grey finalreport provisional
TRANSCRIPT
iv
Predicting Stresses in Cylindrical
Vessels for Complex Loading
On Attachments using
Finite Element Analysis
By:
Charles Grey
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
IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Prof. Ernesto Gutierrez-Miravete, Engineering Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2012
v
Contents
LIST OF FIGURES ............................................................................................................ vi
LIST OF SYMBOLS ....................................................................................................... viii
ABSTRACT ....................................................................................................................... ix
1 - INTRODUCTION ......................................................................................................... 7
2 - BACKGROUND ........................................................................................................... 9
3 - THEORY & METHODOLOGY ................................................................................. 12
4 - RESULTS .................................................................................................................... 31
5 – DISCUSSION ............................................................................................................. 37
6 - CONCLUSION ............................................................................................................ 42
APPENDIX A ..................................................................................................................... 43
APPENDIX B ................................................................................................................... 45
APPENDIX C ................................................................................................................... 54
Bibliography ...................................................................................................................... 66
vi
LIST OF FIGURES
Figure 1 - Example of Guide Lugs on a Piping System [5] ................................................ 7
Figure 2 - Diagram of Example Created to Illustrate the Results of this Report ................ 8
Figure 3 - Uniform Load Placed on the Nozzle Attachment............................................. 10
Figure 4 - Original Data vs. Experimental Extensions [7] ................................................ 10
Figure 5 – Diagram Showing How Pressure Load Method is Applied to Moments [2] ... 18
Figure 6 – Longitudinal Moment Load Distribution [1] ................................................... 18
Figure 7 – Circumferential Moment Load Distribution [1] .............................................. 18
Figure 8 – Detail of Load Distribution [2] ........................................................................ 20
Figure 9 – Detail of Square Nozzle and Stress Locations [7] ........................................... 31
Figure 10 - Schoessow and Kooistra Example [4] ............................................................ 32
Figure 11 – Strain gauge Layout for Schoessow and Kooistra Example [4] .................... 33
Figure 12 - Proof of Concept Max Von Misses Stress in the Circumferential Direction . 33
Figure 13 – New Example Geometry ................................................................................ 34
Figure 14 - New Example Longitudinal Loading Only .................................................... 35
Figure 15 - Displacement Boundary Condition to Mimic Plate Anchor........................... 39
Figure 16 - Meshing Detail around Nozzle ....................................................................... 40
Figure 17 – WRC No.107 Figure 1A – Graph to find Mϕ for External Circumferential
Moment [7] ........................................................................................................................ 46
Figure 18 - WRC No.107 Figure 2A – Graph to find Mx for External Circumferential
Moment [7] ........................................................................................................................ 47
Figure 19 - WRC No.107 Figure 3A – Graph to find Nϕ for External Circumferential
Moment [7] ........................................................................................................................ 48
Figure 20 - WRC No.107 Figure 4A – Graph to find Nx for External Circumferential
Moment [7] ........................................................................................................................ 49
Figure 21 - WRC No.107 Figure 1B – Graph to find Mϕ for External Longitudinal
Moment [7] ........................................................................................................................ 50
Figure 22 - WRC No.107 Figure 2B – Graph to find Mx for External Longitudinal
Moment [7] ........................................................................................................................ 51
Figure 23 - WRC No.107 Figure 3B – Graph to find Nϕ for External Longitudinal
Moment [7] ........................................................................................................................ 52
Figure 24 - WRC No.107 Figure 4B – Graph to find Nx for External Longitudinal
Moment [7] ........................................................................................................................ 53
Figure 25 – Diagram Illustrating Boundary Conditions on New Example ....................... 55
Figure 26 –Diagram Illustrating Meshing For New Example........................................... 56
Figure 27 - Stresses Presented on New Example from Circumferential Loading Only ... 57
Figure 28 - Stresses Presented on New Example from Longitudinal Loading Only ........ 58
Figure 29 – Front View Max Von Misses Stress for New Example ................................. 59
Figure 30 – Rear View Max Von Misses Stress for New Example .................................. 60
Figure 31 - Displacement Vectors for Complex Load ...................................................... 61
Figure 32 - Displacement Vectors for Complex Loading Inside Shell ............................. 62
Figure 33 - Principal Stress 11 for Complex Loading ...................................................... 63
Figure 34 - Principal Stress 22 for Complex Loading ...................................................... 64
Figure 35 - Principal Stresses 12 from Complex Loading ................................................ 65
vii
LIST OF TABLES
Table 1 – Comparison of Results from WRC to FEA....................................................... 35
Table 2 – WRC No.107 Hand Calculation ........................................................................ 44
viii
LIST OF SYMBOLS
a = Mean radius of cylindrical shell (in.)
b = Coordinate to locate the center of the nozzle (in.)
c1 = Half the length of the nozzle in the circumferential direction (in.)
c2 = Half the length of the nozzle in the longitudinal direction (in.)
l = Length of the shell (in.)
l’ = Half the length of the shell (in.)
m,n = Integers involved in the Fourier series calculations
p = An equally distributed load (lbs.)
po = Maximum -ormal load on the shell caused by Circumferential or
Longitudinal Moments (lbs.)
q = Internal Pressure for the cylinder shell (lbs / in.)
s = a*ϕ
t = Shell wall thickness (in.)
u, v, w = Displacements for the X, Y (ϕ), Z directions (in.)
D = ���
�������
E = Modulus of Elasticity (lbs / in.)
L = 2*π*a (in.)
Mx = Bending Moments on the shell for a Longitudinal Moment (in.lbs.)
Mϕ = Bending Moments on the shell for a Circumferential Moment (in.lbs.)
-x = Membrane Forces on the shell for a Longitudinal Moment (lbs.)
-ϕ = Membrane Forces on the shell for a Circumferential Moment (lbs.)
Z = Radial loading per unit area (lbs / in.)
α = Length / a
α’ = α /2
β1 = c1 / a
β2 = c2 / a
γ = a / t
ϕ = Cylindrical coordinate
ϕmn = Expression defined by equation [16]
λ = ��
λ’ = 2*λ
ν = Poisson’s Ratio
ix
ABSTRACT
This report will perform analysis that will illustrate the faults in the Welding
Research Council Bulletin No.107 for predicting maximum stress values from applying
complex loading to a nozzle or attachment. Since 1965, the Welding Research Council’s
Bulletin No. 107 has been the “go to document” for finding maximum stress values for
applying load to an external nozzle or attachment on an imperforated shell. This work
was based off Professor P.P. Bijlaard’s theoretical and some experimental work on the
topic. The issues with his work arise from assumptions that were made. The focus of this
report is on the assumption that the methods will accurately calculate the maximum
stresses on the four major axis locations. This will not take into account the area parallel
to the major axis points. In some cases of multiple complex loading, these stress values
can be un-conservative. To find a more accurate to life prediction of stress values, a finite
element analysis will be taken of the loaded attachment.
By minimizing assumptions in the finite element analysis the real maximum stress
values will present themselves in the results. The goal of this report is to find how
complex loading will affect the analysis of the piping system around the nozzle and how
that will differ from the stress predictions calculated from Welding Research Council
Bulletin No. 107.
7
1 - INTRODUCTION
Pipe supports are a large part of designing a piping system. If a piping system
cannot take the point load at the support location, then the system must be re-evaluated
with new locations to balance and even out loads and movements apparent on the pipe.
Increasing the number of supports on the piping system is another option. Each of these
options decreases the load at each support point and would increase the likelihood that a
nozzle attachment would pass stress analysis. These support attachments are analyzed
using Welding Research Council Bulletin No.107. Pipes that are to be hung in the vertical
direction have shear lugs that are welded to the outer shell of the pipe. This is a type of
support would warrant the use of the Welding Research Council Bulletin No.107 method.
The vertical shear lug support will mimic the support shown in Figure 1as long as the L-
shaped steel guide plates were not installed.
Figure 1 - Example of Guide Lugs on a Piping System [5]
In Figure 1 guide lugs are shown on a piping system. Guide lugs or nozzles will impart
moments or internal pressure back into the shell of the pipe. The lugs portrayed are
taking a longitudinal load as well as circumferential loads simultaneously. This support in
Figure 1 is a very common style used in the piping industry. This reports main focus will
8
be this style of support. Figure 2 shows the example that has been created for the Finite
Element Analysis performed in this report. The geometry will have both circumferential
and longitudinal loads placed on the faces of the attachment simultaneously, as would be
seen in the field.
Figure 2 - Diagram of Example Created to Illustrate the Results of this Report
WRC No.107 has been widely used by the petrochemical and boiler industry to
engineer nozzles/ attachments for their vessels. Many issues from using WRC No.107 are
that engineers do not observe the assumptions that the bulletin takes into account.
Bijlaard’s initial calculations were only valid for a small grouping of examples. These
assumptions excluded certain terms in formulas when computing results, limiting the
types of shells it could be used to analyze. A lot of what is used in the industry now
exceeds those initial values. The Pressure Vessel Research Committee realized this
limitation as well. Many industry leaders pushed the PVRC to extend the curves out to
the best of their ability while still trying to keep those curves conservative. The curves
were extended. However they may not be accurate to the actual stresses that would be
present in the vessel shell. The curve extensions were created from very minimal
experimental data.
9
2 - BACKGROUND
The Pressure Vessel Research Committee sponsored a program in the mid 1960s
that was tasked to find, through experimentation and analysis, a method of determining
the stresses that are present in the shell of an imperforated pressure vessel when external
loading in the form of shear forces or moments are placed upon it. The main portion of
the research was completed by a Prof. P.P. Bijlaard during his time at Cornell University.
He has published a few papers defining his methodology before it was adopted by the
PVRC and made into a bulletin for the Welding Research Council. The research involved
looking at spherical as well as cylindrical pressure vessels. For the means of this paper,
the spherical vessel’s methodology will not be referred to. During the creation of
cylindrical vessels the shallow shell theory was used to derive the correct curves and
data. This is one of the assumptions that drive P. P. Bijlaard’s calculations. Several more
assumptions were made in the creation of these formulas that will be mentioned in the
latter sections. Some of these assumptions could lead to some inaccuracies. The initial
data the Bijlaard calculated was mainly for smaller diameters of pipe. Several large
companies in the piping industry requested larger piping ratios. Bijlaard used limited
experimental data to extend these curves to accommodate several of these larger
companies. He included warnings to the companies that would inform them of the un-
conservativeness of his method if not used properly. Loading was always considered to
be in the middle of the attachment and uniform in nature. This style of loading can be
seen in a diagram from the FEA model used in this report in Figure 3.
10
Figure 3 - Uniform Load Placed on the -ozzle Attachment
Any other type of loading such as a point load would be un-conservative, forcing uneven
loading and uneven torques to be put into the analysis. Because of the complexity of
Bijlaard’s work it became apparent that an easier methodology was needed to help
engineers follow the correct calculations. The Pressure Vessel Research Committee
summarized all of Bijlaard’s work into a “cook book” type format, with an easy to follow
calculation methodology. This method is presented in the Welding Research Councils
Bulletin No. 107. As mentioned in previous sections, the curves were extended to try and
match some of the experimental data that did not match Bijlaard’s original calculations
for larger diameters and ratios. These curves can be seen Figure 4as well as in the graphs
in Appendix B shown with dashed lines.
Figure 4 - Original Data vs. Experimental Extensions [7]
11
The analytical portions that are directly from Bijlaard’s original work are shown
in solid line. All of these curves were discussed in the committees as conservative and
safe to implement from the available experimental data. The purpose of this report is to
try and find some of the limitations of his methods, specifically during complex loading.
This will warrant a finite element analysis to find the maximum stresses in the shell of the
cylinder with more accuracy.
12
3 - THEORY & METHODOLOGY
The method that Professor P. P. Bijlaard uses is to develop the loads and
displacements into a double Fourier series for numerical evaluation. The loadings
considered are those of a uniformly distributed load within a rectangular nozzle. The
types of loading available for analysis using this method are as follows: a pressure load
pushing toward the center of the shell, a moment in the longitudinal direction uniformly
distributed over a short distance in the circumferential direction, and a moment in the
circumferential direction uniformly distributed over a short distance in the longitudinal
direction. For the purposes of this report the latter two types will be considered. In this
case an eight order differential equation is derived in terms of the radial displacement and
tangential load. Using this method the displacements, bending moments, and membrane
forces are found.
Three partial differential equations for the thin shell theory [1] are used to start the
derivations,
02
1
2
1 2
2
2
22
2
=∂∂
−∂∂
∂++
∂∂−
+∂∂
x
w
ax
v
a
u
ax
u υφ
υφ
υ
[1]
( ) 011212
11
2
1
2
122
2
2
2
2
2
32
3
2
3
2
2
2
2
22
22
=
∂
∂+
∂−+
∂
∂+
∂∂
∂+
∂∂
−∂
∂+
∂
∂−+
∂∂∂+
φυ
φφφφυ
φυ
a
v
dx
v
a
t
a
w
x
w
a
tw
a
v
ax
v
x
u
a [2]
( )0
12
1212
2
33
3
2
324
2
=−
+
∂
∂+
∂∂
∂−−∇−−
∂∂
+∂∂
ZEta
v
x
v
a
tw
at
a
w
a
v
x
u υφφ
υφ
υ[3]
u, v, and w are denoted as displacements of X, Y (ϕ), and Z directions respectively while
a is considered the mean radius of the cylinder and t the thickness of the cylinder. The ν
is considered to be Poisson’s ratio and the
13
2
22
2
2
24
∂
∂+
∂
∂=∇
φax
When looking at equation [2] one can consider that the terms containing ��
���� can be
discarded. When dealing with the thin shell theory the ratio of t/a is considered to be very
small. In consideration of this ratio being small it makes the concluding terms
insignificant.
When starting to develop the necessary equations to solve for displacements of
the system, one can simplify the equations [1], [2], and [3] by performing the next
operations. First, the operations ��
��� and ��
���� are applied to equation [1]. This results in
the equation,
������ + ���
��������� + ���
�����
����� − ��
������ [4]
and,
�
�����
������ + ������
������ + ���
������
����� − ���
�������� [5]
respectively. Each of these is to then be solved for ���
������ and ���
������� after which the
latter is to have ��
����� applied to it. Both equations are to be inserted back into equation
[2] which will result in the formula,
∂∂
∂+
∂∂
∂−+
+∂∂
∂−
∂
∂=∇
42
5
23
5
2
2
22
3
3
34
121
1
φφυυ
φυ
xa
w
x
w
a
t
xa
w
x
wua
. [6]
Then by taking equation [2] and applying the terms �
!� and �
" #� to it in the same manner
as the first step it will result in equaling,
14
�����
�������� + ���
������� + �
�����
������ − ���
�������� + ��
���� $ �%������ + �%�
��������& + ��
���� '(1 −*+ ���
��� + �����������, [7]
And
������
�������� + ���
������
������ + ���
������ − �
�������� + ��
���� $ �%������� + �%�
����%& + ��
���� '(1 −*+ ���
������ + ��������, [8]
respectively. Each of these is to then be solved for ���
������ and ���
������� after which the
latter is to have ��
����� applied to it and then both are to be inserted back into equation [1].
This will yield the equation,
( )
∂
∂+
∂∂
∂−−
+∂∂
∂−
−∂
∂+
∂∂
∂+=∇
53
5
32
5
4
5
2
2
33
3
2
34
1
3
1
2
122
φφυυ
φυφφυ
a
w
xa
w
x
wa
a
t
a
w
xa
wva
[9].
These two resulting equations ([6] and [9]), will have the expressions
! and
" # applied
to each respectively and then applying ∇. to the latter equation. These are both inserted
back into equation [3] to acquire the equation,
( ) ( ) ( ) 01
7621112 4
424
6
242
62
66
6
24
4
22
28 =∇−
∂∂
∂++
∂∂
∂−++
∂
∂+
∂
∂−+∇ Z
Dxa
w
xa
w
a
w
ax
w
taw
φυ
φυυ
φυ
[10]
The new terms that are included in order to simplify the written form of equation [10] are
as follows:
( )2
3
112 υ−=
EtD
[11]
and
ww 448 ∇∇=∇ .
15
When evaluating deflections found from equation [10] is has been discovered that
some engineers have left off the third term for shells with larger length to radius ratios
and larger thickness to radius ratios. In some cases it was found that the calculated
displacement values were seen as up to twenty five percent too low when matching
against experimental data. This will cause problems when using this method to evaluate
shells with those characteristics. To avoid those issues this term will remain in the
analysis.
When looking at the differential equations [1], [2], and [3] there is a certain lack
of attention to the products of the resulting forces and moments placed on the shell. If
these were to be accounted for they would cause the differential equations to become
non-linear and would increase the difficulty of the computations. This is discussed in
more depth in latter chapters of the project.
The internal pressure of a pipe and the membrane forces that are part of this
calculation are easily included. However, Prof. Bijlaard did not include them in his
method and therefore, they will not be including in calculations performed for that sole
reason. These differences would eventually skew the results being examined for this
report. This is one of the assumptions that Bijlaard takes in his cookbook calculations that
sometimes are not realized by engineers when making calculations.
Equation [6] contains only derivatives of w; therefore it can be solved by
developing the w deflections and the external loads into a double Fourier series.
( ) xa
mZw mn
∑∑=λ
φ sincos
[12]
( ) xa
mZZ mn
∑∑=λ
φ sincos
[13]
16
/ = 1234
When you take equations [12] and [13] and insert them back into equation [10] you will
extrapolate out,
( ) ( ) ( ) ( )
( )( ) 0sincos
11
76211121
222
4
4
42
2
24
2
6
6
2
4
22
2422
8
=
+−
+−
−+−−+
−++
∑∑ xa
m
ZmaD
wa
m
aa
m
aa
m
aatam
a
mn
mn λφ
λ
λυ
λυυ
λυλ
[14].
When you simplify and solve equation [14] for the displacements you will arrive at the
simple solution,
D
lZw mnmnmn
2
4
φ= [15]
where,
( )( ) ( ) ( ) ( )[ ]222244244422444242222
22222
762112
2
παυπυυααγαπυπα
παφ
nmnmmnnm
nmmn
++−++−−++
+=
[16]
a
l=α
,
and
t
a=γ
.
By combining all present equations and simplifying the equations down even further, the
displacement equation,
( ) xa
mZD
lw mnmn
∑∑=λ
φφ sincos2
4
[17]
is computed. Displacements u and v of the piping system may be expressed using
equation [15].
( ) xa
muu mn
∑∑=λ
φ coscos
[18]
( ) xa
mvv mn
∑∑=λ
φ sinsin
[19]
17
By inserting equations [17], [18], and [19] back into the original equations found, [6] and
[9]. The displacement formulas become,
( )( ) mnmn wmm
a
tm
mu
+
−+
+−+
= 222
2
222
222 1
1
12λ
υυ
υλλ
λ
[20]
( )( ) mnmn wmm
a
tm
m
mv
+−−
+−
++++
= 4224
2
222
222 1
3
1
2
122 λ
υυ
λυ
λυλ [21]
When looking at the equations pictured above there are terms that have been mentioned
previously (5�
��"� ), this term would only still be included for systems that have a thick
shell and that have higher lambda values. For the report examples, thin shells are being
used making the values for these terms insignificant and can therefore be ignored. This
transforms the equations [20] and [21] into equations,
( )( )
xa
mwm
mu mn
+
−∑∑=
λφ
λ
υλλcos)cos(
222
22
[22]
( )[ ]( )
xa
mwm
mmv mn
+
++∑∑=
λφ
λ
λυsin)sin(
2222
22
[23]
where wmn is taken from equation [15].
The basic equations have been formed for finding the three directional
displacements. To properly analyze the load placed on the shell, the method of external
pressure loading on a cylindrical shell will be considered for it is the basis in how the
longitudinal and circumferential moments are calculated. A visualization of the technique
can be seen in Figure 5.
18
Figure 5 – Diagram Showing How Pressure Load Method is Applied to Moments [2]
The external pressure load is considered equal and opposite from the central axis. P. P.
Bijlaard looked at the longitudinal moment and postulated that it is the same thing as a
uniform pressure load considering that the load is varying from the center of the
attachment. The largest pressure load is considered at being applied at the outermost edge
of the nozzle. This is also true for the circumferential moment. Bijlaard considers the
circumferential moment to be a varying load with the highest pressure load being
apparent from the outermost edge of the attachment. Both of these can be seen visually
by Figure 6 and Figure 7.
Figure 6 – Longitudinal Moment Load Distribution [1]
Figure 7 – Circumferential Moment Load Distribution [1]
19
The moments of a system that are the key to Bijlaard’s cookbook are considered
by equations,
( )φυXXDM xx +−= [24]
and
( )xXXDM υφφ +−= [25]
where,
∂∂
+=2
2
2
1
φφw
wa
X
[26]
and
2
2
x
wX x ∂
∂= . [27]
These formulas were taken from reference [8]. Extrapolating out these two equations
with the given terms will create,
∂
∂++
∂
∂−=
2
2
2
22
2 φυ
ww
x
wa
a
DM x
[28]
and
∂
∂+
∂
∂+−=
2
22
2
2
2 x
wa
ww
a
DM υ
φφ
[29]
The membrane forces can also be examined the same way as the moments by using the
following equations as considered from reference [8] as well.
−
∂∂
+∂∂
−=
a
w
a
v
x
uEt� x φ
υυ 21
[30]
∂∂
+−∂∂
−=
x
u
a
w
a
vEt� υ
φυφ 21[31]
Then to get these real values you must take the equations [17], [22], and [23] and insert
them into equations [28] through [30] which will solve out to.
20
( ) ( ) xa
mmn
ZlM mnmnx
−+
∑∑=
λφυ
απ
φα sincos12
1 2
2
2222
[32]
( ) xa
mn
mZlM mnmn
+−∑∑=
λφ
απυ
φαφ sincos12
12
22222
[33]
( )( )
( ) xa
mnm
nmZa� mnmnx
+∑∑−−=
λφ
παφγαυπ sincos16
22222
222522
[34]
( )( )
( ) xa
mnm
nZa� mnmn
+∑∑−−=
λφ
παφγαυπφ sincos16
22222
42424
[35]
These four equations are the basic equations that show the computation of the moment
and membrane forces on a shell when given an outside load. It will now be shown how
the external loading must be modified so that it will accommodate either a
circumferential or a longitudinal moment applied to the system.
Figure 8 – Detail of Load Distribution [2]
The load is examined as being contained in the square that is the attachment as seen in
Figure 8. The external load will be developed into a Fourier series, in which the loading
of the attachment looks like,
21
∑∞
+=1
2cos
2
1)(
L
msaasp mo
π
[36]
where
∫−
=2/
2/
)(2
L
L
o dsspL
a
[37]
and
∫=2/
0
2cos)(
4L
m dsL
mssp
La
π
[38]
Combining these three equations leaves you with the equation,
∑∞
+=
1 11
1
1
1 cossin12
)(l
ms
l
mc
m
p
l
pcsp
πππ
[39]
this then will simplify down to
( ) ( )∑∞
+=1
11 cossin
12)( φβ
ππβ
φ mmm
ppp [40],
This is the final equation that is used for figuring a uniformly distributed load across an
attachment connected to a cylinder. A circumferential or a longitudinal moment load is
not considered an equally distributed load in all directions like an external pressure load.
The differences can be seen in Figure 6 and Figure 7. Over the length of the attachment,
the load is highest at the outermost point and will decrease to zero as it passes the
centroid of the loaded attachment. The load will continue to increase in the opposite
direction as it moves away from the centroid in the opposite direction on the shell. The
load on one side of the shell is therefore in tension while the opposite side of the shell is
equal in magnitude but in compression. This load must be conformed to meet these
distributions.
22
By looking at the uniformly distributed loading one can then turn the loading of
the attachment into an odd function of x. This making the equation,
∑∞
=1
sin),(l
xnbxp n
πφ
[41]
where,
dxl
xnp
lb
cb
cb
n ∫+
−
=2
2
sin)(2
4 πφ
[42]
and
=
l
bn
l
cn
n
pbn
πππφ
sinsin)(4 2
[43]
Therefore making the final loading distribution looks like,
= ∑
∞
l
xn
l
bn
l
cn
nn
pxp
ππππφ
φ sinsinsin1)(4
),(1
2
[44]
With a longitudinal moment, the uniformly distributed load will vary with the
value of x. This is considered to be negatively mirrored from the centroid of the
attachment. This will be represented by
opc
xp
lxx
llx
2
'
'
'
2
=
−=
==
.
This ratio can then be inserted into [44] to form,
( ) ( )∑∞
+=1
1
22
1 cossin1
'2
')( φβππ
βφ mm
mx
c
px
c
pp oo
. [45]
23
The same function can be completed by changing the equation [39] into an odd function
of x; the same can be performed to equation [45].
∑∞
=1
'2sin),(
l
xnbxp n
πφ
[46]
∫=2/
0
''2
sin)(4l
n dxL
mxp
lb
πφ
[47]
By then combining all of these terms the conclusion can be made that
Zxp =),( φ . [48]
Forming the equation,
xa
mZxpZ mn
∑∑=='
sin)cos(),(λ
φφ[49]
where
l
an
l
an ππλλ
2
'2' ===
Making
=
=
−
−=
,...3,2,1
0
'cos
''sin
)1('22222
2
3
1
n
m
nnn
npZ
n
omn βαπ
βαπ
βαπ
βπβα
[50]
=
=
−
−=
,...3,2,1
,...3,2,1
sin('
cos''
sin)1('4
)12222
2
3
n
m
mnnn
mnpZ
n
omn ββαπ
βαπ
βαπ
βπα
[51]
where
2
'
2'
l==
αα
The value of Zmn will be used in conjunction with some of the previous equations to find
the displacements at each point. By using equation [51] in conjunction with [16] and
inserting that into equation [32] and [33], one can find the moments Mx and Mϕ. If
24
equation [51] and [16] are used in conjunction with equations [34] and [35] the values for
membrane forces Nx and Nϕ will result. When a more detailed computation is desired
equations [51] and [16] can be inserted back into the equation [10]. This will result in
exact values for displacements for each u, v, and z directions. By taking equation [15]
and using it to solve [17], [22], and [23] all directional displacement values result.
The same procedure is now completed for a circumferential moment with changes
made to some of the equations. The load is considered to be uniformly placed on the shell
while still being proportional to the angle of ϕ away from the zero angle on the centroid
of the attachment. This will mean that the forces and moments at ϕ from the centroid of
the attachment are equal and opposite at the –ϕ from the centroid.
When taking the generic equation for the load induced into the system use the
equation,
= ∑
∞
l
xn
l
bn
l
cn
nn
pxp
ππππφ
φ sinsinsin1)(4
),(1
2
[52]
Then as was performed in the previous process it is needed to be turned into an odd
function for ϕ.
∑∞
=1
2sin)(
L
smbp m
πφ
[53]
where
∫=2/
0 1
2sin
4l
om dsL
msp
c
s
lb
π
[54]
This extrapolates out to
−
=
L
cm
L
cm
L
cmp
cm
Lb om
111
1
22
2cos
22sin
ππππ
[55]
25
when
6 = 223 8� = 9�3 : = ;
3
Combining all these terms together gives you.
<(:+ = �=>?@A
B �C�
DCE�,�,G… (sin L8� − L8� cos L8�+ sin L∅ [56]
Then by using the same methodology as described earlier we can assume
P = <(Q, :+ = B B PCR sin L: sin S� Q [57]
where
8� = 9�3
Therefore coming to the final conclusion that,
PCR = (−1+TUA
� V?�@A
=>C�R(sin L8� − L8� cos L8�+ sin R?
W 8� $L = 1,2,3 …1 = 1,3,5 … &[58]
We can then take equation [16] and [58] and combine them in the equation,
Z = [�
�\ B B :CR PCR sin L: sin S� Q[59]
The displacement equations below are different than the displacement equation for
longitudinal moments. The change in force calculations drive the terms to become sin
(mϕ) rather than cos (mϕ). This does not causing a sign change in the results of the
equations put in. Therefore your results result in the equations
Z = B B ZCR sin L: sin S� Q [60]
] = B B ]CR sin L: cos S� Q [61]
^ = B B ^CR cos L: sin S� Q [62]
26
For Bijlaard’s final method in the creation of Welding Research Council Bulletin No.107
cookbook the following equations are formulated to use equations [17] and [58] to find
the moments and membrane forces that you will see in the bulletin.
_� = �� `�4� B B :CR PCR '$R�?�
W� & + ^(L� − 1+, sin L: sin S� Q [63]
_� = �� `�4� B B :CR PCR 'L� − 1 + $�R�?�
W� &, sin L: sin S� Q [64]
Because both of these terms have only derivatives of w in the terminology there are no
changes made to them. However, for membrane forces Nx and Nϕ there are values for w,
u, and v.
−
∂∂
+∂∂
−=
a
w
a
v
x
uEt� x φ
υυ 21
[65]
∂∂
+−∂∂
−=
x
u
a
w
a
vEt� υ
φυφ 21[66]
These terms must be broken down by using equations
] = B B S(C���S�+(S��C�+ ZCR sin L: cos S
� Q [67]
and
^ = − B B Ca(���+S��C�b(S��C�+ ZCR cos L: sin S
� Q [68]
Derivatives are taken of these equations and then placed back into the original equations
given ([34] and [35]). These will then result in the equations,
c� = −62�(1 − ^�+`ef�3 B B :CRPCRC�R�
(C�W��R�?�+� sin L: sin S� Q [69]
c� = −62.(1 − ^�+`.f�3 B B :CRPCRR�
(C�W��R�?�+� sin L: sin S� Q [70]
27
Equations [16] and [58] are used to find the values which Bijlaard will use in his
cookbook methodology.
To complete the cookbook (WRC No. 107) Bijlaard would take each of these
equations and solve them for changes in the value β, graphing them accordingly for each
type of loading induced into the system. This is what will be used to calculate out his
methods and compare them to the results from the finite element analysis.
The next section of the analysis is one that explains a hand calculation of the
Welding Research Council method as it is transcribed in the bulletin. There are a few
parameters that must be found in order to use the method properly. The shell parameter
γ is the ratio of the shell’s mean radius to the thickness of the shell. This parameter is
used to read each curve off the chart in which to capture the correct data for input in the
calculation sheet.
f = gCh
The second parameter that is needed is the β term. This term will vary depending on what
type of attachment you are using and the orientation of that attachment. The method has
the possibility of using a round attachment, a square attachment, or a rectangular
attachment. Each of these types have a different formula to calculate β. For the purposes
of this report, the square attachment will be considered. For this we will use the formula,
8 = 8� = 8� = 9�gC
= 9�gC
28
After finding each value in the charts, calculations involving some of these initial terms
will take place. Each of the charts are derived from the original equations that were
described above.
When looking for stresses resulting from a circumferential moment applied to the
attachment, these are the steps that should be followed. First, to find the circumferential
stresses in the shell using Figure 19 in Appendix B will be referenced. Reading the value
from the chart forij
kl/no�@. If the value does not fall directly on a specific γ value then
one must interpolate between the values of the upper and lower limits surrounding the
desired. The next step is to find, in Figure 17 in Appendix B the values forkj
kl/no@. Once
these values are found, the initial conditions are used to solve for the membrane stress
and the circumferential bending stress by using the following equations
[71]
And
[72]
Once each of these are solved for Nϕ/T and 6Mϕ/T2 then they can be combined back into
a general stress equation of,
p� = qRijr ± qt
ekjr� [73]
Where Kn and Kb are stress concentration factors to be considered in cases where there is
a brittle material or a fatigue analysis is to be completed on the attachment and pipe.
=
TR
M
RM
�
m
c
mc **/ 22 ββφ
=
2**
*6
/ TR
M
RM
M
m
c
mc ββφ
29
The same process is then used to find values for iu
kl/no�@ from Figure 20 in
Appendix B and ku
kl/no@ from Figure 18. The values are then input into equations,
[74]
And
[75]
Once each of these are solved for Nx/T and 6Mx/T2 then they can be combined back into
a general stress equation of,
p� = qRiur ± qt
ekur� [76]
When looking for stresses resulting from a longitudinal moment applied to the
attachment, the same steps will be completed using different charts. First to find the
circumferential stresses Figure 23 in Appendix B will be referenced. Reading the value
from the chart for ij kl/no�@. The next step is to find in Figure 21 in Appendix B the values
forkj
kl/no@. As mentioned in previous sections the steps remain the same by inputting the
results from the graphs into
[77]
And
[78]
Then putting the solved values into the final stress equation
p� = qRijr ± qt
ekjr� [79]
=
TR
M
RM
�
m
c
mc **/ 22 ββφ
=
2**
*6
/ TR
M
RM
M
m
c
mc ββφ
=
TR
M
RM
�
m
c
mc **/ 22 ββφ
=
2**
*6
/ TR
M
RM
M
m
c
mc ββφ
30
The next term needed is iu
kl/no�@ from Figure 24 in Appendix B and ku
kl/no@
from Figure 22. They are then input into the equations,
[80]
And
[81]
They can then be combined back into a general stress equation of,
p� = qRiur ± qt
ekur� [82]
The calculation has been simplified down to a single sheet for Bijlaard’s cookbook
computation. This calculation page can be seen in Table 2 in Appendix A. The curves
that are present in Appendix B are directly taken from WRC Bulletin No.107.
=
TR
M
RM
�
m
c
mc **/ 22 ββφ
=
2**
*6
/ TR
M
RM
M
m
c
mc ββφ
31
4 - RESULTS
In the previous chapter the derivatives of Prof. Bijlaard were made clear. In the
computations some of the same assumptions that he had made were continued through
the report’s calculations to make sure that the derivations match exactly what Bijlaard
had used to create the method used in WRC bulletin No.107. Each of these assumptions
will have a weakening effect on his methodology in finding the true to life stress
distribution for a nozzle attached to a cylindrical pipe. These assumptions are as follows:
1.) All stresses are computed at (when looking in a plan view) the up, down, right,
and left midpoints at each side of the attachment. Each of these locations both
interior and exterior of the shell is shown in Figure 9 below.
Figure 9 – Detail of Square -ozzle and Stress Locations [7]
2.) When initially creating the stress equations, the terms for $ ��
����&� were removed
from the equation.
3.) After finding the derivations of the equations the term ��
���� were ignored.
4.) Stresses at the mean radius were to be considered to be zero.
5.) Internal pressure is ignored from the initial sets of equations.
6.) Stresses presented from this method are considered to be equal and opposite for
the stresses considered in the same axis plane.
32
Each of these assumption’s effects will be explained in the discussion section of this
report.
For proof of concept an example from one of Bijlaard’s papers was used. In this
example, he took experimental data to compare his methods to. The work of Schoessow
and Kooistra is described a test cylinder that was 71 inches in length and measured 56
inches at the mean diameter. This cylinder was 1.3 inches in thickness and had an 11.75
inch pipe attached to the side. The cylinder was fixed on the end by a steel plate welded
around the diameter, as shown below in Figure 10.
Figure 10 - Schoessow and Kooistra Example [4]
This system was subjected to 410,000 in. lbs. in both the circumferential and the
longitudinal directions. The strain was recorded from gauges attached at varying
distances from the welded attachment on the pipe. These locations are illustrated in
Figure 11.
33
Figure 11 – Strain gauge Layout for Schoessow and Kooistra Example [4]
The computed values in the circumferential moment are equal to 23,280 psi
computed from Bijlaard’s calculations. Stress values equal to 25,000 psi were recorded
from the extrapolated data obtained from the strain gauges.
Figure 12 - Proof of Concept Max Von Misses Stress in the Circumferential Direction
The difference has a deviation of 6.8%. This is an acceptable deviation. The same
example was completed from a Finite Element Analysis perspective as seen in Figure 12.
The maximum stress values for this model are exactly where Bijlaard predicted them to
be. For a circumferential moment the maximum stresses are located on the major axis on
either side of the attachment. The maximum stress values equal 21,820 psi, giving a
variation of 6.3%. The deviation shows a proof of concept between all three analyses.
The longitudinal moment was performed with the same values. The resulting
stress was 13,910 psi from the Bijlaard’s calculated stress; this would be compared to
34
13,500 psi extrapolated from the experimental data. The difference in the calculations
was only 2.9%. When using the finite element model a difference of 9.3% was achieved.
This shows a proof of concept for the methods that were performed during the analysis in
the finite element analysis program Abaqus.
Using the same methods that have been used for a proof of concept, a separate
example was created in order to facilitate a combined longitudinal and circumferential
moments on a nozzle. The geometry of the example is as can be seen in Figure 13:
Figure 13 – -ew Example Geometry
The mean radius of the cylinder is considered to measure 15 inches with a thickness of
0.3 inches. These dimensions give the example a γ value of 50. The attachment
characteristics are a square attachment having a length of 7.5 inches and a height of 10
inches from the mean radius of the shell. This gives the nozzle a β value of .25. The
loading of the attachment will be in both the circumferential and the longitudinal
direction. The loading of the nozzle consisted of moments equaling 25,000 in. lbs.
In the hand calculation from the Welding Research Councils method in Appendix
A, the calculated values for the combined stress are shown in Table 1. Each of these
35
values were calculated from numbers extrapolated from curves found in WRC bulletin
No.107. The curves were created directly from the method derived in the theory section.
Each of the values In Table 1 match directly from the locations that would be found on
the axis around the nozzle as defined from WRC 107.
Table 1 – Comparison of Results from WRC to FEA
AU AL BU BL CU CL DU DL
WRC
Method 1735.3 -8253.9 -1735.3 8253.9 -11557.5 19553.7 11557.5 -19553.7
FEA -2888.3 2887.4 -12331.7 12464.7
Difference 39.92% 39.90% 6.28% 7.28%
The models presented are accurate to each method. However the finite element
method varies considerably from the WRC methodology as shown in the longitudinal
direction. The longitudinal moment analysis even when performed on its own results in
inconsistent results with Bijlaard’s method.
Figure 14 - -ew Example Longitudinal Loading Only
Figure 14 illustrates that the maximum stress is not located at the major axis of the
nozzle. The maximum stress is concentrated at each corner. The reasoning behind this
will be covered in the discussion section of this report. The circumferential loading
36
however is within an acceptable 10% margin. The major on axis stresses were not the
highest recorded stress value in the finite element model. There is a stress concentration
at the corner of the nozzle that equals 32,680 psi. This value has a differential equal to
67.1% when compared to the maximum calculated stress from Bijlaard’s method. This is
not acceptable for a maximum value when looking for maximum stress results from the
WRC method.
37
5 – DISCUSSION
Bijlaard made a few assumptions in his analysis in order to simplify the computations
and make the final method easy enough for a cookbook type computation.
1.) The assumption was made that each of the midpoints on the exterior and the
interior of the shell would reflect the maximum stress values for loads that are
placed onto a shell from an attached nozzle. This is completely true for simple
systems. It has been shown in the report’s proof of concept. However because of
the neglect of the surrounding parallel planes in an analysis involving complex
multiple loads, the stresses presented on these axes may not be representative of
the maximum stress for the overall system. There are higher stress concentrations
in areas from parallel elements occurring from a longitudinal loading and a
circumferential loading. As is visible in the results and the diagrams in Appendix
C, there is a stress concentration in the corner of the loaded shell. This stress
concentration will not be represented when looking for a maximum allowed stress
on the shell. This absence of this stress concentration can cause the results found
from WRC 107 to be un-conservative.
2.) When first deriving formulas to create the curves for WRC 107 the terms for
$ ��
����&�
were removed. The thickness to shell radius ratio was so low, that it
would make the terms involving this equation nonexistent and insignificant. This
is true for this report for thin shells are used in the examples created. Leaving any
terms out of a calculation however, will decrease its overall true to life accuracy.
38
3.) This again removed terms from the final calculation containing ��
���� because of
the very small values this term would produce. This will have little effect on the
final calculations in this report.
4.) The mean radius of the shell is considered to have zero stress because of the equal
and opposite nature of the loading put into the shell. This assumption would be
invalidated if there was a large enough deflection in the shell that would cause the
central axis to move. This occurrence can be seen in Figure 14. A high
displacement value was achieved with longitudinal loading alone. The results
become skewed and stress concentrations move from their predicted locations at
the major axes. The same finite element conditions were used in the proof of
concept; therefore the assumption can be made that the failure is in the WRC
No.107’s method and not in the FEA analysis.
5.) Internal pressure is ignored from the calculation and derivation of values in
Bijlaard’s methods for the addition of such a force can make the stress calculation
equations become non-linear and would increase the difficulty of the calculation
by hand.
6.) Stresses are considered to be equal and opposite on either side of the nozzle
attachment. This is a very good assumption that can be taken and is proven in the
stresses that were calculated using the finite element analysis method. The
stresses in Table 1show that there is little variation between the stresses calculated
on either side of the attachment.
39
In the example from Schoessow and Kooistra the Stresses calculated are within a
10% acceptable deviation. When creating the example I used a master/ slave physical
constraint for the attachment of the long pipe to the shell of the test pipe, this locked
the pipes together in a manner that would be accurate to the welding shown in Figure
10. A full displacement restraint of the ends of the pipe was made to imitate the plates
that are welded to the end of the pipe to lock the rig in place during testing (Figure
15).
Figure 15 - Displacement Boundary Condition to Mimic Plate Anchor
The same material used in the test example is used for properties throughout both
finite element analyses. The modulus of elasticity considered to be 30E+06 psi and
the Poisson’s ratio to equal 0.3. The meshing as you can see in Figure 16 was
concentrated in the area around the nozzle to increase accuracy around that area of the
shell.
40
Figure 16 - Meshing Detail around -ozzle
The rest of the model was partitioned up to have a more coarse mesh as to not
increase the time and difficulty of the calculation without warrant. Sometimes a
model having an overly fine mesh will not converge properly. The limits to the finer
mesh were determined as where stresses would not deviate and change greatly
between elements. The nozzle attachment was determined to have an equal load
across the entire width of the attachment to keep a uniform load, just as is required
from Bijlaard’s assumptions. Point loads were not used for a few reasons. They
would cause a local deformation of the lug that would skew the final results desired
from the model. This same method used on the creation of the proof of concept was
implemented on the new example created to show the complex loading on a nozzle
attachment.
The loads in the circumferential direction reflect very well on the WRC
methodology. However the longitudinal loads were not accurate to the WRC
methodology. This is caused by some of Bijlaard’s assumptions earlier in this section.
The majority of errors between real life stresses and Bijlaard’s calculations come
from the assumption that all of the calculations assume the maximum stress values
are located at points on the major axes of the nozzle. When the parallel loading planes
41
are not considered and accounted for, adverse effects happen that can cause the final
results to vary from the actual real world stresses.
42
6 - CONCLUSION
The Welding Research council gave the task of creating a cookbook calculation to
Prof. Bijlaard. In the creation of this method, certain assumptions were made. The
method is only to be used for thin shelled cylinders and spheres. The stresses considered
were only to be found for each side of the attachment on the major axes. These
assumptions are not necessarily a good practice for it will not give you accurate
concentrations of stresses formed from complex loading imparted on a nozzle attachment.
The example presented in this report shows the un-conservative nature of the PVRC’s
calculation. However it cannot be said that Welding Research Council Bulletin No. 107 is
un-conservative for all sets of geometry with complex loading. There are many different
combinations that can be made and there is no ability to discuss all of them in this paper.
43
APPENDIX A
44
Ta
ble
2 –
WR
C -
o.1
07
Ha
nd
Calc
ula
tio
n
45
APPENDIX B
46
Figure 17 – WRC -o.107 Figure 1A – Graph to find Mϕ for External Circumferential Moment [7]
47
Figure 18 - WRC -o.107 Figure 2A – Graph to find Mx for External Circumferential Moment [7]
48
Figure 19 - WRC -o.107 Figure 3A – Graph to find -ϕ for External Circumferential Moment [7]
49
Figure 20 - WRC -o.107 Figure 4A – Graph to find -x for External Circumferential Moment [7]
50
Figure 21 - WRC -o.107 Figure 1B – Graph to find Mϕ for External Longitudinal Moment [7]
51
Figure 22 - WRC -o.107 Figure 2B – Graph to find Mx for External Longitudinal Moment [7]
52
Figure 23 - WRC -o.107 Figure 3B – Graph to find -ϕ for External Longitudinal Moment [7]
53
Figure 24 - WRC -o.107 Figure 4B – Graph to find -x for External Longitudinal Moment [7]
54
APPENDIX C
55
F
igu
re 2
5 –
Dia
gra
m I
llu
strati
ng
Bou
nd
ary
Con
dit
ion
s o
n -
ew E
xa
mp
le
56
F
igu
re 2
6 –
Dia
gra
m I
llu
stra
tin
g M
esh
ing
For
-ew
Ex
am
ple
57
F
igu
re 2
7 -
Str
esse
s P
rese
nte
d o
n -
ew E
xam
ple
fro
m C
ircu
mfe
ren
tial
Loa
din
g O
nly
58
F
igu
re 2
8 -
Str
esse
s P
rese
nte
d o
n -
ew E
xam
ple
fro
m L
on
git
ud
inal
Lo
ad
ing
On
ly
59
F
igu
re 2
9 –
Fro
nt
Vie
w M
ax
Vo
n M
isse
s S
tres
s fo
r -
ew E
xa
mp
le
60
Fig
ure
30
– R
ear
Vie
w M
ax
Vo
n M
isse
s S
tres
s fo
r -
ew E
xa
mp
le
61
F
igu
re 3
1 -
Dis
pla
cem
en
t V
ecto
rs f
or
Co
mp
lex
Load
62
F
igu
re 3
2 -
Dis
pla
cem
en
t V
ecto
rs f
or
Co
mp
lex
Load
ing
In
sid
e S
hel
l
63
F
igu
re 3
3 -
Pri
nci
pal
Str
ess
11
for
Co
mp
lex
Load
ing
64
F
igu
re 3
4 -
Pri
nci
pal
Str
ess
22
for
Co
mp
lex
Load
ing
65
F
igu
re 3
5 -
Pri
nci
pal
Str
esse
s 1
2 f
rom
Co
mp
lex
Load
ing
66
Bibliography
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