................................................................... 15
iii
LIST OF FIGURES
Figure 1 – Free Body Diagram of Class 3, Category 1 Flange (Waters and Schneider 1969) ............................ 5
Figure 2 – Free Body and Moment Diagram for Annular Ring Portion of Cover Plate & Flange ...................... 6
Figure 3 – Meshed Solid Element ABAQUS Model of Class 3, Category 1 Appendix Y Joint .......................... 10
Figure 4 – Solid Element Model Boundary Conditions and Loads .................................................................. 11
Figure 5 – COPEN Extraction Path................................................................................................................... 11
Figure 6 – Symbol Plot of CNORMF for 80% Yield Bolt Pre-Stress Case ......................................................... 12
Figure 7 – Meshed Shell/Beam Element ABAQUS Model of Class 3, Category 1 Flange Pair ........................ 14
Figure 8 –Maximum Joint Separation at Sealing Element vs. Bolt Pre-Stress ................................................ 17
Figure 9 –Joint Separation vs. Distance from the Outside Diameter of Flange – Radial Beam Analysis
Method ......................................................................................................................................... 19
Figure 10 –Joint Separation vs. Distance from the Outside Diameter of Flange – Solid Element Finite
Element Model ............................................................................................................................. 19
Figure 11 –Joint Separation vs. Distance from the Outside Diameter of Flange – Shell/Beam Element Finite
Element Model ............................................................................................................................. 20
Figure 12 –Maximum Joint Separation at Sealing Element vs. Bolt Pre-Stress (Different Cover Plate
Thicknesses) .................................................................................................................................. 21
Figure 13 – Maximum Joint Separation at Sealing Element vs. Bolt Pre-Stress (4 and 16 NPS Sizes) ............ 22
Figure 14 – Normalized Contact Distance from Bolt Circle vs Bolt Pre-Stress................................................ 23
Figure 15 – Normalized Contact Distance from Bolt Circle vs Bolt Pre-Stress (16 NPS Flange Pair) .............. 24
Figure 15 – Radial Stress (S11) in Cover Plate vs Bolt Pre-Stress.................................................................... 26
Figure 16 – Tangential Stress (S22) in Cover Plate vs Bolt Pre-Stress............................................................. 26
Figure 17 – Radial Stress (S11) in Cover Plate vs Cover Plate Thickness ........................................................ 28
Figure 18 – Tangential Stress (S22) in Cover Plate vs Cover Plate Thickness ................................................. 28
iv
ACKNOWLEDGMENT
Type the text of your acknowledgment here.
v
ABSTRACT
Type the text of your abstract here.
vi
NOMENCLATURE
a
Width Of The Outermost Region Of The Radial Beam
ASME
American Society of Mechanical Engineers
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
BPVC
Boiler and Pressure Vessel Code
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
FEA
Finite Element Analysis
I
Moment Of Inertia
L
Length of the Flange from Outside Diameter 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
RBT
Radial Beam Theory
Rm
Radius of the Sealing Element
vii
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
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
viii
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
pre-stress carried in the bolts and 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 Opening 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 nonidentical flange pairs is presented below.
4
A free body diagram of a Class 3, Category 1 Appendix Y flange configuration is shown in Figure
1. 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 1 – 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 (1969). A system of balanced forces and moments acts equally on the flange
and cover. This system of forces and moments produces the flange separation behavior identical
to 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 1. 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 2.
Figure 2 – Free Body and Moment Diagram for Annular Ring Portion of Cover
Plate & Flange
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
6
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)
𝑐𝑡 3
12∗(1−𝜈2 )
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)
𝐼𝐵 =
The curvature of the beam can be determined as:
𝜃𝐴 (𝑥) = ∫0
𝜃𝐵 (𝑥) = ∫𝑏
𝜃(𝑥) = {
𝜃𝐴 (𝑥)
𝜃𝐵 (𝑥)
𝑓𝑜𝑟
With the 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
𝐿
𝑣𝐵 (𝑥) = ∫𝑏 𝜃𝐵 (𝑥) 𝑑𝑥
𝑣 (𝑥)
𝑣(𝑥) = { 𝐴
𝑣𝐵 (𝑥)
𝑓𝑜𝑟
0≤x<b
(8)
b≤x≤L
(9)
0≤𝑥<𝑏
𝑏≤𝑥≤𝐿
(10)
With the boundary conditions vA(0) = 0 and vA(b)= vB(b). Although too cumbersome to be presented in this report, the expression for v(x) is dependent on the reaction forces and moments
that must be determined by solving the system of equations for the entire flange assembly. The
7
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 pre-stresses, 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 stress 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.
Waters and Schneider (1969) offers an equation to predict the radial and tangential stress at the
center of the cover also. As in Appendix Y, both the radial and tangential stress at the center of
the cover plate are predicted to be equal. The magnitude of the stresses are given by equation
48 (Waters and Schneider 1969):
3
𝑃∗𝑅𝑚2 ∗(3+𝑣)
−
8
𝑆𝑅𝐼𝐼𝑆𝑊𝑆 = 𝑆𝑇𝐼𝐼𝑊𝑆 = 𝑡 2 ∗ [
𝐼𝐼
8
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. 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:
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, a 11.25° segment is required.
9
Solid models of the joint segment were created using ABAQUS CAE and 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
3.
Figure 3 – Meshed Solid Element ABAQUS Model of Class 3, Category 1 Appendix Y Joint
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
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
10
the cover from the center up to location of the hypothetical sealing element at Rm, the bore of
flange/hub, and also 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 4.
Figure 4 – 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 5.
Figure 5 – COPEN Extraction Path
11
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 6 – Symbol Plot of CNORMF for 80% Yield Bolt Pre-Stress Case
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.
12
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 was determined by:
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 and the dimensions shown
in Table 1. The bolt was represented in the analysis as a B33 (2 node cubic beam in space) element 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 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 noded 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 7.
13
Figure 7 – 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 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
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.
14
Table 3 – Analyses Performed
Analysis
Method
Joint
Separation
Along Length
0
Pressure
Load
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
Radial Beam Theory
Solid Element FEA
Shell Element FEA
15
25%
Yield
t2
2*t1
0.5*t1
t1
Cover Thickness
Bolt Pre-Stress
16"
4"
NPS Size
80%
Yield
0
Pressure
Load
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 by this project. 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 8. It can be seen that the three analysis methods show a similar relationship;
joint separation decreases as bolt pre-stress increases. This relationship is important, as 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
in order to 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,
16
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 separation 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 8 –Maximum Joint Separation at Sealing Element vs. Bolt Pre-Stress
The total joint separation as a function of distance from the outside diameter of the flange using
RBT is shown in Figure 9, solid element FEA models in Figure 10, and shell/beam element FEA
models in Figure 11. The general magnitudes of separation from RBT and solid element FEA were
in good agreement for lower bolt pre-stresses. The shell/beam element FEA models predicted
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 element
17
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 joint separation
when RBT and solid element FEA are used and the flexibility of the cover plate and flange has a
larger influence when when the shell/beam element FEA analysis is used. This difference in results is suspected to be caused by 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 appears to be a limit where there is much smaller variation in the
behavior of the joint at higher bolt pre-stresses. This appears to occur between the 25% Yield
and 80% Yield curves. This limit is not seen when the joint is analyzed using either FEA methods.
Both FEA methods predicted complete separation of the cover plate and flange at the outside
diameter when the bolts were not preloaded. This complete separation was not observed in the
RBT analyses. 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 prestresses.
18
Figure 9 –Joint Separation vs. Distance from the Outside Diameter of Flange –
Radial Beam Analysis Method
Figure 10 –Joint Separation vs. Distance from the Outside Diameter of Flange –
Solid Element Finite Element Model
19
Figure 11 –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 12. 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 prestress 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.
20
Figure 12 –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 13. 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 with the predicted separation for the smaller flange pair.
21
Figure 13 – 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 investigated by this project. 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 14. All of the analysis methods are in good agreement
22
at bolt pre-stresses greater than 20% of the bolt’s yield strength. Agreement between the analysis 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 a lack of capability of the analytical
methods to predict complete separation of the joint at low bolt pre-stress. 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 slope between the flanges of zero.
Figure 14 – Normalized Contact Distance from Bolt Circle vs Bolt Pre-Stress
23
3.2.2
Normalized Location of Contact Due to Nominal Pipe Size
The normalized contact distance from the bolt circle due to bolt pre-stress predicted by the different analysis methods for the 16 NPS flange pair is shown in Figure 15. The contact distance
was calculated using RBT, Appendix Y equations, and solid element FEA models. All three analysis
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 analyses for the 16 NPS flange pair than there was 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 15 – Normalized Contact Distance from Bolt Circle vs Bolt Pre-Stress
(16 NPS Flange Pair)
24
3.3 Stress in the Center of the Cover Plate
The radial and tangential stress 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.
3.3.1
Stress in the Center of the Cover Plate Due to Differing Bolt Preload
Radial (S11) and tangential (S22) stress 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 16
and Figure 17, respectively. The calculated stresses varied greatly between the four analyses
types for both radial and tangential stress. RBT calculated high values for stress that appeared to
approach a limit as bolt pre-stress increased. Appendix Y results calculated a fairly constant stress
over the entire bolt pre-stress range the analysis method could be used in. Both FEA methods
calculated stresses that decreased as the pre-stress approached 25% of the bolt’s 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.
25
Figure 16 – Radial Stress (S11) in Cover Plate vs Bolt Pre-Stress
Figure 17 – Tangential Stress (S22) in Cover Plate vs Bolt Pre-Stress
26
3.3.2
Stress in the Center of the Cover Plate Due to Cover Plate Thickness
Radial (S11) and tangential (S22) stress 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 18 and
Figure 19, 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 18 and Figure 19 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 increased, the results from the different analysis methods
converged. The results of the different analysis methods were showed to be in good agreement
for the standard and double-thickness cover plates.
27
Figure 18 – Radial Stress (S11) in Cover Plate vs Cover Plate Thickness
Figure 19 – Tangential Stress (S22) in Cover Plate vs Cover Plate Thickness
28
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 by this
project. 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.
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 less 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 was no longer decreased at increasing bolt pre-stress.
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 separation for lower bolt pre-stresses were
greater than those predicted for the 4 NPS flange pair, however the separations appeared to
converge as bolt pre-stress was increased.
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
29
normalized location of contact between flange pair sizes were found to not show any direct relationship.
There appeared to be differences in predicted radial and tangential stresses for the four different
analysis types, however all analysis types 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 pre-stresses of 25% and 80% the yield strength of the bolt were caused by excessive
element distortion at the center of the cover plate for the solid element FEA model.
30
5. References
1. ASME Boiler and Pressure Vessel Code, Section VIII – Rules for Construction of Pressure
Vessels, 2013 Edition, Appendix Y.
2. 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.
3. Waters, E.O and Schneider, R.W, “Axisymmetric, Nonidentical, Flat Face Flanges with
Metal-to-Metal Contact Beyond the Bolt Circle,” Journal of Engineering for Industry, Vol
91, 1969, pgs 615-622.
4. 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.
31