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
Contents
LIST OF FIGURES ........................................................................................................... iii
LIST OF SYMBOLS ......................................................................................................... vi
ABSTRACT ...................................................................................................................... vii
1 - INTRODUCTION ......................................................................................................... 1
2 - BACKGROUND ........................................................................................................... 4
3 - THEORY & METHODOLOGY ................................................................................... 7
4 – RESULTS ................................................................................................................... 25
5 – DISCUSSION ............................................................................................................. 34
6 - CONCLUSION ........................................................................................................... 38
REFERANCES ................................................................................................................. 39
APPENDIX A ..................................................................................................................... 40
APPENDIX B ................................................................................................................... 42
APPENDIX C ................................................................................................................... 51
ii
LIST OF FIGURES
Figure 1 – Vertical Lug Support (left) and Guide Lug Support(right) on a Piping System 2
Figure 2 - Diagram of Example Created to Illustrate the Results of this Report ................ 2
Figure 3 - Uniform Load Placed on the Nozzle Attachment .............................................. 5
Figure 4 - Original Data vs. Experimental Extensions [6].................................................. 6
Figure 5 – Diagram Showing How Pressure Load Method is Applied to Moments [2] .. 12
Figure 6 – Longitudinal Moment Load Distribution [1] ................................................... 13
Figure 7 – Circumferential Moment Load Distribution [1] .............................................. 13
Figure 8 – Detail of Load Distribution [2] ........................................................................ 15
Figure 9 – Detail of Square Nozzle and Stress Locations [6] ........................................... 25
Figure 10 - Schoessow and Kooistra Example [4]............................................................ 26
Figure 11 – Strain gauge Layout for Schoessow and Kooistra Example [4] .................... 27
Figure 12 - Proof of Concept Max Von Misses Stress in the Circumferential Direction . 28
Figure 13 – New Example Geometry ............................................................................... 29
Figure 14 - Circumferential and Longitudinal Moments on Attachment ......................... 30
Figure 15 - New Example Circumfrential Loading Only ................................................. 31
Figure 16 - New Example Longitudinal Loading Only .................................................... 32
Figure 17 - Combined Loading Max Von Mises Stress ................................................... 33
Figure 18 - Displacement Boundary Condition to Mimic Plate Anchor .......................... 36
Figure 19 - Meshing Detail around Nozzle....................................................................... 37
Figure 20 – WRC No.107 Figure 1A – Graph to find M∅ for External Circumferential
Moment [7] ....................................................................................................................... 43
Figure 21 - WRC No.107 Figure 2A – Graph to find Mx for External Circumferential
Moment [7] ....................................................................................................................... 44
Figure 22 - WRC No.107 Figure 3A – Graph to find N∅ for External Circumferential
Moment [7] ....................................................................................................................... 45
Figure 23 - WRC No.107 Figure 4A – Graph to find Nx for External Circumferential
Moment [7] ....................................................................................................................... 46
Figure 24 - WRC No.107 Figure 1B – Graph to find M∅ for External Longitudinal
Moment [7] ....................................................................................................................... 47
Figure 25 - WRC No.107 Figure 2B – Graph to find Mx for External Longitudinal
Moment [7] ....................................................................................................................... 48
Figure 26 - WRC No.107 Figure 3B – Graph to find N∅ for External Longitudinal
Moment [7] ....................................................................................................................... 49
Figure 27 - WRC No.107 Figure 4B – Graph to find Nx for External Longitudinal
Moment [7] ....................................................................................................................... 50
Figure 28 – Diagram Illustrating Boundary Conditions on New Example ...................... 52
Figure 29 –Diagram Illustrating Meshing For New Example .......................................... 53
Figure 30 - Stresses Presented on New Example from Circumferential Loading Only ... 54
Figure 31 - Stresses Presented on New Example from Longitudinal Loading Only ........ 55
Figure 32 – Front View Max Von Misses Stress for New Example ................................ 56
Figure 33 – Rear View Max Von Misses Stress for New Example ................................. 57
Figure 34 - Displacement Vectors for Complex Load ...................................................... 58
Figure 35 - Displacement Vectors for Complex Loading Inside Shell ............................. 59
iii
Figure 36 - Principal Stress 11 for Complex Loading ...................................................... 60
Figure 37 - Principal Stress 22 for Complex Loading ...................................................... 61
Figure 38 - Principal Stresses 12 from Complex Loading ................................................ 62
iv
LIST OF TABLES
Table 1 – Comparison of Results from WRC to FEA ...................................................... 30
Table 2 – WRC No.107 Hand Calculation ....................................................................... 41
v
LIST OF SYMBOLS
a
b
c1
c2
l
l’
m,n
p
po
= Mean radius of cylindrical shell (in.)
= Coordinate to locate the center of the nozzle (in.)
= Half the length of the nozzle in the circumferential direction (in.)
= Half the length of the nozzle in the longitudinal direction (in.)
= Length of the shell (in.)
= Half the length of the shell (in.)
= Integers involved in the Fourier series calculations
= An equally distributed load (lbs.)
= Maximum Normal load on the shell caused by Circumferential or
Longitudinal Moments (lbs.)
q
= Internal Pressure for the cylinder shell (lbs / in.)
s
= a*∅
t
= Shell wall thickness (in.)
u, v, w = Displacements for the X, Y (∅), Z directions (in.)
𝑬𝒕𝟑
D
= 𝟏𝟐(𝟏−𝝊𝟐 )
E
L
Mx
M∅
Nx
N∅
Z
α
α’
β1
β2
γ
∅
∅mn
= Modulus of Elasticity (lbs / in.)
= 2*π*a (in.)
= Bending Moments on the shell for a Longitudinal Moment (in.lbs.)
= Bending Moments on the shell for a Circumferential Moment (in.lbs.)
= Membrane Forces on the shell for a Longitudinal Moment (lbs.)
= Membrane Forces on the shell for a Circumferential Moment (lbs.)
= Radial loading per unit area (lbs / in.)
= Length / a
= α /2
= c1 / a
= c2 / a
=a/t
= Cylindrical coordinate
= Expression defined by equation (16)
λ
=
λ’
ν
= 2*λ
= Poisson’s Ratio
𝒏𝝅𝒂
𝒍
vi
ABSTRACT
This report contains results of analysis that illustrate shortcomings in the Welding
Research Council Bulletin No.107 for predicting maximum stress values from applying
complex loading to a nozzle or attachment on a piping system. 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
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 accurately
calculate the maximum stresses on the four major axis locations. This assumption does
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 of the loaded attachment is
required.
By minimizing assumptions in the finite element analysis, more accurate
maximum stress values are predicted. The goal of this report is to show how complex
loading affects the analysis of the piping system around the nozzle and how the results
differ from the stress predictions calculated from Welding Research Council Bulletin No.
107.
vii
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 [6]. Pipes that are to be hung in the
vertical direction have shear lugs that are welded to the outer shell of the pipe as shown
in Figure 1. This is a type of support is a vertical shear lug support and would warrant the
use of the Welding Research Council Bulletin No.107 method.
1
Figure 1 – Vertical Lug Support (left) and Guide Lug Support(right) on a Piping System
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 pressure load as well as circumferential loads simultaneously. The support
shown in Figure 1 is a very common style used in the piping industry. This reports main
focus will 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 often violate 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
2
could be used to analyze. Much of what is used in the industry now exceeds those initial
values and the Pressure Vessel Research Committee realized this limitation. 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.
3
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 present in the shell of an imperforated pressure vessel when external loading
in the form of shear forces or moments are placed upon attachments joined to it. The
main portion of the research was completed by a Prof. P.P. Bijlaard during his time at
Cornell University. He published a few papers ([1], [2], and [3]) 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 purpose of this report, only cylindrical vessels will
be dealt with. In the original analysis of cylindrical vessels, shallow shell theory was used
to derive formulae, curves, and data. This was one of the assumptions behind P. P.
Bijlaard’s calculations. Several more assumptions were made in the creation of these
formulas that will be mentioned in the latter sections and some of these could lead to
inaccuracies. The initial data 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 is shown by the pink arrows in the diagram from the FEA model used in this
report in Figure 3.
4
Figure 3 - Uniform Load Placed on the Nozzle 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 carry out 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 previously, 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 4 as well as in the graphs in
Appendix B shown with dashed lines.
5
Figure 4 - Original Data vs. Experimental Extensions [6]
The analytical portions that are directly from Bijlaard’s original work are shown as a
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
identify some of the limitations of the above methods, specifically for structures
subjected to complex loading. A carefully calculated and validated finite element analysis
is used to find the maximum stresses in the shell of the cylinder.
6
3 - THEORY & METHODOLOGY
The method that Professor P. P. Bijlaard uses consists in developing 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 only 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 from the thin shell theory [1] are used to start
the derivations,
 2 u 1    2 u 1    2 v  w



0
2a x a x
x 2 2a 2  2
1    2 u 1    2 v 1  2 v 1 w
t2

 2


2
2
2a x
2 x
a  12a 2
a 

 3w
3w 
t2
 2  2 3  
2
 x  a   12a
(1)

 2v
 2v 
1    2  2 2   0
dx
a  


(2)

u v w at 2 4
t 2  2    3v
 3v  1   2


 
 w  

Z 0
x a a 12
12  a x 2  a 3  3 
Et
(3)
u, v, and w are denoted as displacements of X, Y (∅), and Z directions respectively while
a is considered the mean radius of the cylinder and t the thickness of the cylinder. ν is
the Poisson’s ratio and
7
 2
2
   2  2 2
a 
 x
4



2
(4)
𝑡2
Since we are dealing with thin shells the terms containing 12𝑎2 can be discarded.
Equations (1), (2), and (3) can be simplified as follows. Applying the operators
𝜕2
𝜕2
and 𝑎𝜕𝜙2 to equation (1) yields,
𝜕𝑥 2
𝜕4 𝑢
𝜕𝑥 4
1−𝜐 𝜕4 𝑢
+ 2𝑎2 𝜕𝜙4 +
1+𝜐 𝜕 4 𝑣
𝜐 𝜕3 𝑤
− 𝑎 𝜕𝑥 3
𝜕𝑥 3 𝜕𝜙
2𝑎
(5)
and,
1
𝜕4 𝑢
1−𝜐 𝜕4 𝑢
1+𝜐 𝜕4 𝑣
𝜐
𝜕3 𝑤
+ 2𝑎4 𝜕𝜙4 + 2𝑎3 𝜕𝑥𝜕𝜙3 − 𝑎3 𝜕𝑥𝜕𝜙2
𝑎2 𝜕𝑥 2 𝜕𝜙2
𝜕4 𝑣
(6)
𝜕4 𝑣
respectively. Each of these is to then be solved for 𝑎𝜕𝑥 3 𝜕𝜙 and 𝑎3 𝜕𝑥𝜕𝜙3 after which the
𝜕2
latter is to have 𝑎𝜕𝑥𝜕𝜙 applied to it. Both equations are then inserted back into equation
(2) which results in the formula,
a 4 u  
3w
3w
1 t 2


x 3 a 2 x 2 1   12a 2
∂2
 5w
5w 
 3 2  2

a x 4 
 x 
.
(7)
∂2
Then by taking equation (2) and applying the terms ∂x2 and a ∂ϕ2 to it in the same manner
as before it will result in,
1+𝜐 𝜕4 𝑢
2𝑎
𝜕𝑥 3 𝜕𝜙
+
1−𝜐 𝜕4 𝑣
2
𝜕𝑥 4
1
𝜕4 𝑣
1
𝜕3 𝑤
𝑡2
𝜕5 𝑤
𝜕5 𝑤
𝑡2
+ 𝑎2 𝜕𝑥 2 𝜕𝜙2 − 𝑎2 𝜕𝑥 2 𝜕𝜙 + 12𝑎2 (𝜕𝑥 4 𝜕𝜙 + 𝑎2 𝜕𝑥 2 𝜕𝜙3 ) + 12𝑎2 [(1 −
𝜕4 𝑣
𝜕4 𝑣
𝜐) 𝜕𝑥 4 + 𝑎2 𝜕𝑥 2 𝜕𝜙2 ]
(8)
and
1+𝜐 𝜕4 𝑢
1−𝜐
𝜕4 𝑣
1 𝜕4 𝑣
1 𝜕3 𝑤
𝜕4 𝑣
𝜕4 𝑣
𝑡2
𝜕5 𝑤
𝜕5 𝑤
𝑡2
+ 2𝑎3 𝜕𝑥 2 𝜕𝜙2 + 𝑎4 𝜕𝜙4 − 𝑎4 𝜕𝜙3 + 12𝑎4 (𝜕𝑥 2 𝜕𝜙3 + 𝑎2 𝜕𝜙5 ) + 12𝑎4 [(1 −
2𝑎3 𝜕𝑥𝜕𝜙3
𝜐) 𝜕𝑥 2 𝜕𝜙2 + 𝑎2 𝜕𝜙4 ]
8
(9)
𝜕4 𝑢
𝜕4 𝑢
respectively. Each of these is to then be solved for 𝑎𝜕𝑥 3 𝜕𝜙 and 𝑎3 𝜕𝑥𝜕𝜙3 after which the
𝜕2
latter is to have 𝑎𝜕𝑥𝜕𝜙 applied to it and then both are to be inserted back into equation (1).
Doing this yields the equation,
a 4 v  2   
3w
3w
t2


ax 2  a 3  3 12a 2
 2a  5 w
3  5w
5w 




4
2
3
a 3  5 
 1   x  1   ax 
∂
(10).
∂
For equations (7) and (10) one applies the operators ∂x and a ∂ϕ respectively and then
applying ∇4 to the latter equation and both of these are inserted back into equation (3)
resulting in the equation,
8 w 


12 1   2  4 w 1

a 2 t 2 x 4 a 2
 6w
6w
6w  1 4
2


2

6





7


 6 6
  Z 0
a 2 x 4  2
a 4 x 2  4  D
 a 
(11)


The new terms included in order to simplify equation (10) are the flexural rigidity of the
shell:
Et 3
D
12 1   2


(12)
and
8 w   4  4 w .
(13)
When evaluating deflections from equation (11) it has been found 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.
9
When looking at 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 nonlinear and would increase the difficulty of the computations. This is discussed in more
depth in latter sections of the report.
The internal pressure inside 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, they will not be including hence as these differences would eventually skew
the results being examined for this report. This is one of the assumptions that Bijlaard
makes in his analysis that sometimes are not realized by engineers when making
calculations.
Equation (7) contains only derivatives of w; therefore it can be solved by
developing the w deflections and the external loads into a double Fourier series.

w    wmn cosm sin   x
a
 
Z    Z mn cosm sin   x
a
𝑛𝜋𝑎
𝜆=
𝑙
(14)
(15)
Inserting equations (14) and (15) into equation (11) yields,
4
4
2
2
4


4
121   2    
1 
m6
1

   m 
2   m
  8 2  m 2  
   2  2 6  6       2  7     4  wmn 
2 2
a t
a 
a
a
a a
 a  a  
 
 a


  
 cosm  sin  a  x  0
2
 1  1 2  m 2  Z 

mn 
 D  a 4


Solving equation (16) for the displacements produces the coefficients,
10
(16).
wmn   mn Z mn
where,
 mn 
m 
2
2
 n 2 2




l4
2D
2 m 2 2  n 2  2

(17)

2



 12 1   2 n 4 4 4  2  m 2 4 2m 4 4  6     2 n 4 4  7   m 2 2 n 2 2 (18)
4

l
a,
and
a
t .
Combining all equations and simplifying produces the displacement equation,
 
l4

w
   mn Z mn cosm sin   x
2D
a
(19)
The u and v displacements of the piping system may be expressed as

u    u mn cosm  cos  x
a
(20)

v    v mn sin m sin   x
a
(21)
By inserting equations (19), (20), and (21) back into equations (7) and (10), the
displacement coefficients become,
u mn 
v mn 


m
2
 m2


2
2
 2

t 2 1 2 2
2
m   m 2  wmn
m   
2
12a 1  


(22)

t 2  2 4 3  2 2

2
2


2




m

 
 m  m 4  wmn

2 
1
12a  1  


(23)
 m2

2


t2
The equations pictured above contain the term (12a2 ). This term is only important for
systems that have a thick shell and high lambda values. For the report examples, only thin
shells are being considered making the values for these terms insignificant and thus they
can therefore be ignored. This transforms the equations (22) and (23) into equations,
11
u  
 m 2  2 

wmn cos( m ) cos  x
a
  m 
m2     m 
v  
w
  m 
2 2
2
2
2
2
mn
2 2

sin( m ) sin   x
a
(24)
(25)
where wmn is taken from equation (17).
Equations (19), (24), and (25) can now be used to find 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.
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 examined 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 as being applied at the outermost edge
of the nozzle. This is also true for the circumferential moment. Bijlaard considered the
circumferential moment to be a varying load with the highest pressure load being
12
apparent from the outermost edge of the attachment. Both of these are shown in Figure 6
and Figure 7.
Figure 6 – Longitudinal Moment Load Distribution [1]
Figure 7 – Circumferential Moment Load Distribution [1]
The moments of a system that are the key to Bijlaard’s approach are represented
by equations [7],
and
M x   D  X x  X  
(26)
M    D X   X x 
(27)
where,
X 
1 
2w 


w

a 2 
 2 
and
13
(28)
Xx 
2w
.
x 2
(29)
Combining the two momentums these equations with the given terms yields,
Mx  

D  2 2w
 2 w 


a


w



a 2  x 2
 2 

(30)
and
M  
D
a2
2

2w
2  w
w



a


 2
x 2 

(31)
The membrane forces can also be examined the same way as the moments by using the
following equations as considered from reference [7] as well.
Nx 
N 
Et
1 2
 u
 v w 
 
   
 a a 
 x
(32)
Et
1 2
 v w
u 
 a  a   x 


(33)
The previously obtained displacement equations ((19), (24), and (25)) can now be
inserted into equations (30) through (32) to give
Mx 
M 
 n 2 2
1 2 2
 l    mn Z mn  2
2
 



   m 2  1  cosm sin   x
a





 n 2  2
1 2 2
 l    mn Z mn m 2  1  
2
2
 



N x  6 1     a   mn Z mn
2
2
5
2
14
m 
2


 cosm sin   x
a

m2n2
2
n 
2

2 2

cosm sin   x
a
(34)
(35)
(36)


N   6 1     a   mn Z mn
4
2
4
2
m 
2
n4
2
n 
2

2 2

cosm sin   x
a
(37)
These four equations are the basic equations for 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, i.e.
p(s) 

1
 2ms 
a o   a m cos

2
 L 
1
(38)
where
L/2
ao 
2
p(s)ds
L  L/ 2
15
(39)
and
4
am 
L
L/2

0
p( s) cos
2ms
ds
L
(40)
Combining these three equations yields the equation,
p( s) 
 mc1   ms 
pc1 2 p  1
 cos


sin 

l1
 1 m  l1   l1 
(41)
1
2p  1
p
 sin m1 cosm 

 1 m
(42),
which then simplifies to
p ( ) 
This equation is used to represent 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 conditions.
By looking at the uniformly distributed loading one can then turn the loading of
the attachment into an odd function of x, i.e. by writing the equation,

 nx 
p ( x,  )   bn sin 

 l 
1
(43)
where,
bc
4 2
 nx 
bn 
p( ) sin 
dx

2l b c2
 l 
16
(44)
and
bn 
4 p( )  nc 2   nb 
sin 
 sin 

n
 l   l 
(45)
Therefore the final loading distribution looks like,
p ( x,  ) 
4 p( )  1  nc 2   nb   nx 
 sin 
 sin 

 sin 
n 1 n  l   l   l 
(46)
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
x  l' 
l
2
x'  x  l
x'
p
po
c2
.
This ratio can then be inserted into (46) to produce,
p( ) 

2p
1 p o
1
x' o x'  sin m1 cosm 
 c2
c2
1 m
.
(47)
By then combining all of these terms the conclusion can be made that
p ( x,  )  Z .
(48)
Forming the equation,
 ' 
Z  p( x,  )    Z mn cos( m ) sin   x
a
where
 '  2 
na 2na

l'
l
leads to
17
(49)
Z mn
2 ' 1
(1) n
 3
po
 2
n2
  n 
n
 n  
 sin 
 2 cos   2 
 2 
'
 '  
  ' 
m  0



 n  1,2,3,...
Z mn 
4 '
 32
po
(50)
(1) n
mn 2
  n
 sin 
  '
n

 n
 2 cos
 2 
'

 '
 
  2  sin( m1)
 
 m  1,2,3,... 


 n  1,2,3,... 
(51)
where
'

2

l'
2
The value of Zmn will be used in conjunction with some of the previous equations to find
the displacements at each point. By using equations (50) and (51) in conjunction with
(18) and inserting that into equation (34) and (35), one can find the moments Mx and M∅.
If equations (50), (51), and (18) are used in conjunction with equations (36) and (37) the
values for membrane forces Nx and N∅ will result. When a computation of displacements
is desired, equations (50), (51), and (18) can be inserted back into the equation (11). By
taking equation (17) and using it to solve (19), (24), and (25) 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 ∅ away from the zero angle on the centroid
of the attachment. This will mean that the forces and moments at ∅ from the centroid of
the attachment are equal and opposite at the –∅ from the centroid.
The generic equation for the load induced into the system is given by,
18
4 p( )  1  nc 2   nb   nx 
 sin 
 sin 

 sin 
n 1 n  l   l   l 
p ( x,  ) 
(52)
Then, proceeding as before this needs to be turned into an odd function of ∅.

 2ms 
p( )   bm sin 

 L 
1
(53)
where
bm 
4
l
l/2
s
c
0
po sin
1
2ms
ds
L
(54)
This extrapolates out to
  2mc1  2mc1
L
 2mc1  
po  sin 
cos

 
2
L
 m c1   L 
 L 
bm 
2
when
𝐿 = 2𝜋𝑎 𝛽1 =
(55)
𝑐1
𝑠
𝜙=
𝑎
𝑎
Combining all these terms together gives
2𝑝
1
𝑝(𝜙) = 𝜋𝛽𝑜 ∑∞
𝑚=1,2,3… 𝑚2 (sin 𝑚𝛽1 − 𝑚𝛽1 cos 𝑚𝛽1 ) sin 𝑚∅
1
(56)
Then, using the same methodology described earlier we can assume
𝜆
𝑍 = 𝑝(𝑥, 𝜙) = ∑ ∑ 𝑍𝑚𝑛 sin 𝑚𝜙 sin 𝑎 𝑥
where
𝛽2 =
(57)
𝑐2
𝑎
Therefore, the final result is,
𝑍𝑚𝑛 = (−1)
𝑛−1
2
8
𝑝𝑜
𝜋 2 𝛽1
𝑚2 𝑛
(sin 𝑚𝛽1 − 𝑚𝛽1 cos 𝑚𝛽1 ) sin
𝑛𝜋
𝛼
𝛽2 (
𝑚 = 1,2,3 …
)
𝑛 = 1,3,5 …
(58)
We can then take equations (18) and (58) and combine them in the equation,
𝑙4
𝜆
𝑤 = 2𝐷 ∑ ∑ 𝜙𝑚𝑛 𝑍𝑚𝑛 sin 𝑚𝜙 sin 𝑎 𝑥
The displacement equations below are different to the displacement equations for
longitudinal moments. The change in force calculations drive the terms to become sin
19
(59)
(m∅) rather than cos (m∅). This does not cause a sign change in the results of the
equations put in. Therefore,
𝜆
𝑤 = ∑ ∑ 𝑤𝑚𝑛 sin 𝑚𝜙 sin 𝑎 𝑥
(60)
𝜆
𝑢 = ∑ ∑ 𝑢𝑚𝑛 sin 𝑚𝜙 cos 𝑎 𝑥
(61)
𝜆
𝑣 = ∑ ∑ 𝑣𝑚𝑛 cos 𝑚𝜙 sin 𝑎 𝑥
(62)
For Bijlaard’s final method in the creation of Welding Research Council Bulletin No.107
guideline, the following equations are formulated to use equations (19) and (58) to find
the moments and membrane forces as found in the bulletin, i.e.
1
𝑀𝑥 = 2 𝛼 2 𝑙 2 ∑ ∑ 𝜙𝑚𝑛 𝑍𝑚𝑛 [(
𝑛2 𝜋 2
𝛼2
𝜆
) + 𝑣(𝑚2 − 1)] sin 𝑚𝜙 sin 𝑎 𝑥
𝑣𝑛2 𝜋 2
1
𝑀𝜙 = 2 𝛼 2 𝑙 2 ∑ ∑ 𝜙𝑚𝑛 𝑍𝑚𝑛 [𝑚2 − 1 + (
Nx 
N 
Et
1 2
𝛼2
𝜆
)] sin 𝑚𝜙 sin 𝑎 𝑥
 u
 v w 
 
   
 a a 
 x
(63)
(64)
(65)
Et  v w
u 
  
2 
x 
1    a a
(66)
These terms must be broken down by using the equations
𝑢 = ∑∑
𝜆(𝑚2 −𝑣𝜆2 )
(𝜆2 +𝑚2 )
𝜆
𝑤𝑚𝑛 sin 𝑚𝜙 cos 𝑎 𝑥
(67)
and
𝑣 = −∑∑
𝑚[(2+𝑣)𝜆2 +𝑚2 ]
(𝜆2 +𝑚2 )
𝜆
𝑤𝑚𝑛 cos 𝑚𝜙 sin 𝑎 𝑥
(68)
Derivatives are taken of these equations and then placed back into the original equations
given ((36) and (37) to obtain the equations,
20
𝑚2 𝑛 2
𝜆
𝑛4
𝜆
𝑁𝑥 = −6𝜋 2 (1 − 𝑣 2 )𝛼 6 𝛾 2 𝑎 ∑ ∑ 𝜙𝑚𝑛 𝑍𝑚𝑛 (𝑚2 𝛼2 +𝑛2 𝜋2 )2 sin 𝑚𝜙 sin 𝑎 𝑥
𝑁𝜙 = −6𝜋 4 (1 − 𝑣 2 )𝛼 4 𝛾 2 𝑎 ∑ ∑ 𝜙𝑚𝑛 𝑍𝑚𝑛 (𝑚2 𝛼2 +𝑛2 𝜋2 )2 sin 𝑚𝜙 sin 𝑎 𝑥
(69)
(70)
Equations (18) and (58) are used to find the values used by Bijlaard.
To complete the guideline (WRC No. 107), Bijlaard took each of these equations
and solved them for changes in the value β, graphing them according to each type of load
induced into the system. This is what was used in this project to calculate his results and
compare them to the results from the finite element analysis.
The next section of the analysis 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.
𝛾=
𝑅𝑚
𝑡
The second parameter that is needed is the β term. This term will vary depending on the
type of attachment used and its orientation. The method has the possibility of using a
round attachment, a square attachment, or a rectangular attachment. Each of these types
has 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,
𝛽 = 𝛽1 = 𝛽2 =
21
𝑐1
𝑐2
=
𝑅𝑚 𝑅𝑚
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 Figure 22 in Appendix B will be used. Reading the value from the
chart for 𝑀
𝑁𝜙
2
𝑐 /𝑅𝑚 𝛽
, 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
𝑀
next step is to find, in Figure 20 in Appendix B, the values for 𝑀 /𝑅𝜙 𝛽. Once these values
𝑐
𝑚
are found, the initial conditions are used to solve for the membrane stress and the
circumferential bending stress by using the following equations
 N
 M c


 2
 
2
 M c / Rm   Rm *  * T 
And
 M   6 * M c 



2 
 M c / Rm   Rm *  * T 
(71)
(72)
Once each of these is solved for N∅/T and 6M∅/T2 then they can be combined back into a
general stress equation of the form,
𝜎𝜙 = 𝐾𝑛
𝑁𝜙
𝑇
± 𝐾𝑏
6𝑀𝜙
𝑇2
(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.
22
The same process is then used to find values for
Appendix B and
𝑀𝑥
𝑀𝑐 /𝑅𝑚 𝛽
𝑁𝑥
𝑀𝑐 /𝑅𝑚 2 𝛽
from Figure 23 in
from Figure 21. The values are then input into equations,
 N
 M c




2
 R 2 *  * T  
M
/
R

 c m  m

And
 M   6 * M c 



2 
 M c / Rm   Rm *  * T 
(74)
(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,
𝜎𝑥 = 𝐾𝑛
𝑁𝑥
𝑇
± 𝐾𝑏
6𝑀𝑥
𝑇2
(76)
When looking for stresses resulting from a longitudinal moment applied to the
attachment, the same steps are followed using different charts. First, to find the
circumferential stresses, Figure 26 in Appendix B is used. Reading the value from the
chart for
𝑁𝜙
𝑀𝑐 /𝑅𝑚 2 𝛽
, the next step is to find in Figure 24 in Appendix B the values
𝑀
for 𝑀 /𝑅𝜙 𝛽. As mentioned previously the steps remain the same by inputting the results
𝑐
𝑚
from the graphs into
 N
 M c


 2
 
2
 M c / Rm   Rm *  * T 
and
 M   6 * M c 



2 
 M c / Rm   Rm *  * T 
Then putting the solved values into the final stress equation
𝑁
6𝑀
𝜎𝜙 = 𝐾𝑛 𝑇𝜙 ± 𝐾𝑏 𝑇 2𝜙
23
(77)
(78)
(79)
The next term needed is
𝑁𝑥
𝑀𝑐 /𝑅𝑚
2
from Figure 27 in Appendix B and
𝛽
𝑀𝑥
𝑀𝑐 /𝑅𝑚 𝛽
from Figure 25. They are then input into the equations,
and
 N
 M c


 2
 
2
 M c / Rm   Rm *  * T 
 M   6 * M c 



2 
M
/
R

R
*

*
T
 c m  m

(80)
(81)
and they can then be combined back into a general stress equation of,
𝜎𝑥 = 𝐾𝑛
𝑁𝑥
𝑇
± 𝐾𝑏
6𝑀𝑥
𝑇2
(82)
The calculation has been simplified down to a single sheet for Bijlaard’s computation
scheme. 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.
24
4 – RESULTS
In the previous chapter Prof. Bijlaard’s methodology was presented. In the
computations, some of the same assumptions that he made were continued through the
report’s calculations to make sure that the derivations match exactly with what Bijlaard
used to create the method incorporated 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 are shown in Figure 9 below.
Figure 9 – Detail of Square Nozzle and Stress Locations [6]
𝑡2
2
2.) When initially creating the stress equations, the terms for (12𝑎2 ) were removed
from the equation.
25
𝑡2
3.) After finding the derivations of the equations the term 12𝑎2 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.
The effects of each of these assumptions 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 describes a test cylinder 71 inches in length and 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]
26
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 various distances
from the welded attachment on the pipe; these locations are illustrated in Figure 11.
Figure 11 – Strain gauge Layout for Schoessow and Kooistra Example [4]
The computed value of the circumferential moment using Bijlaard’s method is
equal to 23,280 psi. Stress values equal to 25,000 psi were recorded from the extrapolated
data obtained from the strain gauges. The difference is about 6.8%. This is an acceptable
deviation.
The same example was simulated using a Finite Element Analysis, as shown in
Figure 12.
27
Figure 12 - Proof of Concept Max Von Misses Stress in the Circumferential Direction
The maximum stress values calculated with 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 among all three
analyses.
The longitudinal moment analysis was performed with the same values. The
resulting stress was 13,910 psi using Bijlaard’s method. This must be compared to 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 calculations that were performed during the analysis
using the finite element method.
Using the same methods that have been used for a proof of concept, a separate
example was created in order to facilitate the modeling of combined longitudinal and
circumferential moments on a nozzle. The geometry is shown in Figure 13:
28
Figure 13 – New Example Geometry
The mean radius of the cylinder is 15 inches with a thickness of 0.3 inches. These
dimensions give 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 shown in Figure 14.
29
Figure 14 - Circumferential and Longitudinal Moments on Attachment
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
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 (see Figure 9 ). The values for each
column represent the locations where the stress was measured in the experiment.
Table 1 – Comparison of Results from WRC to FEA
AU
WRC
Method
FEA
Difference
AL
BU
BL
1735.3 -8253.9 -1735.3 8253.9
CU
CL
DU
-11557.5 19553.7 11557.5
-2888.3
2887.4
-12331.7
12464.7
39.92%
39.90%
6.28%
7.28%
30
DL
-19553.7
The finite element method mimics Bijlaard’s methods and theories closely when
the circumferential moment is analyzed on its own as can be seen in Figure 15. The
stresses created from the circumferential load create a stress concentration around the
major axis of the attachment. This case containing the loading in the circumferential
direction is testament as to the results shown in Table 1 only having a deviation equal to
6.28% and 7.28% at locations CU and DU respectively.
Figure 15 - New Example Circumfrential Loading Only
The finite element method varies considerably from the WRC methodology in the
case of loading along the longitudinal direction. The FE analysis with the longitudinal
moment produces results inconsistent with Bijlaard’s method. Figure 16 shows that the
maximum stress is not located at the major axis of the nozzle. The maximum stress is
concentrated at each corner of the attachment. The results have a staggering difference of
39.9%, almost four times the accepted variation.
31
Figure 16 - New Example Longitudinal Loading Only
The results from finite element calculations for the combined loading model are
shown in Figure 17. 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
(seen in Figure 17) that equals 32,680 psi. This value has a difference of 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. How these types of stress are ignored in Bijlaard’s guideline will be
discussed in the latter sections.
32
Figure 17 - Combined Loading Max Von Mises Stress
33
5 – DISCUSSION
Bijlaard made a few assumptions in his analysis in order to simplify the calculations
and produce a final method easy enough for 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 very reasonable for simple
systems and 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 shown in the finite element 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 using the simplified approach. The absence of this
stress concentration can cause the results found from WRC 107 to be unconservative.
2.) When first deriving formulas to create the curves for WRC 107, the terms for
𝑡2
2
(12𝑎2 ) were neglected, since the thickness to shell radius ratio was low. This is
approximately true for the thin shells used in the examples considered. Leaving
34
any terms out of a calculation however, will decrease its overall true to life
accuracy.
𝑡2
3.) The terms containing 12𝑎2 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 16. 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. The finite element model is not subject to this limitation.
6.) Stresses are considered to be equal and opposite on either side of the nozzle
attachment. This is a 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.
35
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 pipe and the attachment 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 in the original experiment (Figure 18).
Figure 18 - 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 geometry of the shell was created as shell
elements. These elements use the thin shell theory as their base calculation. Shell
elements produced the best results when compared against experimental data. The
geometry of the attachment was modeled as hex elements. Solid structures like the
attachment produce more accurate results than tetrahedral elements when modeling
the examples used in this report. The mesh (shown in Figure 19) was concentrated in
the area around the nozzle to increase accuracy.
36
Figure 19 - Meshing Detail around Nozzle
The rest of the model was partitioned up to have a coarser mesh and to not increase
the time and difficulty of the calculation. The limits to the finer mesh were
determined as where stresses would not deviate and change greatly with further
refinement. The nozzle attachment was determined to have an equal load across the
entire width of the attachment to represent a uniform load, just as is required from
Bijlaard’s assumptions. Point loads were not used since they would cause a local
deformation of the lug that would skew the final results. The 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 agree very well with those obtained
from the WRC methodology. However the longitudinal loads were not in good
agreement. 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 consider that the maximum stress
values are located at points on the major axes of the nozzle. When the parallel loading
planes are not considered and accounted for, adverse effects happen that can cause the
final results to vary from the actual real world stresses.
37
6 - CONCLUSION
The Welding Research council gave the task of creating a simplified calculation
method for piping systems with welded attachments to Prof. Bijlaard. In creating this
method, certain assumptions were made. The method was 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 always satisfied in practice and
they will not give accurate concentrations of stresses resulting from complex loads
imparted on a nozzle attachment. The examples presented in this report show the unconservative 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 loading combinations and only a few of them
were discussed in this paper, but they could well be the subject of further study using the
finite element program developed herein.
38
REFERANCES
1. Bijlaard, P. P. "Stresses From Local Loadings in Cylindrical Pressure Vessels."
Transactions Of The ASME (August, 1955): 805-816.
2. —. "Stresses From Radial Loads and Eternal Moments in Cylindrical Pressure
Vessels." The Welding Journal (December, 1955): 608-s - 617-s.
3. —. "Stresses from Radial Loads in Cylindrical Pressure Vessels." The Welding
Journal (December, 1954): 615-s - 623-s.
4. F., Schoessow G. J. and Kooista L. "Stresses in a Cylindrical Shell Due to Nozzle
or Pipe Connection." Transactions of The ASME (June, 1945): A-107 - A-112.
5. Gould, Phillip L. Analysis of Shells and Plates. New York, New York: SpringerVerlag New York Inc., 1988.
6. K. R. Wichman, A. G. Hopper, and J. L. Mershon. "Local Stresses in Sperical and
Cylindrical Shells Due to External Loading." Welding Research Council Bulletin
No. 107 (July, 1970): ii - 69.
7. Timoshenko, S. Theory of Plates and Shells. New York, New York: McGraw-Hill
Book Company Inc., 1940.
39
APPENDIX A
40
Table 2 – WRC No.107 Hand Calculation
41
APPENDIX B
42
Figure 20 – WRC No.107 Figure 1A – Graph to find M∅ for External Circumferential Moment [6]
43
Figure 21 - WRC No.107 Figure 2A – Graph to find Mx for External Circumferential Moment [6]
44
Figure 22 - WRC No.107 Figure 3A – Graph to find N∅ for External Circumferential Moment [6]
45
Figure 23 - WRC No.107 Figure 4A – Graph to find Nx for External Circumferential Moment [6]
46
Figure 24 - WRC No.107 Figure 1B – Graph to find M∅ for External Longitudinal Moment [6]
47
Figure 25 - WRC No.107 Figure 2B – Graph to find Mx for External Longitudinal Moment [6]
48
Figure 26 - WRC No.107 Figure 3B – Graph to find N∅ for External Longitudinal Moment [6]
49
Figure 27 - WRC No.107 Figure 4B – Graph to find Nx for External Longitudinal Moment [6]
50
APPENDIX C
51
Figure 28 – Diagram Illustrating Boundary Conditions on New Example
52
Figure 29 –Diagram Illustrating Meshing For New Example
53
Figure 30 - Stresses Presented on New Example from Circumferential Loading Only
54
Figure 31 - Stresses Presented on New Example from Longitudinal Loading Only
55
Figure 32 – Front View Max Von Misses Stress for New Example
56
Figure 33 – Rear View Max Von Misses Stress for New Example
57
Figure 34 - Displacement Vectors for Complex Load
58
Figure 35 - Displacement Vectors for Complex Loading Inside Shell
59
Figure 36 - Principal Stress 11 for Complex Loading
60
Figure 37 - Principal Stress 22 for Complex Loading
61
Figure 38 - Principal Stresses 12 from Complex Loading
62