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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



i

© Copyright 2015

by

Christopher Wowk

All Rights Reserved



ii

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 Theory ............................................................................................................................ 4

2.1.1 Joint Separation Behavior ................................................................................. 4

2.1.2 Stress in the Center of the Cover ...................................................................... 8

2.2 Methodology .................................................................................................................. 9

2.2.1 Solid Element Finite Element Model ................................................................ 9

2.2.2 Shell and Beam Element Finite Element Model ............................................. 13

2.3 Evaluation vs Analysis Type ......................................................................................... 14

3. Results and Discussion ........................................................................................................... 17

3.1 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

3.2 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

3.3 Stress in the Center of the Cover Plate ........................................................................ 25



iii

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



iv

LIST OF TABLES

Table 1 – Flange Component Geometric Parameters ..................................................................... 3

Table 2 - Material Summary ............................................................................................................ 3

Table 3 – Analyses Performed ...................................................................................................... 16



v

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 .................. 10Figure 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



vi

ACKNOWLEDGMENT

I would like to thank my project advisor, Professor Norberto Lemcoff, for his support and under-

standing 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, with-

out which completion of this project and degree program would not have been possible.



vii

ABSTRACT

Guidelines for sizing the flange geometry and bolting requirements for flanges with contact out-

side 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 investi-

gates 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 re-

sults 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 thick-

ness 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 separa-

tions 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 ra-

dial beam theory method was unable to predict complete separation of the joint at low bolt pre-

stress.



viii

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



ix

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



x

ACRONYMS

ASME American Society of Mechanical Engineers

BPVC Boiler and Pressure Vessel Code

FEA Finite Element Analysis

RBT Radial Beam Theory



1

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 cir-

cle are not stressed until pressurization. The behavior of these flanges are dependent on the pre-

stress 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 main-

tained 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-Manda-

tory 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).



2

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 anal-

ysis (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.



3

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 compo-

nents, 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

Number of Bolts 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

Table 2 - Material Summary

Flange

Component

Material

Specification

Young’s

Modulus

Yield

Strength

Poisson’s

Ratio

Cover Plate &

Flange/Hub

ASTM A515

Grade 6030,000 ksi 32 ksi 0.3

BoltsASTM A193

Grade B730,000 ksi 105 ksi 0.3



4

2. Theory/Methodology

2.1 Theory

This project will investigate the joint separation behavior and cover plate stresses for Class 3, Cat-

egory 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. Alt-

hough 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 in-

vestigation 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 modi-

fied 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.



5

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.



6

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



7

The moment along the length of the radial beam can be expressed as:

"+"ℓ 0 ≤ x < b (1)

( ′

ℓ) ′

() 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.

∗(−) 0 ≤ x < b (3)

∗(−) b ≤ x ≤ L (4)

The curvature of the beam can be determined as:

() ∫ 0 ≤ x < b (5)

() ∫

b ≤ x ≤ L (6)

() {() 0 ≤ < () ≤ ≤ (7)

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:

() ∫ () 0 ≤ x < b (8)



8

() ∫ () b ≤ x ≤ L (9)

() {() 0 ≤ < () ≤ ≤ (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 de-

termined 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 pro-

ject. 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 stresses at the center of the cover plate to be equal. The stresses can be

calculated using equation 38 of Appendix Y:

.394∗∗

6∗

∗∗ (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.



9

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 ∗ [∗∗(3+)

8 2 ∗ ] (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 calcula-

tions. 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 seg-

ment 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 de-

termined by:



10

− 8

(13)

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 symmet-

rical 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



11

bolt appeared in the model, only half of the preload was required to be applied in order to gen-

erate 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.



12

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 con-

tact 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 out-

puts 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



13

∑ 1 ∗= (14)

∑ = (15)

(16)

(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 tan-

gential 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 ele-

ments 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:

−ℎ 36

(18)

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



14

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 inde-

pendent variables investigated by this project, not all combinations of nominal pipe size, bolt



15

preload, and cover plate thickness were evaluated with each analysis method. Instead, the influ-

ences 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.



16

Table 3 – Analyses Performed

NPS Size

Cover Thickness

Bolt Pre-Stress 0Pressure

Load

25%

Yield

80%

Yield0

Pressure

Load

25%

Yield

80%

Yield0

Pressure

Load

25%

Yield

80%

Yield0

Pressure

Load

25%

Yield

80%

Yield

Radial Beam Theory

Solid Element FEA

Shell Element FEA

Radial Beam Theory

Appendix Y Equations

Solid Element FEA

Shell Element FEA

Radial Beam Theory

Appendix Y Equations

Solid Element FEA

Shell Element FEA

Radial Beam Theory

Solid Element FEA

Shell Element FEA

Joint

Separation

Along Length

Radial

Location of

Contact

Stress In

Center of

Cover

Maximum

Joint

Separation

Analysis

Methodt1 0.5*t1 2*t1

4"

t2

16"



17

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 maxi-

mum 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 ser-

vice, 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,



18

however the results of this project indicate that a preload at least 6x the operating load can sig-

nificantly 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 capabil-

ities 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 ele-

ment 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



19

element FEA results, whereas the separation appears to be exponential for the shell/beam ele-

ment 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 diam-

eter 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.



20

Figure 10 – Joint Separation vs. Distance from the Outside Diameter of Flange –

Radial Beam Theory Analysis Method


Solid Element Finite Element Model



21


Shell/Beam Element Finite Element Model

3.1.2 Maximum Joint Separation Due to Cover Plate Thickness


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.



22

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


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 con-

verges to the predicted separation for the smaller flange pair.



23

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 con-

tact 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 dif-

ferent 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



24

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



25

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 thick-

nesses. 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.



26

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 pur-

view 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.



27

Figure 17 – Radial Stress (S11) in Cover Plate vs Bolt Pre-Stress

Figure 18 – Tangential Stress (S22) in Cover Plate vs Bolt Pre-Stress



28

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 thick-

ness 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 in-

vestigating 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 stand-

ard and double-thickness cover plates.



29

Figure 19 – Radial Stress (S11) in Cover Plate vs Cover Plate Thickness

Figure 20 – Tangential Stress (S22) in Cover Plate vs Cover Plate Thickness



30

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 Wa-

ters 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 effi-

cient 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 deter-

mined 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.



31

Predictions for the radial location of contact between the cover plate and flange for the four anal-

ysis 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 rela-

tionship.

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 anal-

ysis 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% yield strength of the bolt were caused by excessive element distortion

at the center of the cover plate for the solid element FEA model.



32

5. References

ASME Boiler and Pressure Vessel Code, Section VIII – Rules for Construction of Pressure Ves-

sels, 2013 Edition, Appendix Y.

Galai, H and Bouzid, A.H, “Analytical Modeling of Flat Face Flanges with Metal-to-Metal Con-tact Beyond the Bolt Circle,” Journal of Pressure Vessel Technology, Vol 132, 2010, 061207-

061207-8.

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 Metal-

to-Metal Contact Beyond the Bolt Circle,” Journal of Engineering for Industry, Vol 91, 1969,

pgs 615-622.



A1

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.



A2



A3



A4



A5



A6



B1

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 psi

nu 0.3

P 1500 psi

Flange Properties

t1 2.12 in

t2 2.12 in

Bolt Hole

Diameter1.38 in

Flange Outside

Diameter12.25 in

Pipe Bore 4.6 in

Diameter of Seal 5.49 inBolt Circle

Diameter9.5 in

Pipe Outside

Diameter6.38 in

g0 0.89

g1 0.89

re1 1

Bolt Properties

Tensile Diameter 1.11 in

Stress Area 0.969 in^2

Number of Bolts 8

Bolt Length 5.35 in



Calculated Values

Estar1 285843840

Estar2 285843840

X 0.5

AR 0.369909594

rB 0.039464491a 1.980874317

β 1.365209472

F 0.9082

V 0.550103

Fprime 5.685849602

H 35507.96878

HD 24928.53771 hD 2.005

HT 10579.43107 hT 2.2275

h0 2.023363536

hc or "b" 0.103 0.25 0.36 1 1.375JS 0.668475969 0.681993 0.692108 0.750959 0.7854415

JP 0.400964295 0.414482 0.424596 0.483447 0.5179298

MP 73547.40081 73547.4 73547.4 73547.4 73547.401

C1 -0.204855533 -0.202947 -0.201543 -0.19374 -0.189443

C2 -7474.244217 -8089.674 -8542.753 -11059.3 -12445.25

C3 -0.12541143 -0.122282 -0.119964 -0.10686 -0.099474

C4 -12579.73298 -12929.27 -13188.21 -14652 -15477.14

θrb1 -1.52528E-05 -1.45E-05 -1.4E-05 -1.1E-05 -9.39E-06

θrb2 1.52528E-05 1.45E-05 1.4E-05 1.1E-05 9.389E-06

Ms1 -12032.94804 -12421.71 -12708.72 -14315.61 -15210.16Ms2 -8367.399809 -8932.051 -9348.314 -11669.12 -12953.7

Mu1 -1,833 -1,745 -1,680 -1,323 -1,128

Mu2 1,833 1,745 1,680 1,323 1,128

Mb1 -10,200 -10,677 -11,029 -12,992 -14,082

Mb2 -10,200 -10,677 -11,029 -12,992 -14,082

θB1 0.00012 0.00013 0.00013 0.00015 0.00015

θB2 0.00015 0.00016 0.00016 0.00017 0.00017

HC 615,022 251,482 173,664 60,555 43,248

Wm1 650,530 286,990 209,172 96,063 78,756

Sigb 83,918 37,021 26,983 12,392 10,159Si 83,916 37,013 26,966 12,265 9,924

% YS 80% 35% 26% 12% 9%

Fbi 81,315 35,866 26,130 11,885 9,617

SR2 c 3 760 3 804 3 836 4 016 4 115

effects of analysis method, bolt pre-stress, and cover plate thickness on the behavior of bolted...

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