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ANALYSIS OF A PRESSURE VESSEL JUNCTION
BY THE FINITE ELEMENT METHOD
by
MAHADEVA SIVARAMAKRISHNA IYER, B.Sc. in C.E., M.Sc. in C.E,
A DISSERTATION
IN
CIVIL ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for
the Degree of
DOCTOR OF PHILOSOPHY
Approved
Accepted
December, 1972
ACKNOWLEDGMENTS
I am deeply indebted to Dr. C. V. G. Vallabhan for his guidance
and counseling during this investigation and also for serving as
Chairman of the Advisory Committee. I also wish to express my deep
appreciation to Dr. Ernst W. Kiesling for his guidance and encour
agement throughout my graduate studies at Texas Tech University. I
am also grateful to Dr. James R. McDonald, Dr. Jimmy H. Smith and
Dr. Robert A. Moreland for their helpful criticisms and valucQ le
suggestions.
11
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
LIST OF TABLES vi
LIST OF FIGURES vii
LIST OF SYMBOLS x
I. INTRODUCTION 1
1.1 Definition of the Problem 1
1.2 Scope of the Present Research 5
1.3 Selection of a Suitable Finite Element 5
1.4 Analysis for Local Stress Concentrations 6
1.5 Analysis for Dynamic Loads 7
II. THE FINITE ELEMENT METHOD 8
2.1 The Displacement Method 11
2.2 Displacement Field Requirements 12
2.3 Applied Load Vector 17
2.4 Initial Strain Problems 18
2.5 Analysis for Dynamic Loads 20
2.6 Assembly of the Stiffness Matrix for a Continuum. . 20
2.7 Boundary Conditions 21
2.8 Solution of the Equilibrium Equations 22
2.9 Development of Computer Program for the Pressure
Vessel Analysis 22
III. Review of Plate and Shell Elements 24
3.1 Plane Triangular Elements 24
3.2 Curved Triangular Elements 27
iii
iv
3.3 Plane Quadrilateral Elements 29
3.4 Curved Quadrilateral Elements 30
3.5 Selection of Finite Element for the Pressure Vessel Analysis 31
IV. EVALUATION OF ELEMENT STIFFNESSES 34
4.1 Area Coordinates 35
4.2 Derivation of the Constrained Linear Strain Triangle Element Stiffness 39
4.3 Thin Plate Theory 42
4.4 Plate Bending Stiffness Matrix for the Triangular Element 44
4.5 Assembly of the Stiffness Matrix for the Quadrilateral Element 58
4.6 Coordinate Transformations 60
4.7 Volume Coordinates 62
4.8 Stiffness Matrix for a Tetrahedral Element 65
4.9 Stiffness Matrix for the Octahedral Element . . . . 67
V. RESULTS OF STATIC ANALYSIS 69
5.1 Bending of a Square Isotropic Plate 69
5.2 Pressurized Pipe 84
5.3 Cylindrical Shell Roof 86
5.4 Analysis of a Folded Plate Structure 86
VI. ANALYSIS FOR DYNAMIC LOADS 94
6.1 Ecjuations of Motion 94
6.2 Mass Matrix 97
6.3 Damping Matrix 99
6.4 Response to Dynamic Loading 101
6.5 Step by Step Numerical Integration Procedure 102
6.6 Selection of the Time Interval At 105
6.7 Results of Dynamic Analysis - Simply Supported Rectemgular Plate under Dynamic Loading 106
VII. RESULTS OF PRESSURE VESSEL ANALYSIS 112
7.1 Results of Static Analysis using the Shell Element. . 112
7.2 Analysis for Local Stress Concentrations 124
7.3 Results of Dyneunic Analysis 131
VIII. CONCLUSIONS AND RECOMMENDATIONS 137
8.1 Conclusions 137
8.2 Reccxnmendations for Further Research 139
LIST OF REFERENCES 140
LIST OF TABLES Table Page
4.1 Interpolating Functions for Lateral Displacement w. . . . 50
4.2 Coefficients for the Interpolating Functions 51
4.3 Derivatives of Interpolating Functions 54
4.4 Values of the Constants in the Derivatives of the Interpolating Functions 55
5.1 Bending of a Square Isotropic Plate 71
5.2 Comparison of Results - Bending of a Scjuare Isotropic Plate 73
5.3 Analysis of a Pressurized Pipe 84
6.1 Simply Supported Rectangular Plate under Dynamic Loading Case 1 - Constant Forcing Function 107
6.2 Simply Supported Rectangular Plate under Dynamic Loading Case 2 - Triangular Forcing Function 108
7.1 Pressure Vessel Analysis - Stress Distribution along the Vertical Section 115
7.2 Pressure Vessel Analysis - Radial and Tangential Stress Distribution in the Cover Plate 116
7.3 Dynamic Analysis of Pressure Vessel 132
VI
LIST or FIGURES Figure Page
4.1 Area Coordinates 37
4.2 lunear Strain Triangle with 12 degrees of freedom
Sign Convention for Forces and Displacements 37
4.3 Sign Convention for Inplane Stress Components 48
4.4 Sign Convention for Plate Bending McMnent Components . . 48
4.5 Triangular Plate Element - Positions of Regions 1, 2, 3 49
4.6 Triangular Plate Element - Sign Convention for Forces
and Displacements 49
4.7 Planar Quadrilateral with 33 degrees of freedom . . . . $7
4.B Global and Local Coordinate Systems 57
4.9 Tetrahedral Element - Volume Coordinates 68
4.10 OctjJiedral Element and its Subdivision into Tetrahedrons 68
5.1 S<iuare Plate - Element Subdivision 77
5.2(a) Simply Supported Square Plate
Compaurison of Results - Central Concentrated Load . , 78 5.2(b) Simply Supported Square Plate
Coo^arison of Results - Uniformly Distributed Load. . 79
5.2(c) Clamped Square Plate Comparison of Results - Central Concentrated Load . . 80
5.2(d) Claiiv>ed Squ2u:e Plate Comparison of Results - Uniformly Distributed Load. . 81
5.3 Distribution of Bending Moment along the Center Line of a Uniformly Loaded Scjuare Plate 82
5.4 Singly Sv5)ported Scjuare Plate - Distribution of Reactions along the Supporting Edge (F.E.M. Analysis) . 83
5.5 Pipe under Internal Pressure 85
5.6 Cylindrical Shell Roof 88
5.7(a) Cylindrical Shell - Convergence of Results 89
Vll
viii
5.7(b) Cylindrical Shell - Comparison of Results 90
5.8 Folded Plate Analysis 91
5.9 Results of Folded Plate Analysis 92
5.10 Results of Folded Plate Analysis 93
6.1 Simply Supported Plate under Dynaunic Loading 109
6.2 Response of a Simply Supported Rectangular Plate Load Case 1 110
6.3 Response of a Simply Supported Rectangular Plate Load Case 2 Ill
7.1 Schematic Diagram of a Junction at a Manhole in a Pressure Vessel 119
7.2 Stresses around a Meuihole in a Pressure Vessel Schematic Diagram of Finite Element Idealization . . . . 120
7.3 Stress Distribution in the Cover Plate on a Vertical Section through the Center of the Manhole 121
7.4 Pressure Vessel Analysis Direct Stress Tangent to the Manhole at 40 psi Pressure. 122
7.5 Pressure Vessel Analysis Direct Stress Radial to the Manhole at 40 psi Pressure . 123
7.6 Finite Element Idealization for Three Dimensional Analysis 126
7.7 Hoop Stress Distribution on the Cover Plate at the Junction 127
7.8 Hoop Stress Distribution across the Thickness of the Cover Plate at the Junction 128
7.9 Hoop Stress Distribution across the Thickness of the
Manhole Plate at the Junction 129
7.10 Stress Distribution at the Inside Edge of the Junction . 130
7.11 Dynamic Analysis of a Pressure Vessel 133
7.12 Dynamic Analysis of a Pressure Vessel 134
IX
7.13 Dynamic Analysis of a Pressure Vessel of Displacements
- Propagation 135
7.14 Dynamic Analysis of a Pressure Vessel Variation of Hoop Stress at Node 28 . 136
A4
LIST OF SYMBOLS
A: Area.
[B]: Matrix of the coefficients of strain displacement relationships.
[C]: Damping matrix for the complete structure.
[c]: Dampinq matrix for an element.
[D]: Matrix of the coefficients of stress strain relationships
E: Modulus of elasticity.
(P}t Vector of equivalent nodal forces.
{F)-» Force vector ciue to damping forces. a
(P) : Force vector due to inertia and damping forces.
{P}^: Force vector due to inertia forces.
(P) : Force vector due to surface forces. P
(P) : Force vector due to body forces.
{P} 0! Force vector corresponding to initial strcdns. e
h: Thickness of the plate element.
[K] : Stiffness matrix for the ccmplete structure.
[k ] : Stiffness matrix of an element with respect to me local ccxDrdinates.
[k ] : Stiffness matrix of an element with respect to the global coordinates.
[k]: Stiffness matrix of an element.
L , L , L-: Area ccx>rdinates.
L,, L , L , L.: Volume coordinates.
(M): Vector of bending moments.
(M): Mass matrix for the complete structure.
xi
Im]: Mass matrix for an clement.
[Nj: Matrix of interpolating functions.
n: Number of unknowns in a problem.
{p}: Vector of the components of surface tractions.
R: Radius of cylindrical shell.
[Tl: Transformation matrix.
(T^l: Matrix of direction cosines, c
t: Thickness
U: Strain energy of deformation.
u: Displacement in the x direction.
u: Velocity.
u: Accelera t ion .
{u}: Vector of ncxial displacements .
V: Volume.
v: Displacement in the y direction.
w: Displacement in the z direction.
{w}: Vector of nodal displacements and rotations for the plate element.
o: Nondimensional parameter for deflection.
a : Proportionality constant for viscous damping, m
a, : Proportionality constant for structural damping.
P: Newmark's 3 parameter.
3, Y- Nondimensional parameters for bending moments.
A: Nondimensional parameter for reaction.
{A}: Vector of generalized displacement parameters.
xii
At: Time interval.
{e}: Vector of strain componehts.
e
(c }: Vector of elastic strain components.
{c®}: Vector of initial strain components.
(a): Vector of stress components.
v: Poisson's ratio.
•: Total potential energy function. 9 : Rotation about x aocis. x
6 : Rotation about y axis
\x: Dancing coefficient.
p: Mass per unit volume.
{p} : Vector of t h e ccxnponents of the bcxiy f o r c e s
CHAPTER I
INTRODUCTION
1.1 Definition of the Problem
One of the most important problems in pressure vessel design is
that of adequate reinforcement at openings. Solution to this problem
is achieved by the use of codes developed frcan experimental and simpli
fied analytical methods. A brief description of the methods specified
by several codes and some additional analytical and experimental infor
mation are given in the following paragraphs.
There sire three design methods in common use, and they may be
related to the typical codes as follows (1).
1. A S M E VIII, Divisions 1 and 2 - Area for area
2. B.S. 1515 - Controlled maximum stress
3. German code - Experimental yield
The first method has been in practice for many years cuid has
proved both simple in application and essentially relicible in service.
The amount of material removed from the opening is replaced by ein equal
amount of reinforcement around the opening.
The secx>nd method is based on the adoption of a Stress Concentra
tion Factor ( SCF ), defined as the ratio of the maximum stress in the
shell near the opening to the design stress for the shell without any
opening. B.S.1515 uses a SCF of 2.25. This approach is based on the
work of Leckie and Penny (2). Although the analysis was made for a
cylinder/sphere junction, the method is also used for cylinder/cylinder
junctions. The method is usually restricted to a nozzle/vessel diameter
ratio less than 1:3 (1). A comparison with the "area for area" method
shows that substantial economy of material is achieved by the SCF
methcxl (3) .
Method 3 is based on pressure tests for a series of cylinder/cylin
der junctions. The pressure ( PQ 2 ^ ° produce 0.2% residual strain at
the junctions is used to define "weakening factors". The design pressure
is PQ 2^^'^' '^^^^ ^ ® nozzle has the same reserve on yield as the main
body of the vessel, when the design stress ecjuals Oyjj/1.5 iOy^ is the
yield stress of the material). In general, this method yields about
the same reinforcement design as method 2 using a SCF of 2.25 (1). It
should be noted that in the pressure tests used as the basis for this
methcxi, strains were measured only on the outside surfaces of the models.
It is known that strains can be higher on the inside corner.
The practice of designing reinforcement based on ccxies is mostly
empirical and gives little information about the actual stress condi
tions that may exist around the openings. Some theoretical and experi
mental studies have recently been made of the stress distributions
curound openings in pressure vessels. A brief description of these works
is given below.
Elastic stress auialysis of single radial openings in spherical
pressure vessels was done by Leckie and Penny (2) and by Waters (4).
Tables and graphs are given that can be used for simplified calcnilation
of stresses at any point in the vicinity of a sphere/cylinder junction.
Palmer (5) has presented a design technique for determining the recjuired
reinforcement around an opening in a spherical pressure vessel. In his
method a search is made for a stress field in equilibrium with the load
and then the parts of the shell are proportioned so that the material
yield condition is not achieved. A simple design chart is also presented.
The problem of two normally intersecting cylindrical shells was
considered by Pan (6), Lind (7), Maye and Eringen (8), Jones and Hansberry
(9), and D.H.van Campen (10). In the work of Pan (6), the differential
ec^ations for the two intersecting shells were solved numerically subject
to the boundary conditions imposed along the intersection of the two
cylinders. Lind (7) has also presented a similar analysis, but under the
assumption that the vessels are thin walled. In the problem presented
by Naye auid Eringen (8), the solution for each shell was obtained by the
superposition of the solution for a shell without intersection and the
solution for the shell aurbitrarily loaded along the intersection. Donnel's
shell theory was used for the case of loading along the intersection and
the exact solution was obtained as an infinite series. The analysis
presented by Jones and HeUisberry (9) is believed to be valid for nozzle
to cylinder ratios of less theui 1/3. In the methcxi developed by
D.H.van Campen (10), both mechanical and thermal stresses in cylinder to
cylinder intersections of ecjual or nearly equal diameters were determined.
A finite element method using a tricuigular ring element is presen
ted by D.H.van Campen (11). This method is used to determine peak stresses
at nozzle to cylinder intersections for sufficiently small ratio of
diameters.
In addition to elastic analyses, theoretical limit analyses have
been done by Schroeder and Rangarajan (12) and by Cloud and Rodabaugh (13)
The latter report forms the basis of nozzle design procedure in the
U.S.A. (1).
Extensive experimental work has also been performed in this field
by Hardenberg, et al (14), Taylor and Lind (15), Leven (16), Taniguchi,
et al (17), Decock (18), Kitching, Davis and Gill (19), Fidler (20),
and others. These include three dimensional photoelastic analyses (15,
16, 17, 20), fatigue tests (18), and strain gage measurements.
Clearly, a large amount of theoretical and experimental work on
pressure vessel openings has been carried out. A consideraQ^le amount of
it is classified and remains proprietory to pressure vessel mcuiufacturers.
The research works referenced here are those published in readily avail
able journals. It is evident that the theory is limited to the analysis
of pressure vessel junctions having regular geometry only. Most of the
cx>nventional openings euid connections fall in this category and hence
it may appear that the theory is sufficiently developed. However, it
should be noted that none of these analytical methcxls is general enough
for the analysis of pressure vessel junctions under all circumstances.
There is a mass of uncorrelated experimental data obtained by
researchers interested only in particular applications.These are of value
in a limited field only. Certain qualitative results are common to all
tests : the region of disturbed stress is confined to a circle of twice
the diameter of the opening; the stress concentration increases as the
ratio of the opening to vessel diameter increases; the effectiveness of
the reinforcement is a function of it's proximity to the edge of the
opening. A reasoneibly accurate numerical value for the stress rise caused
by either reinforced or unreinforced openings in pressure vessels is not
presently available.
1.2 Scope of the present research
This dissertation presents a rational approach to the analysis of
stresses around openings in pressure vessels. The method of analysis
used is the Finite Element MethcxJ (21). The finite element method is
basically a special form of the Ritz analysis (22) and as such it pro
vides a means for the discretization of continuum problems. The special
feature of the finite element method which makes it so efficient in
digital computer analysis is that the structure may be divided into a
system of appropriately shaped finite elements and the properties of the
vi ole structure may be derived from the properties of the individual
elements by suitable superposition. The major advantage of this methcxi
of disc:retization is the ease that structures of arbitrary shapes and
with vauriations of properties can be approximated as a system of finite
elements having simple shapes and uniform ( or simply varying ) proper
ties. Since the ccanplex geometry euid varying thicknesses of the different
parts Ceui be easily teiken into account in the analysis, this advantage
is very significant in the investigation of stresses around pressure
vessel openings.
1.3 Selection of a Suitcdjle Finite Element
The amount of work that has been accomplished in the application
of the finite element technicjue is so extensive that it was considered
prudent to review the literature for applicable developments. The
research during the early stages of this investigation was directed
tcswards finding the most appropriate finite element technique for the
present problem. A detailed study was made of the different types of
shell elements that can be used to solve the pressure vessel problem.
These include triangular, rectangular, and quadrilateral elements, some
of which are flat while others are curved. After reviewing the proper
ties and applicabilities of the different elements, it was found that
the more sophisticated types of shell elements, including doubly curved
elements, did not contribute significantly to the accuracy of the analy
sis. The results of the survey indicated that the cjuadrilateral shell
element developed by Clough and Johnson (23) would be the most appropriate
for the pressure vessel analysis.
A computer progrcun using the quadrilateral shell element was
written specifically for the analysis of pressure vessel junctions.
Many problems with known solutions were solved to check the accairacy of
the program and the efficiency of the particular shell element.
A pressure vessel which had been instrumented and tested experi
mentally by the Chicago Bridge and Iron Company (24) was chosen for
analysis in order to investigate the applicability of the finite element
methcxi and the particular element to the pressure vessel problem. The
results of the finite element analysis and the experimental investigation
agreed very closely, indicating that the finite element method using the
peurticular cjuadrilateral shell element can be confidently applied for
the analysis of pressure vessel problems.
1.4 Analysis for Local Stress Concentrations
The analysis using the shell element yielded valuable insight into
the stress conditions around the opening. From the membrane stresses and
bending moments obtained by the analysis, the stress gradient across the
thickness of the coverplate and other parts of the structure were deter
mined. But along the line of intersection of the pressure vessel wall
and the manhole, the stresses and moments obtained by the shell analysis
represented average conditions only. Because of the abrupt discontinuity
of the shell geometry along the above line, it was not possible to
predict the nature of the stress variation across the thickness of the
plates. To determine the precise nature of the stress variation across
the thickness and the stress concentration at the corners and fillets,
a detailed analysis of a small portion of the structure was performed
using a three dimensional finite element analysis. Two shell elements,
one of vrtiich is part of the main body of the pressure vessel and the
other a part of the meuihole, were analyzed by the finite element method
using octahedral elements. The boundary forces and displacements for
this analysis were those obtained frcxn the shell analysis.
1.5 Analysis for Dynamic Loads
The finite element method for static analysis was extended to the
analysis for dynamic loads (25). The computer program developed for the
static cuialysis was modified to analyze for dynamic loads using a step
by step integration procedure. The pressure vessel junction was analyzed
for an impact load using the program. The results obtained indicate that
the finite element method is as versatile and powerful in dynamic analysis
as it is for static analysis.
CHAPTER II
THE FINITE ELEMENT METHOD
The concept of finite element methcxis for solid continua was
developed in the mid 1950's and has been attributed to Turner, Clough,
Martin, cuid Topp (26). The matrix displacement method was applied to
solve plane stress problems using triangular and rectangular elements.
In 1955, Argyris (27) in his well known treatise on matrix structural
analysis, showed a derivation of the stiffness matrix of a plane stress
rectangular peuiel. The formulations of the element stiffness matrices
by these authors and other early investigators of finite element methods
were not based on the field equations of the entire elastic continuum.
Only since the early 1960's has it become apparent that the finite
element method can be interpreted as an approximate Ritz method associ
ated with a variational principle in continuum mechanics. Courant (28),
as early as 1943, and Prager and Synge (29) in 1947 proposed methods
which were essentially identical to those in current use. Using varia
tional principles in solid mechanics, it is possible to derive numerous
finite element mcxiels which may lead to either a displacement method,
a force method or a mixed method. Furthermore, the variational principles
CcUi be applied to initial strain problems, elastic stability problems,
or plasticity problems in order to formulate the corresponding system
ecjuations for the finite element models for these problems. Finally,
finite element methods can also be derived for many field problems in
mathematical physics and engineering other than those in solid mechcuiics.
The analysis of a complex structural system generally requires
8
transformation into a discrete mathematical system. The finite element
method provides such a transformation, and is the most general one
available. The simplified structural model consists of various types
of discrete or finite structural elements. The approximate behavior of
each element can be expressed in terms of selected generalized stress
and strain variables using elasticity theory. The elements are then
assembled by enforcing equilibrium of forces and compatibility of
displacements at a finite number of locations (ncxles) on the model.
These cx>nditions are expressed as a set of nonhomogenous linear equa
tions in which the variables are element forces and structural displace
ments and the constant terms are the applied loads and initial strains.
Depending upon the assumptions made to express the approximate
behavior of each element, there are mainly three approaches in the
finite element methcxi:
1. The Displacement methcxi
2. The Force method
3. Mixed ( or Hybrid ) method
In the displacement method, displacement functions are assumed
inside each element which are compatible along the interelement boun
daries. The elements are then assembled by enforcing equilibrium of
forces at the nodes. The resulting equations are mcxiified for the
boundary conditions and solved for the nodal displacements. The strains
and stresses inside each element are then computed using the assumed
displacement functions.
The force method assumes a stress condition within the element
which maintains equilibrium of boundary tractions. The elements are
10
assembled by enforcing compatibility of displacements at the ncxies.
The resulting equations are modified for the boundary conditions and
solved for the ncxial forces. The stresses within the elements are then
computed using the assumed force functions.
In the hybrid method, there are three models: hybrid I, hybrid II,
and mixed model. Hybrid I assumes stresses inside each element and
ccxnpatible displacements along the boundaries. In hybrid II, displace
ments inside the element and equilibrating tractions along the bounda
ries aure assumed. In the mixed methcxi, continuous displacements and
stresses are assumed within each element maintaining displacement
cxxi^atibility along the boundaries.
Of all these approaches, the displacement methcxi has received
the most attention frcxn researchers because of simplicity of euialysis.
The displacement methcxi is used in the present research.
The basis for the derivation of the stiffness matrix of a finite
elatient and the requirements for the selection of the displacement
functions are discussed in the next two sections. Section 2.3 deals
with the derivation of the equivalent force vector due to bcxiy forces
and applied loads. Sections 2.4 and 2.5 present a brief description of
the procedure for handling initial strain problems and dynamic problems.
The assembly of the stiffness matrix for the complete structure,
incorporation of the boundary conditions, and the solution of the
ecjuations are dealt with in sections 2.6, 2.7, and 2.8. A brief descrip
tion of the computer program developed for the pressure vessel analysis
is presented in section 2.9.
11
2.1 The Displacement Method
The displacement methcxi is based on the principle of minimum
potential energy of a deformable bcxiy which can be stated as follows:
" Among all displacements of an admissible form, those which satisfy the equilibrium conditions make the total potential energy function • assume a stationary value. For stable ecjuilibrium • is a minimum " (30).
The total potential energy function is defined as
• = U - V , (2.1)
where U is the strain energy of deformation and V is an energy function
which is the inner prcxiuct of the prescribed forces and the correspon-
ding displacements. V can be expressed as {F} {U} where {F} denotes the
vector of the ecjuivalent nodal forces corresponding to the prescribed
forces euid {u} is the vector of the nodal displacements.
For a linear structural system, neglecting effects of initial
strain and bcxiy forces, the strain energy of deformation is given by
U = J J {e}' [D] {e} dvol (2.2) vol
where the integral is teUcen over the volume of the element, {e} repre
sents the vector of the strain components at any point, and [D], the
matrix of the coefficients of the stress-strain relationships.
If the displacement field can be defined, then {z} is related to
the joint displacements {u} by the strain-displacement relationships:
{e} = [B] {u} (2.3)
where [B] is a matrix of the coefficients of the strain-displacement
12
r e l a t i o n s h i p s .
S u b s t i t u t i n g i n ecjuation (2 .2)
1 fr ,T,^,T U - Y J ^"^ [B] [D] [B] {u} dvol
vo l
1 r tT r _ , T . _ . [B] [D] IB] dvol {u} . (2 .4)
^ v o l
According to the principle of minimum potential energy
5 •
Substituting for • frcjm equation (2.1) and noting that V = {F}' {u}
JJ^ [ U - {F}' {u} ] = 0 . (2.5)
3 U Therefore j ^ = {F} . (2.6)
Substituting for U frcxn equation (2.4) and differentiating
d U / , ,T a{u} J [ B ] M D ] [B] dvo l {u} = {F} . (2 .7)
v o l
The e lement s t i f f n e s s m a t r i x fo l lows from ecjuation (2.7) and i s g iven
by / - ' [k] = J [B]"[D] [B] dvol . (2.8)
vol
Thus, the essential requirements for the formulation of the
element stiffness matrix are the selection of the displacement field
within the element for the establishment of ecjuation (2.3) and the
performeuice of the integrations as indicated by ecjuation (2.8) .
2.2 Displacement Field Requirements
It is evident that the accuracy which may be obtained by the
13
finite element methcxi depends directly on the accuracy with which the
deformation patterns are selected. The assumed deformation patterns
should, as closely as possible, reproduce the distortions actually
developed within the element. If the deformation patterns are not proper
ly chosen, the deformations will not necessarily converge to correct
values v^en the mesh size is decreased. On the otherhand, very gooi
results may be obtained with a very coarse mesh if the element defor
mation patterns selected closely correspond to the actual patterns. Thus
the most critical factor in the entire finite element analysis is the
proper selection of the element displacement field.
The usual method of representing the displacement field is by
selecting interpolation functions and generalized displacements at a
finite number of ncxial points of each element. To fulfill the conditions
of the principle of minimum potential energy, the interpolation functions
must be such that the displacements along the interelement boundcuries
cire compatible. In matrix form the assumed displacements are expressed
as {A} = [N] {u} (2.9)
where {A} is a column matrix of the generalized displacement parameters,
{u} is a column matrix of the generalized displacements at the
boundeury nodes of the element, and
[N] is the matrix of interpolating functions.
The strain distribution may be derived from equation (2.9) by the
strain-displacement relationships as
(e) = [B] {u} (2.10)
Once the matrix [B] has been defined, equation (2.8) enables us to
14
derive the element stiffness matrix.
Most finite element formulations are based on the assumed dis
placement approach. By recognizing the similarity between such an
approach and the Ritz methcxi, many authors have provided convergence
proofs of the assumed displacement finite element methcjd (31, 32, 33).
Irons and Draper (34) have pointed out the existence of three conditions
for the assumed displacement functions which assure convergence of the
solutions to the exact values when the element size is progressively
decreased. These conditions are:
1. Representation of all rigid body displacements.
2. Representation of states of constant stress.
3. Compatibility at the interelement boundaries.
It is generally agreed that the first two conditions ( referred
to as the " ccampleteness " requirement ) are the necessary conditions
for convergence. Investigators have indicated that finite element mcxiels
based on nonccxnpatible interelement displacements can result in conver
ging solutions. Irons, et al (34, 35) have pointed out the difficulties
of achieving all three of the conditions in the case of a tricuigular
plate element in bending and have demonstrated that, by using noncom-
patible elements, solutions converging to correct answers can be
obtained under one element mesh arrangement while diverging or incorrect
solutions cu:e obtained under another arrangement.
For analyzing plane or three dimensional elasticity problems, for
which only the continuity of the displacement components is recjuired
at the interelement boundaries, it is a relatively simple matter
15
to construct interpolation functions to fulfill the above three condi
tions. In the case of a plane elasticity problem using triangular ele
ments or a three dimensional problem using tetrahedral elements, the
triangular(area) or tetrahedral(volume) coordinates are most convenient
for the formulation of the stiffness matrices, and linear displacement
functions are the simplest choices (36). For rectangular and right
prismatic elements with all ncxies at the corners, the bilinear and
trilinear interpolation functions are again the simplest. In all of the
above elements higher order interpolation functions ceui be used if
additional ncxies along the edges are introduced, or if derivatives of
the displacements at the ncxies are also used as generalized displacement
coordinates (37, 38, 39). Finally, arbitrary (juadrilateral elements with
straight or curved edges and arbitrary hexagonal elements with flat or
curved faces cam be mapped into corresponding rectangular and right
prismatic elements. Investigators (39, 40) have shown that, if the
interpolation functions for the displacements of these arbitrary elements
are identical to the transformation functions used for the mapping of
the rectangular or right prismatic element, then the rigid body and
constamt strain modes and the interelement compatibility conditions are
all satisfied. Such type of element representation is named isoparametric
transformation.
The interpolation functions for the element displacements may
include displacements at internal nodes also. The displacement compo
nents corresponding to such internal nodes can be eliminated from the
final degrees of freedom of the element by the so-called static conden
sation (23, 41, 42).
16
For the solution of the plate bending problem, the continuity of
the lateral displacement as well as the normal slopes is required at
the interelement boundaries. In this case, the construction of the
interpolation functions is no longer a simple task. Although escperi-
ences (35, 43) have shown that converging solutions may be obtained
using noncompatible elements, many compatible elements have been cons
tructed either by dividing the element into subregions each using
different interpolation functions (23, 35, 41, 44) or by using deri
vatives of order higher than the first for the lateral displacement
at the ncxies (39, 45) .
Shell structures can be modeled by the finite element methcxi by
the superposition of membrane and flexural behavior. Examples of this
type of application are described in the literature by Clough and
Johnson (46). However, shell analysis using curved shell elements
presents some difficulties, except in the case of shells of revolution.
Shells of revolution under axisymmetric and asymmetric loadings have
been amalyzed successfully (47, 48, 49, 50) using axisymmetric elements
which are either conical or meridionally curved frusta. Interelement
cxxnpatibility is automatically achieved by making the displacements
along the ncxial lines compatible.
The difficulties faced in the development of curved shell ele
ments are discussed in detail by Gallagher in reference 30. For a
curved element, the membrane and flexural behavior cannot be decoupled
as for a flat element, which demands an equality of the order of poly
nomial representation for each displacement. Construction of inter
polation functions to fulfill the three requirements already cited
17
increases the degrees of freedom of the elements considerably. Studies
have been made on the effect of disregarding one of the requirements
such as rigid body mcxies (51) or interelement compatibility (52). In
all cases excellent results are obtained when the element size is
reduced.
As part of this investigation, a detailed study was made of
recently developed finite elements for analyzing plate and shell prob
lems. The purpose of the study was to select a suitable element for the
analysis of pressure vessel problems. A brief review of the properties
of these elements is presented in chapter III.
2.3 Applied Load Vector
The applied loading on the structural elements can be any loading
condition that can be approximated by concentrated loads at ncjdal points.
Actual concentrated loads at nodal points can be easily applied. Their
components in the direction of the generalized ccx^rdinates form the
corresponding force vector. However, for other types of loads such as
concentrated loads within the elements and distributed loads such as body
forces or pressure loading, it is necessary to determine the equivalent
concentrated loads at the ncxies in the direction of the generalized
coordinates. These equivalent nodal forces can be derived directly
frcxn the variational principles employed for the derivation of the
stiffness matrices of the individual structural elements.
Load Vector due to Body Forces
The vector of the components of the body forces in the direction
of the generalized coordinates for the element is denoted by {p}. The
18
potential energy due to the body forces undergoing displacements
corresponding to the displacement field { } defined by equation (2.9)
has to be included in the potential energy function 4 iMfore applying
the priciple of minimum potential energy. The modified function * is
given by equation (2.11).
• - J {^f J [Bl' lDj [Bl dvol {u} - J {u}' [N]' {p} dvol Vol vol
- {F}'^{u} . (2.11)
Applying the principle of minimum potential energy by equating ^ ^ 3{u}
to 0 we get
[k] {u} - {F} - {F} = 0 (2.12) P
where t {¥} = J [N]'^{p}dvol . (2.13)
P vol {F} gives the load vector corresponding to the body forces,
P
Load Vector due to Surface Forces
The vector of the ccxnponents of the surface tractions in the
direction of the generalized coordinates for the element is represented
by (p). Then, in a manner similar to the derivation of the load vector
due to the bcxiy forces, the following equation for the load vector due
to the surface forces can be derived.
{F} = J [N]' {p} dA (2.14) A
where the integration is carried out over the surface area over which
the forces act.
2.4 Initial Strain Problems
Initial strain problems such as thermal elasticity problems,
19
elastic-plastic problems, and creep problems can be analyzed by the
finite element method by including in the variational treatments,
respectively, the thermal strains, the plastic strains, and the creep
strains as initial strains (41). For example, by expressing the elastic
e strain e , as the difference between the total strain e and the initial
strain e°, the expression for the total potential energy function
bec:omes
• = 1 J {e®} [D]{e^} dvol - {F}'^{u} vol
= T ( { G - G°} [D] { e - e° } dvol - {F}'^{U} vol
= - \ UflD] {e} dvol - - f uflD] {e°} dvol 2 ^ 1 2 ^«J,i
- ^ r^e'^l'^ED] (e) dvol + - J {e°}'^[Dl {e°} dvol
- {F}' {u} . (2.15)
Substituting for {e} from equation (2.3) as {e} = [B] {u} and
applying the principle of minimum potential energy, we get
J [B]' [D] [B] dvol {u} - I J [B] [D] {e°} dvol vol vol
- - J {e°}' [Dl [B] dvol - {F} = 0 (2.16) 2 vol
Since [D] is symmetric, ecjuation (2.16) can be rewritten as
[k] {u} - {F} . - {F} = 0 (2.17)
where (F) Q = J [B]' [Dl {e°} dvol . (2.18) e vol
20
Thus, in the finite element formulation, the terms involving the
initial strains can be reduced to a column matrix of equivalent ncxial
forces. This approach has been used in the finite element auialysis of
thermal stress problems (50, 53).
2.5 Analysis of Structures for Dynamic Loads
When dynamic loading is applied to an elastic body (or structure),
the elastic displacements are functions of not only the structural
characteristics, but of the time as well. For the determination of
stresses aund displacements under dynamic loading conditions the inertia
properties, in addition to the structural stiffnesses, must be included
to describe the dynamic chauracteristics of the structure. The finite
element methcxi for static amalysis of structures cam be extended to
dynamic amalysis with some modifications. The derivation of the
ecjuations of dynamic equilibrium and their solution are presented in
detail in Chapter VI.
2.6 Assembly of the Stiffness Matrix of the Continuum
The stiffness matrix of the complete structure is assembled most
c onveniently by the direct stiffness methcxi. The two essential steps
in this procedure are the coordinate tramsformations and the subsequent
superposition of the element stiffnesses. In general, the element
stiffness matrices are evaluated using a local ccx)rdinate system differ
ent frcxn the coordinate system of the entire continuum (the latter
being referred to as the global coordinate system). Hence the stiffness
matrix of the element is to be transformed by appropriate transforma
tions to the global ccxjrdinate system before assembling. The assembly
21
is accomplished by the superposition of individual terms in the element
stiffness matrix according to the nodal point numbers of the element.
The details of the transformations required for the quadrilateral shell
element used in this investigation are presented in Chapter IV.
2.7 Boundary Conditions
The assembled stiffness matrix [K] relates the equivalent nodal
forces {F} acting on the structure to the nodal displacements {u}.
[K] {u} = {F} (2.19)
Equation (2.19) must be modified to account for the actual displace
ment boundary conditions. Since the stiffness matrix [K] is symmetric
only one half of it is usually stored in the computer. In such a case
the following techniques can be used to effect the necessairy modifica
tions for incxjrporating the boundary conditions. If the specified
boundary conditions have zero values, they can be easily accounted for
by striking out the rows and columns of the degrees of freedom associ
ated with the boundary constraints and by replacing the corresponding
diagonal elements with unit value. The force vector should then be
mcxiified by making the components corresponding to the specified dis
placements equal to zero. If the specified displacements are nonzero,
[K] must be postmultiplied by a vector consisting of the specified
values with all other values equal to zero, and the resulting vector
must be subtracted frcxn the force vector {F}. Then the procedure for
the case where the specified boundary conditions are zero should be
applied to the stiffness matrix [K]. The force vector should then be
modified by making the ccxnponents corresponding to the specified
displacements equal to the specified values of the displacements.
22
2.8 Solution of the Equilibrium Equations
The stiffness matrix [K], in which the displacement boundary
cx>nditions have been accounted for, may be characterized in general as
1. Symmetric
2. Banded ( if properly arranged ), amd
3. Positive definite.
Various methcxis and. computer algorithms are availadsle for the
solution of the simultaneous equations represented by equation (2.19)
(54, 55, 56). In the present investigation Gaussian elimination and
bac:ksubstitution are used for the solution of the equations. The three
properties mentioned a±>ove are very significant in connection with the
solution of the large system of equations involved in the finite element
analysis. Symmetry permits a reduction of approximately one half in the
nxmtber of calculations amd also in the amount of storage required in
the computer. The bamded property permits consideration of only the
cxjefficients contained within the bamdwidth for elimination at amy
stage. The positive definiteness ensures that the solution may be
obtained without pivoting (57).
2.9 Development of Computer Program for the Pressure Vessel Analysis
The computer program for the amalysis of a structure by the
finite element method may, in general, be divided into the following
steps:
1. Computation of element stiffnesses
2. Assembly of the stiffness matrix for the complete structure
3. Incorporation of the boundary conditions
23
4. Solution of the resulting equations for the displacements
5. Computation of element stresses.
A computer program was developed specifically for the pressure
vessel analysis. One of the features of the program is that disconti
nuities at the junctions - such as between a manhole amd the main bcxiy
of the pressure vessel - which require the definition of two systems
of surface c(x>rdinates for the same point, can be accomodated within
the program. The final results for moments amd forces at the junction
aure obtained with respect to both coordinate systems, thus getting the
nature of the stress distribution in both parts of the structure at
the joint. Another feature is that auxiliary storage units aure used,
thus permitting greater subdivisions of the structure even with a
con^uter of limited core storage. The stiffness matrix, reduced by
Gauss elimination, is stored in the auxiliary storage for the back-
substitution, permitting analysis for various loading conditions
without the necessity of solving the complete set of equations each
time. The program permits anaU.ysis for dynamic loadings using Newmark's
3 parameter method (82), the details of which are presented in
(3iapter VI.
CHAPTER III
REVIEW OF PLATE AND SHELL ELEMENTS
A brief review of the finite elements developed during recent
yeaurs is presented here. Tlie purpose of this study is to select a
suitadDle element for the analysis of pressure vessels with cutouts and
branches. Because of the complex geometry of the structure to be ana
lyzed, aucisymmetrical and rectangular elements are not suitadDle for the
analysis. Hence, only triangular and quadrilateral elements were
studied, and their properties are discussed in the following paragraphs,
The various elements are numbered in sequence and are referred to as
element 1, element 2, etc.
3.1 Plane Triangular Elements
Element 1 Bazeley, Cheung, Irons, and Zienkiewicz (35) discuss plane
triangular elements based on nonconforming as well as conforming dis
placement functions. The use of a polynomial in cartesiam coordinates
X and y to define the transverse deformation of a triangular element
with 9 degrees of freedom involves arbitrary elimination of certain
terms of the complete cubic polynomial which contains 10 terms. In
order to avoid this difficulty, the use of so-called "area coordinates"
is made to represent the transverse displacement w as a polynomial
function of degree 3. Linear polynomial equations are used to represent
the membrane displacements u and v, resulting in a constant strain
triangle for the membrane action. The final degree of freedom per node
is 5. Results of studies on plates with conforming as well as noncon
forming types of displacement functions are reported. It was concluded
24
25
that a simple nonconforming type function is capable of giving greater
accuracy provided such a function sati-sfies the so-called "constant
strain" criterion. It was also concluded that solution to all plate
and shell problems can be achieved with reasonable accuracy using non-
cx}nforming elements.
Element 2. Clough and Tocher (43) have developed a triangular element
which provides for full compatibility along the edges of adjacent ele
ments. The triangular element is divided into three triangular subele-
ments. Independent polynomial displacement expressions are assumed for
each subelement. An incx mplete cubic polynomial is used for the trans-
2 verse displacement w. The term x y is excluded, so that the normal
slope may vary only linearly along the exterior boundary, which ensures
slope compatibility. The displacements (in bending) for the cx>mplete
element involve a total of 27 generalized coordinates. Eighteen of
these are eliminated by the conditions to satisfy internal compatibility
recjuirements between adjacent subelements, resulting in 9 degrees of
freedcxn for the complete plate element in bending. Combining with the
membrane action the total degree of freedom is 5 per ncxie. Since ele
ments 3 and 11 are improvements on this basic element, the accuracy of
this element does not warrant discussion.
Element 3. An improvement on element 2 is reported by Clough and
Felippa (58), in which additional nodes are introduced at the midpoints
of the sides of the triangle resulting in 12 degrees of freedom (in
bending) for the element. The formulation is similar to that of element
2; the only difference is that area coordinates are used for the dis
placement functions of the three subelements. Even though this element
26
anploys an optimum compatible cubic displacement field, and therefore,
will yield the best possible results for a given triangular element
mesh involving compatible cubic displacements, it's midpoint nodes are
a somewhat undesirable feature.
Element 4. Melosh (59) uses a "pyramid" function to describe the dis
placements u, V, and w. The function permits linear variation of the
displacements over the area of the element, along any edge and through
the thicdcness. The element structure is assumed to be made up of two
independent elastic responses, one, a direct response involving strech-
ing amd shearing of the plane and the other involving shearing only.
The stiffness matrix is composed of the sum of the stiffnesses associ
ated with these responses. Successful application of this element to
pure bending and pure shearing cases is reported.
Element 5. Senol Utku (60) describes the calculation of stresses in
lineair thin shells of aeolotropic material using the deflections
obtained by the finite element methcxi. Two methcxis for defining the
transverse displacement field as a complete cubic polynomial are presen
ted. For the evaluation of the 10 constants in the complete cubic poly-
ncxnial, in addition to the 9 geometric boundary conditions, the minimum
strain energy condition for the element is used. The method is somewhat
tedious and the formulations of the stiffness matrices are not presented
in the reference.
Element 6. Argyris et al (39) have formulated a family of fully com
patible triamgular elements which they call TUBA set. The TUBA family
is based on complete polynomial functions for the deflection w of the
order greater than or ecjual to 5. The selection of nodal freedoms at
27
the vertices include not only w and it's first derivatives, but also
all the second derivatives. Additional ncxial points are placed on the
boundary to ensure complete ccxnpatibility for w and for the slope
normal to the edges. Interior ncxies are also selected in some cases to
define uniquely the constants in the polynomial functions. Examples
are given which demonstrate that the TUBA set is highly efficient in
the analysis of plates of arbitrary shapes under static and dynamic
loading.
3.2 Curved Triangular Elements
Element 7. Strickland amd Loden (61) have derived the stiffness matrix
for a doubly curved triangular shell element. The formulation is based
on the shallow shell theory expounded by Novoshilov (62). The surface
of the element is approximated by a second degree polynomial. The
displacement comp>onents tangential to the surface of the element are
assxamed to vary linearly; the normal displacement is assumed to vary
cubically in the manner expounded by Bazeley, Cheung, Irons, amd
Zienkiewicz (35) for plate bending. Area coordinates are used in the
formulation. The final degree of freedom is 5 per node.
Element 8. The complete cjuintic polyncamial is used in the representa
tion of all three displacement fields in the SHEBA-6 element of Argyris
et al (39). Area coordinates are used. The element has a total of 63
degrees of freedom, 18 at each vertex (the function and all of it's
first and second derivatives for each of the three functions) and 3 at
the midpoint of each side (the angular displacement in the direction
of the normal to the side for each function). No numerical results are
presented in the a±>ove reference.
28
Element 9. Dhatt (63, 64) has presented three types of compatible
elements referred to as SI, KCM, and KLM. For the element SI, the bend
ing stiffness matrix is obtained by including the shear energy and
equating the shear deformations at the corner nodes of the element to
zero. For the elements KCM and KLM, the bending stiffness matrix is
obtained by ignoring the shear energy and by constraining the shear
deformation to zero at certain points along the sides of the element.
These constraints on the shear deformations reduce the shear deforma
tion theory of shells to a "discrete" Kirchoff theory. The rotations
9x amd 6y are represented by quadratic polynomials over the surface
of the element. The transverse displacement w is defined either by a
cubic polynomial or by a linear polynomial. The membrane displacements
aure represented by cubic polynomials over the element in order to
include all rigid body motions. The true geometry is approximated by a
shallow cjuadratic surface. The element is divided into three triamgular
subelements with nodes at the midpoints of the interior edges also. The
additional degrees of freedom inside the element are eliminated by
static condensation. The final degree of freedom for all the three
elements is 9 per node and in total there are 27 degrees of freedom
per element. Numerical results are also presented which show the high
efficiency amd precision of the KCM and KLM elements.
Element 10. Recently Lindberg and Olson (65, 66) have developed a
highly successful refined triamgular shallow shell element. The element
uses as generalized displacements, the tangential displacements amd
their first derivatives plus the normal displacement and its first
29 \ I
and second derivatives at each vertex madcing a total degree of freedom
of 12. The transverse displacement function for the element contains
a cxxnplete quartic polynomial plus some higher degree terms and allows
a cubic variation of the normal slope along each edge. The tangential
displacement functions are complete cubic polynomials. Results show
that this element is extremely accurate in predicting stresses as well
as displacements.
3.3 Flame Quadrilateral Elements
Element 11. Clough and Felippa (58) describe the formulation of a
fully ccfflipatible general cjuadrilateral plate bending element. The ele
ment is assembled from four partially constrained lineau: curvature
compatible triangles arranged so that no midside ncxies occur on the
external edges of the quadrilateral; thus, the resulting element has
only 12 degrees of freedom (in bending). This is am improvement to the
plane triangulau: element developed by the same authors (element 3).
Extensive use of this element for thin shell analysis is reported by
Clough amd Johnson (46).
Element 12. Philip Johnson (23) used the fully compatible triangular
element after Clough and Tocher (43) (element 2) for the bending action
amd a linear strain triangle for the membrane action in the formulation
of a quadrilateral element. The quadrilateral is made up of four trian
gular subelements. Each of the four triangles is assigned independent
membrane and bending displacement functions. The stiffness matrix for
the cjuadrilateral element is obtained by superposition of the stiffnesses
of the subelements. The assembled element has 9 ncxies and 33 degrees
30
of freedom. Five nodes are in the interior to the element and the
degrees of freedcxn at these ncxies are eliminated by static condensa
tion. The resulting condensed cjuadrilateral has 20 degrees of freedom,
five at each exterior ncxie. A modification of this plane element for
the analysis of doubly curved shells is accomplished by considering
this element as a substructure made up of four individual triangles.
The nonplanar formulation of the cjuadrilateral element results in
additional translational degrees of freedom at the interior midside
nodes, cxxnpared to the plane cjuadrilateral. These are eliminated by
cxjnstraining the normal component of the displacement at each interior
midside ncxie to be the average of the displacement of the central node
amd the corresponding exterior node. Several examples of the applica
tion of this element are presented, which show that excellent approxi
mations to the exact solutions are obtained for domes, circular cylin
ders, and folded plates.
3.4 Curved Quadrilateral Elements
Element 13. Greene, Jones, McLay, and Strome (67) give the applications
of a doubly curved quadrilateral shell element in the dynamic analysis
of shells. The derivations of the stiffness matrix are not presented
in the above reference.
Element 14. Key amd Beisinger (68, 69) have developed a doubly curved
cjuadrilateral shell element called 'KB6' based on a discrete Kirchoff
hypothesis. Independent displacement assumptions are made for the
membrane displacements u amd v and the transverse displacement w and
also the rotations 0 and 6.,. Hermitian interpolation functions are X y
31
used for the displacement functions and the integration for the strain
energy of the element is performed using a five point Caussiam cjuadra-
ture. Transverse shear deformation shell theory is modified using
constraints on the fiber rotations, thus modifying it to a Kirchoff
shell theory which is called the 'discrete' Kirchoff hypothesis. The
sheau: deformations are made equal to zero at the nodes and the shear
strain energy is neglected. The joint degree of freedom is 9. Applica
tion of this element for bending analysis of plates shows very gcxxi
results. The problem of am infinite cylinder under an axial load amd
with a circular cutout is also analyzed amd results compared with those
obtained by other authors.
In addition to the adxjve plate amd shell elements, a number of
el^nents have been developed (30, 51, 52, 70) which are rectangulau: -
both plame and cylindrical, or are suitable for axisymmetric problems.
Since these are not suitable for the analysis of pressure vessels with
cutouts amd bramches, they are excluded frcxn the present study. A survey
of these elements is reported by Gallagher (30).
3.5 Selection of Finite Element for the Pressure Vessel Analysis
Frcxn the study of the plate and shell elements, it is seen that
a number of elements are available for the analysis of pressure vessels
using the finite element method. The first choice for this study is a
curved triamgular element which can discretize the curved surface of
the pressure vessel more precisely than plane elements and can also
take care of the discontinuities at the junctions. Elements 8, 9, and
10 make possible excellent analyses since they take into account all
32
the recjuirements cited in section 2.2 in their formulation. However,
the application of these elements recjuires the use of a computer having
a large storage capacity since the degrees of freedom per node for
these elements aure 18, 9, and 12 respectively.
Another criterion in the selection of a suitable element is the
size of the half band width of the resulting stiffness matrix for the
structure being analyzed. The final stiffness matrix of the complete
structure has to be assembled from the stiffness matrices of it's
elements in the most efficient manner. The assembled matrix is gene
rally a banded symmetric matrix, the band width depending on what is
called the "maximum node difference" in any element. The maximum node
difference is made as small as possible by suitaJaly numbering the nodes,
usually along the direction of the least dimension of the structure.
In the case of a pressure vessel with bramches, a reduction to any
great extent of the maucimum ncxie difference is not possible since the
dimensions of the structure will be almost the same in all directions.
Hence, it is necessary to make a choice between the following two
alternatives for the analysis. One is to make use of a larger number
elements having lesser joint degree of freedom and the other is to
employ a lesser number of higher order elements having greater joint
degree of freedom. From the study of the results of the analyses in the
various references it is found that the accuracy obtainable by decrea
sing the element size is better than the use of higher order elements
of bigger size. For this reason, elements 8, 9, and 10 as well as 6
and 14 were not considered for the selection. Element 7 has all the
33
desirable characteristics such as limited number of joint degrees of
freedom and the curved triangular shape. However, the results of analysis
done with this element reported in reference 61 show that it is only
slightly more accurate than a plane triangular element. Since better
results are reported by the use of element 12, which is the final choice
in this study, element 7 was also not selected for the analysis.
The formulative procedure for elements 4, 5, and 13 is not
availadDle in the references and hence it was not possible to consider
these. For the formulation of element 12, the concepts used for elements
2, 3, amd 11 are used. It may be remarked that element 12 is an improve
ment on elements 2, 3, and 11 and hence these are also eliminated from
the choice. The final choice is between elements 1 and 12. Element 12
is definitely superior to element 1 since it allows linear variation
of the straiin components as compared to the constant strain components
in element 1. Further, the concept of considering the element as a
substructure made up of four triangular subelements permits better
representation of a cxirved surface with this element compared to the
flat triamgular element. The examples presented in the reference (23)
demonstrate the versatality and accuracy of this element for the analy
sis of shells and folded plates. Considering the above, it was decided
to use the quadrilateral element developed by Johnson (element 12) for
the pressure vessel analysis.
CHAPTER IV
DERIVATION OF ELEMENT STIFFNESSES
The details of the derivation of the stiffness matrices for the
finite elements used in the present research for the analysis of pres
sure vessel junction problems are presented in this chapter. The
elements used are a cjuadrilateral shell element and a three dimensional
octahedral element.
The stiffness matrix of the quadrilateral shell element is
derived by a procedure similau: to that developed by Johnson (23) .
In this derivation, the membrane and bending actions for the shell
element are decoupled. This simplification of the shell action is
justified in the finite element analysis due to the small size of am
individual element and also due to the small deformations in amy
practical problem. The stiffness matrices for the membrane and bending
actions are derived separately and superposed to obtain the shell
action. Because of this procedure, it is possible to use simple plate
theory for the bending action, instead of rigorous shell theories in
which all actions are combined. The Kirchoff theory for small deflec
tions in thin plates is used for the bending stiffness derivation. The
assumptions in this theory are presented in section 4.3. The governing
ecjuations aure also derived in the same section.
In the Kirchoff theory for small deflections in thin plates, the
shear distortion due to transverse shear stresses in neglected. More
refined theories which still retain the assumption of small deflections
(i.e. coupled membrane action is ignored ) and which include the effect
34
35
of transverse shear stress have been developed using Reissner's
variational principle (71). Shell elements which include the energy of
shear distortion have been developed by Key and Beisinger (68) and
Dhatt (63). The properties of these elements have already been dis
cussed in Chapter III. The increase in the accuracy of the results is
not ccDmparable to the increase in the computational effort. Furthermore,
due to the assumption of small deflections, the energy of shear distor
tion is negligible compared to the bending energy. The effect of shear
distortion is neglected in the present development.
For the derivation of the stiffness matrix, the quadrilateral
element is divided into four triangles. The stiffness matrices for the
triangles are derived separately and assembled for the quadrilateral
element. The derivation of the stiffness matrix for the membrane
action of a triangular element is described in section 4.2, and for
the bending action in section 4.4.
The division into triangles results in internal nodes for the
quadrilateral. The corresponding displacement coordinates are elimi
nated by static condensation. The procedure is explained in section 4.5.
In the following derivations, the position of any point within
a triangulau: area is specified in terms of its "area coordinates".
The introduction of area coordinates results in simplified expressions
for the displacements, strains, etc. In the following section, the
definition of the area coordinates and certain useful relations invol
ving them are presented.
For the derivation of the stiffness matrix, the octahedral
36
element is divided into five tetrahedral subelements. The stiffness
matrix for each tetrahedral subelement is derived independently and
assembled to obtain the stiffness matrix of the octahedral element.
The derivation of the stiffness matrices for the tetrahedral and
octahedral elements is explained in sections 4.8 and 4.9 respectively.
Similar to the use of "area cxjordinates" for triamgular elements,
"volume coordinates" are used for the tetrahedral element. The defini
tions and certain useful relations involving volume coordinates are
presented in section 4.7.
4.1 Area Ccxjrdinates (21, 61)
To define the area coordinates for a point inside a triamgle
suppose that the triangle is referred to cartesian coordinates with
vertices at the points (x-j,y2) , (X2,y2) / ^^3*^3^ ^^ shown in
Figure 4.1. An arbitrary point (x,y) within the triamgle divides it
into three smaller triangles which can be numbered according to the
vertex each is opposite. If the areas of these triangles are A^, A^,
and A3, amd if the area of the total triamgle is A, the area coordi
nates of the point are
L^ = A^/ A i = 1, 2, 3. (4.1)
The three area coordinates are not completely independent.
Because of the relation among the areas
A-, + A2 + A3 = A
the area coordinates are connected by the relation
Lj + L2 + L3 = 1 . (4.2)
If the expressions for A^, in terms of the cartesian coordinates
37
O^
3 (X3,y3)
1 (xi,yi)
2 (x2/y2)
-•- X
y»v
FIGURE 4.1 AREA COORDINATES
06.
lFy3/V3
X4,U4
Fxi,ui
•>-x,u
FIGURE 4.2 LINEAR STRAIN TRIANGLE WITH 12 DEGREES OF FREEDOM
SIGN CONVENTION FOR FORCES AND DISPLACEMENTS
38
are substituted into equation 4.1, the following expressions for the
area coordinates can be obtained.
For instance
\ = I I X ( y^ - y^ ) + x^ ( y^ - y ) + x^ ( y - y^ ) 1
" I ^ ^ " 2 3 " "3 2 ^ " ' 2 " 3 ^ " ^ ""3 " ""2
so that
^1 ^ ^ ^ ^ ' 2^ ( c^ H- b^ X + a^ y )
where
b^ = Yj - Xj (4.3)
Similar expressions for L and L can be obtained by cyclic interchange
of suffixes 1, 2, 3.
It may be noted that the area coordinates have been defined so
that L. has the value 1 at the vertex (x.,y.) and zero at each of the 1 1 1
other vertices.
Expressions for the derivatives of the area coordinates with
respect to the cartesian coordinates follow from equation 4.3 and it's
analogs.
L. = b. / 2A ; L. = a / 2A ; i = 1, 2, 3. (4.4) IX 1 ly i
The suffixes x and y denote the partial derivative of L with respect 1
to X and y respectively.
In deriving the stiffness and other matrices, the products of the
area coordinates have to be integrated over the area of the triangle.
39
It is possible to derive the following formula for the integration.
/ L J ' L ^ L P dxdy 2 A m n I p I
( m + n + p + 2 ) (4.5)
4.2 Derivation of the Constrained Linear Strain Triangle Element
Stiffness
The displacements for membrane action are assumed to vaury accor
ding to a second degree polyncxnial in x and y. In terms of area coor
dinates this polynomial can be expressed ais
u = c L2 + C L2 + C L2 + C L L + C L L + C L L 1 1 2 2 3 3 4 1 2 5 2 3 6 3 1
(4.6) V = C L 2 + C L 2 + C L 2 + C L L + C L L + C L L
7 1 8 2 9 3 10 1 2 11 2 3 12 3 1
u amd V are the displacements of any point (x,y) in the x and y direc
tions. The constamts c to c are evaluated in terms of the ncxial 1 12
displacements, so that the above polynomials satisfy the boundary
conditions at the six nodes, (see Figure 4.2). The resulting displace
ment functions are
u = {N}' {ui} , and v = {N}'^{V^} (4.7)
where N
N
{N} = < - : .
N.
N.
6
<
2 L^- L^
2^2-^2
2 1-3-^3
"^1^2
4L2L3
4L3L^
(4.8)
J
40
{u^}
r -N
u„
u,
u,
u.
and (v^} (4.9)
The strains e , £ , and y are given by (72) X y xy
y
xyj
3 u 3 x
3 V 3 y
3 u + 3 V
>
3 y 3 X
N, N^ Ix 2x
0 0
N. N-ly 2y
.N- 0 6x 0
0 N- N_ ly 2y
,N N, N-6y Ix 2x
0
,N 6y
.N 6x
(4.10)
where
amd
N. IX
N.
_^N. 3x ^
-IN. 3y 1
i = 1, 2, , 6 ;
or in other words equation (4.10) is
{e} [B] {u} (4.11)
where
{u}
{u,}
{V.}
41
The stresses 0 , 0 , and T^^ are given by (72) A y xy
<
( 1 - xi )
xy 1 - V
i
r "^
X
y
Y xy ^ J
>(4.12)
where E is the mcxiulus of elasticity, and v is the Poisson's ratio
of the material.
Equation (4.12) may be written as
ia) [D] {e} (4.13)
Substituting for {e} from equation (4.11)
{0} [D] [B] {u} (4.14)
The straiin energy of the element is given by
U
1 2
J {e}' {a} dvol vol
J {u}' [B]' [D] [B] {u} dvol vol
- {u}* [ t / [B]' [D] [B] dA ] {u} 2 A
(4.15)
where t is the thickness of the element, and the integration is carried
over the area of the triangle.
The stiffness matrix relating the nodal forces and displacements
is obtained by the application of the principle of minimum potential
energy (ecjuation 2.8) as
[ k ] = 12x12
• . / [ B ] M D ] [ B ] dA A 12x3 3x3 3x12
(4.16)
In the above integration, the functions to be integrated are L. ,
42
L , and L^Lj , i « 1, 2, 3 and j » 1, 2, 3 . These integrations are
performed by the use of equation (4.5).
Finally, the stiffness matrix for the constrained linear strain
triangle is obtained by modifying the above matrix by adding one-half
of column 4 to columns 1 and 2 and by adding one-half of column 10 to
cxJlumns 7 and 8. This is the constraint on node 4, namely u, and v. 4 4
are assumed to be equal to the average of the corresponding displace
ments of the ncxies 1 and 2. The resulting matrix will be of 10x10 size.
4.3 Thin Plate Theory (41, 71, 73)
The triangular plate element is referred to a right handed local
cartesian coordinate system x,y,z , the x and y axes being located on
the middle surface of the element and the z axis positive upwaurds (see
Figure 4.4).
The Kirchoff theory for small lateral deflections of thin plates
is based on the following assumptions.
1. Membrane or in-plane straining effects are uncoupled from the
bending effects.
2. The deformed state of the plate is determined entirely by the
deformed configuration of the middle surface.
3. Each layer in the thickness of the plate is in a state of
plane stress (or strain). The transverse normal stress (or strain) is
neglected.
4. Transverse shear stresses may exist, but the associated shear
distortion is neglected.
These assumptions lead to the following equations for the dis
placements .
43
w (x,y,:) •«
u
w (x,y,0)
3 w 3 X 3 w 3 y
- z
- z
w (x,y)
(4.17)
The strain field associated with the above displacements is
3 u 3 X
3 V 3 y
= - z
= - z
^2 3 w
3 x
(4.18)
3 y
xy
3 u 3 y
3 V 3 X
= - 2 z 3 w 3x 3y
Usually, plane stress condition is assumed in the derivations
for plate analysis. For an isotropic plate under plane stress condition
> =
xy
V
0
V
1
0
0
0
1 - V
2
<
'- - \ e
X
e y
Y ' xy
(4.19)
The resulting moments may be obtained from the following ecjuations
(see Figure 4.4 for sign convention).
^h/2
M = ~ I ^ z dz -h/2
M = - I a y J_u/-5
z dz (4.20)
-h/2
and M xy /
h/2
-h/2 T z dz xy
Ecjuations (4.20) lead to the well known moment curvature
44
relationships
M
M E h'
M xy
12( 1 - V*) 0
1 - V
<
-2
(4.21)
4.4 Plate Bending Stiffness Matrix for the Triangular Element
The evaluation of the plate bending stiffness matrix for the
triangular element is based on the works of Felippa (41) and Johnson (23)
The triangular element is divided into three regions as shown in
Figure 4.5, with the point O having area coordinates (1/3, 1/3, 1/3).
Independent displacement functions for the lateral displacement w are
assigned to the three regions in terms of the ncxial point displacements
amd rotations. These displacement functions are in terms of area coor
dinates amd they account for internal compatibility conditions between
the three regions. The integration for the strain energy is done sepa
rately for each region and is added to obtain the final stiffness
matrix for the element.
The displacement function for region 3 is given by equation (4.22)
where the interpolating functions N , N , , Ng are those
presented in TadDle 4.1, and {w} is the vector of the nodal point
displacements and rotations.
w {N}" {w} (4.22)
where
and
{N}^ = ^ 1 ' 2 ' 3 ' ' ""S ' \ ' ' ' ^
{w}T = ( w, , 8 , 6 , w^ , e , 6 , w^ , e , e ) ^1 yi ^2 y2 X3 73
45
The strains due to the bending action are given by
•^
xy
3 x'
32w
32w "2 3x 3y
(4.23)
where e represents the curvature instead of longitudinal strain.
Substituting for w from equation (4.22) we get
> =
xy
where
amd
N N N Ixx 2xx 3xx
N, N^ N^ lyy 2yy 3yy
•2N -2N -2N Ixy 2xy 3xy
N. ixx
N. lyy
N ixx
S N. 1
3 x 2
32N. 1
3 y^
1
3x ay
,N N 8xx 9xx
.N N 8yy 9yy
-2N -2N 8xy 9xy
{w} (4.24)
i =* 1, 2, 3, ...., 9
In other words equation (4.24) may be written as
{e} = [B] {w} (4.25)
The expressions for N , evaluated by differentiating the N ixx i
in Table 4.1, are given in Table 4.3. The procedure for obtaining
N au*d N from these expressions is also explained in Table 4.3. lyy ixy
The moments M , M , amd M are given by ecjuation (4.21) x y xy
46
v^ich may be written as
{M} = [Dl {e} . (4.26)
Substituting for {c} from equation (4.25)
{M} = (Dl [B] {w} . (4,27)
The strain energy U due to the bending action is given by
k! T, " = T J (c}'{M} dA 2 •'A
J J {w}' [Bl' [Dl [B] {w} dA A
[Bl' EDl [B] dA {w} (4.28)
The stiffness matrix relating the nodal point moments amd the
ncxial displacements and rotations is obtained by the application of
the principle of minimum potential energy (Ecjuation 2.8) as
[ k ] = / [ B ] ' ^ [ D 1 [ B ] dA (4.29)
9x9 A 9x3 3x3 3x9
In the above formulation, since independent displacement functions
are assumed for the three regions, the integration is to be performed
separately for each region and the stiffness matrix may be obtained as
[k] = J [B,l' [D] [B ] dA + J [B2]' [D] [B2] dA •'AI A2
I [B3]' [D] [B ] dA (4.30)
^^3
where the subscripts refer to the region in the triangular element.
In the above integrations, the functions to be integrated are L^
2 L. , amd L. L. , 1 = 1 , 2, 3 , j = l , 2, 3. These integrations are
performed by the use of equation (4.5).
The expressions given in Tables 4.1 and 4.3 are for region 3 of
+ ''A-
47
the triangular element. The displacement function and the [B] matrix
for regions 1 and 2 are obtained by cyclic permutation of the quantities
a and b as follows.
For region 3 *l " *i b » b
*2 " *2 ^2 ' ^2
*3 - *3 ^3 = ^3
tor region 1 *l " *•> b = b
•2 - »3 "2 " "3
•3 " *! t>3 - "1
For region 2 aj = a, ''1 " ''3
•2 - h "2 = "1
aj = a^ bj = b^
where the a's amd b's to the right side of these equations are the
values shown in Figure 4.5, while the a's and b's on the left side of
the ecjuations refer to those in Tables 4.1 through 4.4.
Similarly, the ncxial point displacements and rotations should
also be rotated when considering regions 1 amd 2 as follows,
{w}"' for region 3 = ( "i . 6 . e^^, w^ , 6 , 6 , W3 , e^^, 6 ^ )
{wf for region 1 = ( «2 ' «X2' S2' "3 ' xj' S^' \ ' «xi' '
(w)"" for region 2 = ( "3 - 6 3 - 6 , w , e^^, 6 , w^ , 6 ' Sy^ > •
Hence, in the evaluation of the final stiffness matrix [k],
the rows and columns in the three matrices for the three separate
regions are to be suitaibly interchanged before adding.
48
\
J xy T
\
xy
FIGURE 4.3 SIGN CONVENTION FOR INPLANE STRESS COMPONENTS
Mxy
Mxy
Mxy
9-
FIGURE 4.4 SIGN CONVENTION FOR PLATE BENDING MOMENT COMPONENTS
49
y 32 +
z ^
-V
>2 Regicin 2 (1/3,1X3/1/3)
• • - X
FIGURE 4.5 TRIANGULAR PLATE ELEMENT - POSITIONS OF REGIONS 1, 2, 3
z (^
Fey3,ey3 F0y,0y
Fex,0x
F6x2,6x2
Z2/W2
Fzi,wx Fexi,exi
FIGURE 4.6 TRITU^GULAR PLATE ELEMENT - SIGN CONVENTION FOR FORCES AND DISPLACEMENTS
tc o fu
W Z o
c:> z M
a o
u Z
(N
0)
+ r"
n
OJ
CN
CM
<N
m
rg CM
in 0)
+ 0)
+
CN n •O
0)
+
T3
0)
+
CM
V£>
CM
0)
+
in •O -o
V£>
(1)
+
TJ
CM
00
+
-o cc
CM ro
n
CM r o
n
CM CO
n
CM r o
n
CM n
en
CM m
n
c>j n
n
CM <N CM CM CM CM CM
CM
cy» 01
+
-o c
CM ro
en
CM
CM
0)
^
b^ z g u a 04
cn M Q
^
B a
. T
HE
—
•H 1-3
CM
I
n -.
CM iH •-)
+
m ^
•H 14-1
+
ro tJ
.0
1
CM
PO ia *«*'
CM iH ^
+
n >A CM
(M
+
m >J CM
IT)
1 CM
•J m 10
'*«—'
CM r-i ^
+
CO •-3 m
«M
+
'-
CM •J CM
1
ro *-
CM CM ^ +
ro t-^
^ UH
+
H vA ro
ia 1
ro
iH
J3 "•-
CM CM ^
+
ro •J in
«4-l
+
rH l-q
ro Id
1
ro 1-3
r-i (d
^ i ^
CM CM ^
+
ro 1-
VO (M
+
ro •
r* (M
+
ro iJ
CX) <4-l
+
ro ^ <Ts
*U
+
« EH
C 0) > •H tr m Id
0) M (d
cr»
^
CM
II
IM
<U
TJ
50
•J
o 11
CM O
CM
ro O
II
ro
O
II
in u
II
in
vD o
II
VO
O 00
00
cr» o
II
a>
u a; u 0)
TABLE 4 . 2
COEFFICIENTS FOR THE INTERPOLATING FUNCTIONS IN TABLE 4 . 1
51
C i - 6 Q. c . - a^ - S^ a , + 4 R 6 2 3 3 3
^2 = Q3 ^ 3 - b ^ + 4 P 7 - °
•3 ' ^3 ^3 " ^ 1 " 4 R3 C3 - 0
C4 - 6 S3 c , - 0
^5 = ^2 - S3 ^3 " ^ ^
h = 3 ( ^2 " C3 )
d2 = ( 9 b^ + 3 Q2 b2 + 3 S3 b3 ) / 6 + 2 P^ - 2 P
^3 = ( 9 a^ + 3 Q2 a2 + 3 S3 a ) / 6 + 2 R - 2 R,
^4 = - 3 S3
dg = ( 3 b^ + 6 b3 - 3 Q^ b3 ) / 6 - 2 P
dg = ( 3 a^ + 6 a3 - 3 Q^ a^ ) / 6 - 2 R
d^ = 3 ( 1 + Q )
cig = ( 3 b ^ + 12 b ^ - 3 S^ b ^ ) / 6 + 2 P ^
d^ = ( 3 a , + 12 a^ - 3 S^ a ) / 6 + 2 R y 1 2 2 2
e , = - 3 Q.
52
TABLE 4 . 2 ( c o n t i n u e d )
®2 ** ( - 3 b j - 6 b3 + 3 S3 b3 ) / 6 - 2 P.
e j « ( - 3 a j - 6 a3 + 3 S3 a3 ) / 6 - 2 R,
e 4 - 3 ( Q^ - S3 )
eg = ( - 9 b2 - 3 Sj^ b^ - 3 Q3 b3 ) / 6 + 2 P^ - 2 P.
eg = ( - 9 a2 - 3 Sj a ^ - 3 Q3 a3 ) / 6 + 2 R^ - 2 R,
ey = 3 ( 1 + S^ )
e g = ( - 12 b^ - 3 b2 + 3 Q^ b^ ) / 6 + 2 P^
e g = ( - 12 a^ - 3 a2 + 3 Q^ a^ ) / 6 + 2 R^
f l = 2 Q3 - S2
f2 = ( 4 b2 + 5 b3 - Q2 b^ - 2 S^ b^ ) / 6 + ( 4 P^ - 2 P^ ) / 3
f3 = ( 4 a2 . 5 a3 - Q2 a^ - 2 S3 a3 ) / 6 + ( 4 R^ - 2 R^ ) / 3
f4 = 2 S3 - QjL
f^ = ( - 4 b^ - 5 b3 + S^ b^ + 2 Q3 b^ ) / 6 + ( 4 P3 - 2 P^ ) / 3
f = ( - 4 a - 5 a + S a + 2 Q a ) / 6 + ( 4 R - 2 R ) / 3 6 ^ 1 3 1 1 3 3 3 1
f , = i - s ^ - Q 2
fg = ( S^ b^ - Q2 b^ ) / 6 - ( 2 P^ . 2 P^ ) / 3
^9 = ^ ^1 ^ - ^2 ^2 ^ / ^ - ( 2 R + 2 R^ ) / 3
53
Table 4 .2 (continued)
where 2 2 2 T •« a + b
1 1 1
2 2 2 T • a + b
2 2 2
2 2 2 T - a + b
3 3 3 2
S = - ( a a + b b ) / T 1 1 3 1 3 1
2 S = - ( a a + b b ) / T
2 2 1 2 1 2 2
S » - ( a a + b b ) / T 3 3 2 3 2 3
Q = i
% "
% "
P - -1
P « -2
P - -3
R = A 1
R = A 2
R = A 3
- S 1
- S 2
- S 3
2 A a / T
1 1
2 A a / T
2 2
2 A a / T
3 3
2 b / T
1 1
2 b / T
2 2
2 b / T
3 3
A = ( a b - a b ) / 2 3 2 2 3
54
TABLE 4.3
DERIVATIVES OF INTERPOLATING FUNCTIONS
^xx = 4 ^ ^ il ^ i2 S ^ 11 12 " 13 S / ' ^
i = 1, 4, 7.
N = - TT2\. C . , L, + C . L + ( C + C + C ) L / 3 ] i x x 4 A^ i l 1 12 2 11 12 13 3
i = 2 , 3 , 5 , 6 , 8 , 9 .
The values of the constants C , C , and C ; 11 12 13
i = 1, 2, 3, , 9 are given in TadDle 4.4
In Table 4.4, the quantities b represent b . b . pq P q
The purpose of using this notation is for establishing N. amd N.
N. and N. may be obtained from the ecjuations for N. by lyy ixy •' ^ ixx
modifying the constants C , , C.^ , and C . as follows. ^ ^ il i2 i3
For N. replace b by a (i.e. a . a ) lyy ^ pq ^ pq p q
in the expressions for C. , C.^ , and C.^ ; 1 = 1, 2, 3, ,9
in Table 4.4 .
For N. r e p l a c e b b y ( a b + b a ) / 2 i n ixy ^ pq p q p q
the expressions for C.. , C.2 » and C.3 ; i = l, 2, 3, ,9
in Table 4.4
55
TABLE 4.4
VALUES OF THE CONSTANTS IN THE DERIVATIVES OF THE INTERPOLATING FUNCTIONS
^11 " 2 b23 c^ + 2 b33 d^ - 6 b^^
Ci2 - 2 bi3 c^ + 2 b33 e^ + 6 b^^
^13 - 2 bi2 1 + 4 b^3 d^ + 4 b23 e^ + 6 b33 f + 6 b^^
^21 - 2 b23 C2 + 2 b33 d2 + 4 b^2 ^3 - "^ ^3 b2
C22 - 2 b i 3 C2 + 2 b33 e2 + 2 b^^ b3
C23 » 2 b i 2 C2 + 4 b i 3 d2 + 4 b23 6 2 + 6 b33 f2 - 2 b^^ b2
C3I = 2 b23 C3 + 2 b33 d3 + 4 b^2 ^ - 4 b^3 a2
C32 - 2 b i 3 C3 + 2 b33 e3 + 2 b^^ a3
C33 . 2 b i 2 ^3 ^ 4 b^3 d3 + 4 b23 63 + 6 b33 f3 - 2 b^^ a^
i '
C41 - 2 b23 C4 + 2 b33 d + 6 b22
C42 ' 2 b]_3 C4 + 2 b33 e^ - 6 b22
C43 = 2 b i 2 C4 + 4 b^3 d^ + 4 b23 e^ + 6 b33 f^ + 6 b22
56
TABLE 4.4 (continued)
= 51 - 2 b23 O5 + 2 b33 dj. - 2 b^^ b^
=52 - 2 b,3 05 + 2 b33 e^ . 4 b^3 b^ - 4 b^^ b3
Ceo - 2 b,^ c^ + 4 b,, d^ + 4 b^ e^ + 6 b f + 2 b b 53 12 5 13 5 23 5 33 5 22 1
C = 2 b c + 2 b d - 2 b a ^61 23 6 33 6 22 3
^62 - 2 3 ^6 " ' 3 % " ^ 3 ^ ^ ' 2 ^
C » 2 b c + 4 b d + 4 b e + 6 b f + 2 b a 63 12 6 13 6 23 6 33 6 22 1
S » 2 b c + 2 b d 1 ^^23 7 33 7
C , = 2 b c + 2 b e 72 13 7 33 7
C = 2 b c + 4 b d + 4 b e + 6 b f 73 12 7 13 7 23 7 33 7
C^ = 2 b c + 2 b d 81 23 8 33 8
C ^ ^ = 2 b c + 2 b e 82 13 8 33 8
C = 2 b c + 4 b d + 4 b e + 6 b f 83 12 8 13 8 23 8 33 8
C « 2 b c + 2 b d 91 23 9 33 9
C^ = 2 b c + 2 b e 92 13 9 33 9
C = 2 b c + 4 b d + 4 b e + 6 b f 93 12 9 13 9 23 9 33 9
57
y
•^ ex-
FIGURE 4.7 PLANAR QUADRILATERAL WITH 33 DEGREES OF FREEDOM
FIGURE 4.8 GLOBAL AND LOCAL COORDINATE SYSTEMS
58
4.5 Assembly of the Stiffness Matrix for the Quadrilateral Element
In sections 4.2 and 4.4 the procedure for the derivation of the
membrane and bending stiffness matrices for a triangular element has
been described. The c[uadrilateral element is considered as a substruc
ture formed by four triamgular elements as in Figure 4.7. The stiffness
matrices for each triangular element are derived separately and the
direct stiffness method is applied to assemble the ccxnplete stiffness
matrix for the cjuadrilateral element. The assembly of the coefficients
in the final stiffness matrix is accomplished by adding the individual
terms in the stiffness matrices of the triangular elements according
to the nodal point numbers of the triangle.
The resulting matrix will be a 33 x 33 matrix corresponding to
the five degrees of freedom (u, v, w, 9 , 6 ) for the nodes 1, 2, 3, X y
4, and 5 amd two degrees of freedom (u, v) for the nodes 6, 7, 8, and
9. The degrees of freedom for the internal nodes 5, 6, 7, 8, amd 9 are
eliminated by static condensation (23, 41, 42) so that they do not
appear in the final stiffness matrix. The procedure for the elimination
is easily illustrated as follows.
Consider the general ecjuilibrium ecjuation for the element with
9 nodes amd 33 degrees of freedcam
[k] {u} = {F} (4.31)
where [k] is the 33 x 33 stiffness matrix, {u} is the displacement
vector with 33 elements, and {F} is the force vector. The above equa
tion may be partitioned as
59
20
t 13
i
[kl
r -\
{u}
L J
{F} (4.32)
Assumed to be 0
v^ere the shaded area represents the coefficients corresponding to the
internal degrees of freedcxn. To uncouple the influence of the internal
degrees of freedcxn on the coefficients corresponding to the external
ncxies, it is necessary to reduce the coefficients in the shaded area
in ecjuation (4.32) to zero. This can be conveniently done by inverse
( uss elimination which gives the following ecjuation
20
13
11
<
r
L ^ J
(4.33)
in which the condensed cjuadrilateral element stiffness matrix (20 x 20)
is represented by the coefficients in the shaded area in equation (4.33)
It should be noted that in performing the above elimination, the nodal
forces and moments at all the interior nodes are assumed to be.zero
so that the force vector {F}remains unaltered during the condensation.
60
4.6 Coordinate Transformations
The procedure described in sections 4.2 and 4.4 is for the deri
vation of the stiffness matrices for the membrane and bending actions
of a triangular element when the xy plane of the coordinate system
cx>incides with the plame of the triangle. The procedure described in
section 4.5 for the assembly of the stiffness matrix for the quadri
lateral element is applicable only if the corner nodes of the quadri
lateral are in the same plane so that the planes of the quadrilateral
and of the triangular subelements coincide. Hence, in order to form
the stiffness matrix for a cjuadrilateral element whose corner ncxies
may be located at randcxn in space so that they may not necessarily be
in a plane, definition of coordinate system for each element as well
as some ccx>rdinate tramsformations is necessary.
In the amalysis two systems of coordinates are used. The first
ccx rdinate system (x,y,z) is referred to as the " Global coordinate "
system amd is a fixed set of cartesian coordinates. The second is the
" Element coordinate " system (x',y*,z') the directions of which depend
on the positions of the four corner ncxies of each quadrilateral ele
ment. The x'y' plame is the plane which "best fits" the cocrdinates
of the exterior ncxies of the quadrilateral, and is estadalished by
minimizing the sum of the squares of the normal distamces from this
plame to the ncjdes. This plane is established from the coordinates of
the comer ncxies and the z* axis is taken perpendicular to this plane.
The x' aucis is taken parallel to the side 1-2 of the quadrilateral
amd the y' aixis is determined such that it is orthogonal to the x' and
y' axes (see Figure 4.8).
61
The transformation of the cx)ordindtes of the nodal points from
the global system to the element coordinate system is a routine matter.
Now, referred to the element coordinate system, all the nodal points
lie in one plane and the stiffness matrix of the element with reference
to the element coordinate system may be derived by the procedures
described in sections 4.2, 4.4, and 4.5.
Before assembling the stiffness matrix for the ccxnplete struc
ture, it is necessary to perform one more transformation. The stiffness
matrix calculated for the cjuadrilateral element will be with respect
to its elonent ccx>rdinate system. As the element coordinate system will
be different for different elements, the stiffness matrices should be
transformed to the common global coordinate system before they can be
ass^nbled to obtain the total stiffness matrix for the structure.
The cx)ndensed stiffness matrix for the quadrilateral element as
obtained by the procedure described in section 4.5 is a 20 x 20 matrix
corresponding to the five degrees of freedom (u, v, w, 0 , 0 ) for the X y
four nodal points. The above five degrees of freedom are with respect
to the element cx)ordinate system. When the stiffness matrix is trans
formed to the global system, the two rotational degrees of freedcxn for
the element with respect to its element coordinates may result in
rotation ccxnponents adx)ut all the three global coordinate axes, so
that due to the transformation, the degrees of freedom for each node
is increased to six: three translations in the direction of the global
coordinate axes and three rotations about these axes. Hence, before
the stiffness matrix is transfoirmed, it is expanded into a 24 x 24
62
matrix including the degree of freedom about the normal to the plane
of the element. Since the element is relatively very stiff against
rotation in its own plane the stiffness coefficients corresponding to
this degree of freedom may be taken as zero. Thus in the expanded
matrix the 6, 12, 18, and 24th rows and columns will have zeros as
the elements and this expanded matrix is transformed to the global
cx>ordinate system before being assembled for the complete structure.
If T represents the matrix of the direction cosines of the
element coordinate system with respect to the global coordinate system
and [k 1 is the expanded stiffness matrix with respect to the element
cxx>rdinate system, then the transformed stiffness matrix [k ] with
respect to the global ccxjrdinate system is given by
I gl tT]'[kg] [T] (4.34)
where
[Tl (4.35)
4.7 Volume Coordinates (21, 74)
Figure 4.9 illustrates a tetrahedral element with nodes 1, 2, 3,
4 in space defined in the x,y,z coordinate system, {X^,YJ^,Z^) being
the ccxjrdinates of node 1, 1 = 1 , 2 , 3 , 4 .
Any arbitrary point P having coordinates (x,y,z) within the
63
tetrahedron divides it into four smaller totrahedrons which are numbered
according to the vertex each is opposite . If the volumes of these tetra
hedrons are Vj , V2 , V3 , and V4 and if the total volume of the complete
tetrahedron is V, the volume coordinates of the point P are defined as
Li - V^ / V i - 1, 2, 3, 4. (4.36)
The volume coordinates of a point are not completely independent.
Because of the relation among the volumes
Vj + Vj + V3 -I- V4 = V
the volume cK>ordinates are connected by the relation
(4.37)
L-j + L2 + L3 + L4 = 1 (4.38)
If the expressions for the volumes V. , in terms of the cartesian
ccx>rdinates are substituted into equation (4.36), the following expre
ssions for the volume ccx^rdinates cam be obtained (21, 74).
For instance
L^ = V^ / V
= i 7 " < ^1 - ^1 ^ - 1 y ^ ^1 ^ ^
where a^ , b, , c, , and d, are constants given by
^1
^1
= det
= -det
X2
^3
^ 4
1
1
1
y2
^3
^4
^2
^3
^4
(4.39)
(4.40)
(4.41)
64
c. = -det
1
1
1
(4.42)
det
1
1 (4.43)
amd 6V = det
1
1
1
1
(4.44)
Similau: expressions for L , L , and L can be obtained by cyclic
intercrhange of the suffixes 1, 2, 3, 4.
It may be noted that the volume coordinates have been defined
so that L. has the value 1 at the vertex i and zero at each of the 1
other vertices.
Expressions for the derivatives of the volume coordinates with
respect to the cartesian coordinates follow immediately from ecjuation
(4.39) and its analogs.
L. = b. / 6V , L. = c. / 6V , L. = d. / 6V IX 1 ly 1 iz 1
i = 1, 2, 3, 4. (4.45)
The suffix X denotes the partial derivative of L. with respect to x, etc,
As per ecjuation (4.39) the volume coordinates are linearly
related to the cartesian coordinates. It follows that the latter are
given in terms of the former by the following relationships.
6 r;
X « 4 "l * 4 "2 * 3 "3 * 4 \
y - L^ y^ + I.J V2 * 1.3 73 + L^ Y^ (4.46)
^ = 4 ^1 * ^2 ^2 - '-3 3 * ^4 ^4
More generally, any quantity S with nodal values & , S , S , and ^ ^ .J
5^ which varies linearly with x, y, and z may be written in the form
6 = ^1 " 1 " 2 ^ 2 •*• 3 *3 " 4 ^ 4 • (4.47)
In deriving the stiffness matrix, the products of volume coordi
nates have to be integrated over the volume of the tetrahedron. The
following formula (74) can be used for the integration.
!^l T^ r^ TS -, 6 V pi ql rl si , tA AO\ L^ L, L. dv = ^ ^ /^—7 r—o \ (4.48) 2 3 4 (p + q + r + s + 3)i V
4.8 Stiffness Matrix for a Tetrahedral Element
The state of displacement of a point in the element is defined
by three displacement cxxnponents u, v, and w in the direction of the
three ccx>rdinate axes x, y, amd z. Referring to the tetraOiedral element
of Figure 4.9, each displacement component has four nodal values at
the four nodes of the element. A linear variation of each displacement
component can be defined by equation (4.47). Thus the displacement
functions may be chosen as
U = L^ u^ + Lj Uj + L3 U3 * L^ u^
V = L^ v^ + Lj Vj + L3 V3 + L4 v^ (4.49)
w = Li «^ + Lj Wj + L3 W3 + L^ w^
66
Six strain components are relevant in a full three dimensional
analysis. The strain matrix can now be derived as
{e} X y z xy yz zx
3 u , 9 V , 9 w 9 X 9 y 9 z
9 u 9 V 9 y " 9 X
9 V 9 w 9 z " 9 y
9 w 9 u 9 X " 9 z ^ (4.50)
Using ecjuat ions (4.49) and (4.50) i t i s easy t o v e r i f y t h a t
{ e } [B] {u} (4.51)
where {u} = ^ ^ i ' ^ i ' ^1 ' ^2 ' ^2 ' ^2 ' ^3 ' ^3 ' ^3 '
^4 ' ^4 ' ^ ^ ( 4 .52 )
and [B] [ B^ , B^ , B3 , B^ ] (4 .53 )
i n which
6v
0
0
c, 1
0
d. L
0
0
b.
0
0
0
1 = 1 , 2 , 3 , 4 . (4.54)
The s t r e s s s t r a i n m a t r i x [D] for an i s o t r o p i c m a t e r i a l i n terms
of t h e Yo\ing's mcxiulus E and P o i s s o n ' s r a t i o v cam be w r i t t e n as (72)
67
[Dl E (1 - V) (1+v)(l-2v)
1 V V 0 1-v 1-v
1 V 0 1-v
0
SYMMETRIC
(1-2V) 2(l-v)
0
0
(l-2v) 2(1-v)
0
(1-2V) 2(1-v)
(4.55)
Knowing the matrices [B] and [D], the stiffness matrix [k] for
the tetraihedral element can be derived from the principle of minimuia
potential energy (equation 2.8) as
[kl / V
[Bl [Dl [Bl dv (4.56)
In this case, since all the elements in the matrices [Bl and [Dl are
constamts
[kl [Bl [Dl [B] X V (4.57)
4.9 Stiffness Matrix for the Octaihedral Element
The octahedral element is divided into five tetrahedral elements
as shown in Figure 4.10. The stiffness matrix for each tetrahedral
element is derived separately according to the procedure explained
in the previous section, and the direct stiffness procedure is
employed to assemble the stiffness matrix for the octaihedral element.
r,8
|Z,W
V Vi
(1,2,3,4) (P,2,3,4) (1,P,2,4)
(1,2,P,4)
(lr2,3,P)
•I** x,u
FIGURE 4.9 TETRAHEDRAL ELEMENT - VOLUME COORDINATES
-l** y
FIGURE 4.10 OCTAHEDRAL ELEMENT AIID ITS SUBDIVISION INTO TETRAHEDRONS
CHAPTER V
RESULTS OF STATIC ANALYSIS
The accuracy and efficiency of the quadrilateral shell element
described in the previous was studied by applying it for the analysis
of a few plate and shell problems with known solutions. The following
problems were analyzed and the results are discussed.
5.1 Bending of a Square Isotropic Plate
The relatively simple problem of a square isotropic plate under
various conditions of support and loading is a convenient test example,
as numerous 'exact' solutions are available in standard texts on plates
and shells.
The finite element solutions were so designed that the effect of
the mesh size could be studied. Figure 5.1 shows the six different
subdivisions used in a quarter of the plate. Only symmetrical cases of
support and loading were considered. Edges of the plate were taken
either as simply supported or as clamped, and the loads represented
either a single concentrated load P at the center or a uniformly
distributed load q. The allocation of the distributed load to the
nodes was done by assigning one-fourth of the load on any element to
each of it's nodes.
Tables 5.1 (case 1 and case 2) show the results of the analysis
for the two types of support conditions and the two types of loading
using different mesh sizes. The values of the nondimensional parameters
(X for the central deflection, and 3 and y for the bending moments are
tabulated. The exact values are also shown for comparison. In all cases
69
70
excellent convergence of the results was obtained as the element size
was reduced.
The results are compared with those of other investigators in
order to evaluate the relative efficiency of this particular quadri
lateral shell element. Tables 5.2 - cases 1, 2, 3, and 4 - summarize
the various results. Figures 5.2(a), 5.2(b), 5.2(c), and 5.2(d) show
the plots of these results. In these taO les and figures QD represents
the quadrilateral element used in the present investigation. Q-19 is
a cjuadrilateral element due to Felippa (41) and HCT is a triamgular
element due to Clough and Tocher (43). The above three elements have
a joint degree of freedom ecjual to 6 when applied to shell analysis.
KCM is a curved triamgular element developed by Dhatt (63) and KB6 is
a curved quadrilateral element developed by Key amd Beisinger (68).
These two elements have degree of freedom of 9 per node. From the
figures it cam be seen that the quadrilateral element QD gives as
gcx>d results as some higher order elements as the element size is
reduced.
Figure 5.3 shows the actual distribution of the bending moments
along the center line of the plate as obtained from the 8 x 8 mesh.
The graph of the exact values is also plotted for the simply supported
edge condition. Figure 5.4 shows the distribution of reactions along
the edges for the simply supported case. It cam be seen that the
element is efficient in evaluating moments amd forces also.
71
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6 x 6 8 x 8
1
FIGURE 5.1 SQUARE PLATE - ELEMENT SUBDIVISION
78
12.0
EXACT 11.6 <X. ^ICCM
11.0
8 Vi
10.0
? ^ • - O ::5i!:i:^ - ^
,---UJ
/
4 _ HCT
9.0
8.0
IQ ± \
50 100 n ( number of unknowns )
J 500 1000
3 2 10 X w D / P £ max / ' **
FIGURE 5.2(a) SIMPLY SUPPORTED SQUARE PLATE
COMPARISON OF RESULTS - CENTRAL CONCENTRATED LOAD
79
4.5
PXACT 4.062 4.0-
3.5-
fe g 3.0
2.5
Q-19
-I-C71-.T; r-'-'r - - • •» , - . o-
2.Q 10 50 100
J 500 1000
n ( number of unknowns )
10^ ^ ^max D / ^
FIGURE 5.2Cb) SIMPLY SUPPORTED SQUARE PLATE
COMPARISON OF RESULTS - UNIFORMLY DISTRIBUTED LOAD
BO
6.0
a ^ 5.0
I
4.0
EXACT 5.6 , < > • - <
10
Q-19 P:^^/
/
.4-
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QD
-—n
50 100 500 1000
n ( number of unknowns )
^ = 1° ^ ^max D / P A
FIGURE 5.2(c) CLAMPED SQUARE PLATE
COMPARISON OF RESULTS - CENTRAL CONCENTRATED LOAD
81
1 . 4 0 ^ •o-
/ \
/ \ *r<- KCM
1.30 /
V. \
EXACT 1 . 2 6 I o.
1.20
Pu O
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1.00 -
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I Q-19
0 .90
10 50 100 500 1000
n ( number of unknowns )
3 , 4 a = 10 x w D / o A
max ^ / ' •*'
FIGURE 5.2(d) CLAMPED SQUARE PLATE
COMPARISON OF RESULTS - UNIFORMLY DISTRIBUTED LOAD
82
- 5 0 0
OQ
c
D
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E-2 pq M
u M PL4 pL. o u
- 3 0 0 -
- 1 0 0 -
+100 -
+ 300 -
+ 5 0 0 ^
.213 Exact 213
Exact 479
F.E.M.496
3 = lo'* X B.M. / q il^
FIGURE 5.3 DISTRIBUTION OF BENDING MOMENT ALONG THE CENTER LINE OF A UNIFORMLY LOADED SQUARE PLATE
83
-700e
ro O r-i
X
Pu
o CO
-500
§ -300
Oi
o EH 2 pq M U
-100-
;:; 100-lu pq
Exact = 650
. Exact = ^-'487.6
UNIFORMLY DISTRIBUTED LO.
CENTRAL CONCENTRATED LOAD . /^.J
/
8
300*-
A = Reaction / (P/4) for concentrated load
A = Reaction / q A' for uniformly distributed load
A Positive for upward reaction
FIGURE 5.4 SIMPLY SUPPORTED SQUARE PLATE - DISTRIBUTION OF REACTIONS ALONG THE SUPPORTING EDGE ( F.E.M. ANALYSIS )
84
5.2 Pressurized Pipe
A relatively simple problem requiring a shell element is a pipe
under internal pressure. Using the symmetry about the axis of the pipe,
one cjuarter of the pipe was analyzed using three different mesh sizes.
The results are presented in Table 5.3 and Figure 5.5. The results
indicate the high degree of accuracy of the shell element for solving
shell problems where the bending moments are insignificant.
TABLE 5.3
ANALYSIS OF A PRESSURIZED PIPE
Mesh Size
4 Elements
6 Elements
8 El^nents
Percentage Error in Radial Displacement.
-1.5
0.0
0.33
Percentage Error in Hoop Stress
-2.0
-0.85
-0.48
85
FINITE ELEMENT SUBDIVISION - 4 ELEMENTS
-2
1-1
pq
pq
Z
pq 04
+1-
ERROR IN HOOP STRESS
ERROR IN RADIAL DISPLACEMENT
i 4 6 8 MESH SIZE ( Number of elements
i n one cjuaurter )
FIGURE 5 . 5 PIPE UNDER INTERNAL PRESSURE
86
5.3 Cylindrical Shell Roof
A cylindrical shell having the dimensions shown in Figure 5.6
was analyzed using three different mesh sizes. The loads are also shown
in the above figure. This problem has been analyzed by Scordelis amd
Lo (75) using elasticity theory and by Siddiqi (76) using the finite
strip method. Only one quarter of the shell was analyzed because of
double symmetry. Figure 5.7(a) shows the convergence of results as the
element size is decreased. Figure 5.7(b) shows some comparisons of the
results. It nay be seen that the results of the present study using
the quadrilateral shell element compare very well with those of other
investigators.
5.4 Analysis of a Folded Plate Structure
A typical folded plate cross-section (Figure 5.8) was ainalyzed
for a span of 70 feet. This problem has been analyzed by Cheung (77)
using the finite strip methcxi, and by DeFries-Skene amd Scordelis (78)
using elasticity method. For the saike of comparison of results, the
loads used in the amalysis aure ridge line loads, the same as those used
in the above references. Uniformly distributed loads present no problem
for the amalysis by the finite element method. The element siobdivision
is also shown in Figure 5.8. The comparison of results is presented in
Figure 5.9. It cam be seen that very good agreement is obtained for the
longitudinal stress as well as bending mcanents. The results for the
vertical deflection at midspam are slightly higher tham those by the
finite strip method, but the relative displacements of the various
parts of the structure at the section are the same. Better agreement
for all the results can be obtained by finer mesh selection.
87
During the study of this problem, it was found that the boundary
conditions specified at the supports of the folded plate have consider
able influence on the results. The usual assumptions regarding the pro
perties of the diaphragm - that the diaphragm is very rigid in it's own
plame and perfectly flexible in the plane normal to it - suggest the
following boundary conditions at the support of the folded plate. The
conditions are that the vertical displacement for all the ncxies along
the supporting edge is zero with all other deformations permitted. The
results presented in Figure 5.9 are those with the above boundary con
ditions. Since the computer program developed enadDles the amalysis of
the folded plate including the diaphragm, a study was made aibout the
effect of the diaphragm. It was found that the diaphragm reduces the
horizontal displacement of the parts of the folded plate in the plame
of it's cross-section and also creates fixing moments at the junction
of the folded plate and the diaphragm. Two more analyses were done for
the folded plate without the diaphragm, but with fully fixed and partly
fixed boundaury cx^nditions. The boundary conditions for partial fixity
were to allow longitudinal movement - as for a roller support - with
all other displacements specified as zero. For the fully fixed condi
tion the longitudinal displacement at the support was also specified
ais zero. The results of all the four studies are presented in Figure 5.10.
The results indicate that the effect of the diaphragm is to decrease
the vertical deflections and also to increase longitudinal moments both
at midspan and the support. Longitudinal stresses are not affected.
The above observations are for the particular problem studied.
88
E = 3.0 X 10 p.s.i. V = 0.0 t = 3.75 in. Loading D.L. 47 psf of shell
surface L.L. 25 psf of horiz
ontal projection
4 x 5 MESH 8 X 12
FIGURE 5.6 CYLINDRICAL SHELL ROOF
89
1 2 . 0
o
1 1 . 0 -
1 1 . 8 SIDDIQI (Reference 76)
10.OL X 4 x 5 6 x 8
MESH SIZE
1 8 X 12
V - MAXIMUM VERTICAL DEFLECTION (AT POINT A)
FIGURE 5.7(a) CYLINDRICAL SHELL - CONVERGENCE OF RESULTS
90
-r:r-zd
-2.0
LONGITUDINAL STRESS T AT MIDSPAN
-3.0 r
ro O
*s
Xi t-i I •P MH
-2.0
-1.0
0.0
TRANSVERSE MOMENT Mx AT MIDSPAN
O FINITE ELEMENT ANALYSIS 8x12 MESH A REFERENCE 75
FIGURE 5.7(b) CYLINDRICAL SHELL - COMPARISON OF RESULTS
91
I 382 lb/ft
535 lb/ft
1.75 ft
5.00 ft
00 ft
% 8.67 ft 4- 9.83 ft J 9.83 ft J 8.67 ft J
SECTIONAL DIMENSIONS AND LOADING
FINITE ELEMENT SUBDIVISION
PLATE 1
FIGURE 5.8 FOLDED PLATE ANALYSIS
92
RIB PLATE 1 PLATE 2 r
•p
pq
2.5
2.0
RIB PLATE 1 PLATE 2
T"
1.5 I}- -..: -
in o
X
UH
IV, 1.0
— S 0.5 'I
-0.1
^-—'•A 0 .0 —
-0.5 -
-0.2
' • J - . N
r
- 1 . 0 "•
-
--''' . -.-^A--.J
MAXIMUM VERTICAL DEFLECTION w AT MIDSPAN
LONGITUDINAL STRESS 0 AT MIDSPAN ^
4.0
CM o 3.0
4J IM \ i3 rH 1
4J
2.0
1.0
0.0
-1.0
PLATE MEMBRANE SHEAR T AT THE SUPPORTS
LONGITUDINAL MOMENT M AT MIDSPAN
o REFERENCE 77 A FINITE ELEMENT ANALYSIS
FIGURE 5.9 RESULTS OF FOLDED PLATE ANALYSIS
93
RIB PLATE 1 _ „ PIiAIE..2
0.4
0.3
0.2 i . . . > * v ' .- - — — - ' • • — - —
RIB PLATE 1 PLATE 2
I.5O--in \> o
1.0
YN ^\^
0.01
-0.1
•4
MAXIMUM VERTICAL DEFLECTION w AT MIDSFAN
4 0
0.5
> ^ 0.0
-0.5
r.,1
-1.0 '
V
ej> s 9" "'•-<-"-• ,r',-'-!/v-'-<>
«?,•-><•_ —
LONGITUDINAL STRESS 0 AT MIDSPAN
-5. On 1
cN^-4.0
.•*J-3.0
i:-2.o
LONGITUDINAL MOMENT M AT MIDSPAN
r^ -1.0
H
""A -.
-o 0. or- 4---T r ^ ^ i: -Jt:- : :— - x --4
S LONGITUDINAL MOMENT M^
AT SUPPORT
O WITH DIAPHRAGM SIMPLY SUPPORTED
D FULLY FIXED A PARTLY FIXED
FIGURE 5.10 RESULTS OF FOLDED PLATE ANALYSIS
CHAPTER VI
ANALYSIS FOR DYNAMIC lOADS
The basic concept of the finite element method is to consider
any structure as made up of a series of "finite elements" with known
stiffness characteristics and which are interconnected at a finite
number of "nodes". This concept reduces the dynamic analysis of a
continuous structure to the analysis of a discrete system with a finite
number of degrees of freedom. The distributed inertia forces and damp
ing forces involved in the dynamic analysis are replaced by their
static equivalents at the nodes where the conditions of ecjuilibrium
amd compatibility are applied. Numerical methods utilizing digital
computers can thus be used for the dynamic analysis of complex struc
tures by the finite element method.
Two basic problems are immediately posed. The first is the
determination of the stiffness characteristics of the individual
elements. The second is how to assign the inertia and damping forces
to the ncxies. The generation of the element stiffness matrix and the
assembly for the ccxnplete structure has been discussed in Chapter IV.
The inertia and damping forces are accounted for by constructing what
aure known as "mass" and "damping" matrices and incorporating these
into the ecjuations of equilibrium. The derivation of these matrices
amd the solution of the resulting equations - which are second order
differential equations - are presented in the following sections.
6.1 Ecjuations of Motion
When the displacements of an elastic body vary with time, two
94
05
sets of additional forces are called into play. The first is called
the inertia force. The inertia force is proportional to the acceler-2
ation i_^ ' ^ and can be replaced by its static equivalent
2
at
2 3 {F} = - p £_2 {A} (6.1)
m 3t
using the well known D'Alembert principle (79). {A} is the generalized
displacanent, and p is the mass per unit volume. This is a force with components in the directions identical to those of the displacement
{A} and is given per unit volume.
The secx)nd force is that due to resistances opposing the motion.
These may be due to friction, microstructure movements, air resistance,
etc; and are generally related in a nonlinear way to the displacement
velocity -r— {A} (21). It is often assumed in the theory of structural at
dynamics that the damping forces are of the "viscous" type, i.e. they
are lineau:ly related to the velocity. Hence the resulting equivalent
static force due to the resistances is
{F}^ = - M |-{A} (6.2) d 31
where y is a constamt friction (or damping) coefficient. This force
is also given per unit volume.
The ecjuivalent static problem, at any instant of time, can now
be discretized as follows. The derivations are similar to those for
the distributed loads which have been explained in section 2.3. The
element force vector due to bcxiy forces was derived (equation 2.13) as
96
{F} - J [N]T{p} dvol P , vol
where {p> corresponds to the vector of the body force components in
the direction of the generalized coordinates. The element force vector
^^^dyn ^^^ ^° ® inertia and damping forces is given in a similar
manner by
" " d y n 3
=
{F} + {F}^ m d
vol at {A} dvol - J [N]* M TT {A> dvol. (6.3)
, 8t vo l
Replacing (A) by lN]{u} from ecjuation (2.9)
^^^dvn = - j [N]'^ P ^ [ [N]{u} ] dvol ^ vo l a t
/ . . . ,T a [N] y •— [ [N]{u} ] dvol (6.4)
, 8t vol
Since [N] is the matrix of interpolating functions which do not vary
with time, amd {u} is the vector of the nodal displacements which are
independent of the integration
{F} = - { U } J [N]" P [N] dvol - { u } J [N]" y [Nl dvol ^y^ vol vol
= - [m] {u} - [c] {u} (6.5)
where ^ M = — 2 {u} (6.6)
9t
9_ 3t
{u} = f- M (6.7)
. / -[m] = J [N] p [N] dvol (6.8) vol
97
/ '
and Icl = J iNl' y [N] dvol (6.9)
vol
The matrix [mj is known as the element mass matrix and [c] the element
damping matrix. The mass matrix for the complete structure [M] and the
damping matrix (CJ for the complete structure are obtained by assemb
ling the element matrices in the same manner as for the stiffness
matrix.
TaJcing into account the equivalent static force, the general
ecjuations of ecjuilibrium for the ccxnplete structure get altered as
[Kl {u} - {F} - {F}^y^ = 0 (6.10)
where {F} is the force vector representing the external applied forces
as a function of time. {F}, is the assembled force vector for the ^ dyn
cxxnplete structure due to inertia and damping forces.
By amalogy with ecjuation (6.5) for an element
{FK = - [M] {u} - [C] {u} (6.11) dyn
Substituting in ecjuation (6.10) we get the general equations of motion
(or equations of dynamic equilibrium at any instant) as follows.
[Ml {u} + [Cl {u} + [K] {u} = {F} . (6.12)
6.2 Mass Matrix
The element mass matrix [m] has been derived in the preceding
section as
[m] J [N] p [N] dvol (equation 6.8)
vol
where [N] is the matrix of interpolating functions used in the
98
ov.Uuation of the stiffness matrix. As the stiffness matrix relates
Uio strain energy of an clonuMit to its nodal displacements, the mass
matrix relates the kinetic energy of t)ie element to its nodal veloci
ties. When equation (6.8) is used to evaluate the mass matrix, the
kinetic energy of the element will be consistent with its potential
energy, and the mass matrix so obtained is known as the "consistent
mass matrix". If the displacement functions used in the formulation
of the stiffness matrix of the element are compatible, the bounding
properties of the Ritz process will apply; i.e. the consistent mass
matrix will lead to am upper bound on the lowest mode frequency (25) .
In general, however, it is not desirable to use the consistent
mass, because excessive computational effort is usually required by
this approach to obtain a desired degree of accuracy(25). Other mass
matrices may be derived by using different interpolating functions in
equation (6.8). The simplest results are obtained by letting [N] have
imit value over a specified portion of the element volume with zero
values elsevrhere. In this case, each diagonal term of the [m] matrix
represents the mass associated with the corresponding nonzero value
of [N) and the offdiagonal terms are zero if the nonzero volumes do
not overlap. This method is known as the "lumped mass" approach. The
advantages of this approach are that the evaluation of the matrix [m]
is very easy and also the mass matrix for the entire structure is
diagonal which leads to great simplifications in the analysis. While
the consistent mass matrix provides an upper bound to the lowest mode
frecjuency, the lumped mass result may be either low or high.
99
Zienkiewicz, Irons, and Nath (80) claim that the use of the
consistent mass matrix results in higher accuracy; while Clough and
Felippa (58) have shown that the lumped mass result is more accurate
tlian consistent mass result* In the present investigation lumped mass
approach is used since for the derivation of the stiffness matrix of
the quadrilateral element static condensation was employed and there
is no known equivalent process of condensation for the mass matrix.
Further, the results obtained indicate that the lumped mass approach
gives fairly gcjod results. The lumped masses are associated only to
the translational degrees of freedom (41).
6.3 Damping Matrix
The element damping matrix [c] has been derived (equation 6.9)
as [c] = J [N] * y [N] dvol
vol / -
where y is a friction coefficient. If the appropriate internal damping
characteristics are specified to evaluate y, the daunping matrix can be
derived frcxn ecjuation (6.9) .
The nature of damping is usually described as one of the follow
ing (81).
1. Viscous damping
2. Structural damping
Viscous damping results when a system vibrates in a fluid (air,
oil, etc). In viscous damping it is assumed that distributed damping
forces are developed which are proportional to the velocity of each
mass point in the element. The damping forces are given by
100
^^^d = - «m P Ir <^> (6.13) ot
where a^ is a proportionality constant. Comparing equations (6.2) and
(6.13) we see that y = a^ p in this case. Substituting for y in
ecjuation (6.9)
[cl = /
vo l
%i
%
[ N ^
/
vo l
[ml
°4n P
T
[Nl
[Nl
dvol
dvol
(6.14)
Structural damping is due to the internal friction within the
structural system. In structural damping it is assumed that damping
stresses are developed which are proportional to the strain velocity.
The damping matrix can be derived as follows. Using the general nota
tions, the elastic stresses are given by
(o) = [Dl {e} = [Dl [Bl {u} (6.15)
The damping stresses are given by
{a}^ = aj [Dl {e} = a, [D] [B] {u} (6.16)
where CL is a proportionality constamt.
Similar to the derivation of the stiffness matrix, by writing
the expression for the strain energy and applying the principle of
minimum potential energy, we cam derive the damping matrix as
[cl = ajc J [B]' [Dl [Bl dvol = aj [kl . (6.17) vol
In practice, however, it is seldom feasible to establish the
101
local damping properties needed to derive the element damping matrices
in this way. Generally, the d»\mping j rof crties are determined by tests
on typical complete structures, and these experimental results are
expressed as modal damping ratios. A technique of deriving the damping
matrix frcxn a set of modal damping ratios is described in reference(25).
It can be seen from equations (6.14) and (6.17) that the damping
matrix is proportional either to the mass matrix or the stiffness
matrix. A linear ccxnbination of both can also be assumed. A dancing
matrix proportional to the stiffness matrix will result in increasing
damping in higher modes. Since this is found to be the case in most
of the structures, this approach using [cl = Cu [ 1 is adopted in
the present investigation.
6.4 Response to Dynamic Loading
The dynamic response analysis of a finite element system is
performed by solving the ecjuations of motion (equation 6.12). Two
different techniques aure available. The simultaneous differential
ecjuations may be solved directly using step by step integration; or
the equations may be first decoupled by transforming to the normal
(mcxle shape) cx>ordinates, solved independently mcxie by mode amd then
the modal results superposed to obtain the total response. The latter
method (mcxle superposition) is advantageous if the essential dynamic
response of the structure is contained in the first few mode shapes.
This will be the case if the applied loads cam be approximated reason
ably well by inertia force patterns associated with the first few
mode shapes amd if the frecjuency content of the input is largely
102
represented by the corresponding lowest frequencies. In cases where
the applied load distribution is extremely complex, and/or the time
variation contains significant high frequency components, it will be
necessairy to include many mode shapes of vibration to obtain adequate
accurac:y by mcxle superposition. In this case the step by step proce
dure is more efficient. Moreover, if the structural response is non
linear due to either material properties or geometric effects, the
step by step integration procedure cam only be used. In the present
study the step by step integration procedure is used.
6.5 Step by step Numerical Integration Procedure
The step by step analysis is based on the powerful iterative
tecrhnicjue developed by Newmark (82) known as Newmark's 3 parameter
methcxi. The acceleration at the end of a time step is estimated, and
the velocity and displacement are then calculated by
u. = u + ^ ( u + u ) (6.18) 1+1 i 2 i 1+1
. 0 1 .. 7 ..
u = u + A t u + A t ( e ) u + 3 A t u i+1 i 1 2 i i+1
(6.19) • ..
u, u, amd u represent tlie displacement, velocity, amd acceleration
of amy point, the suffixes i and i+1 representing the ends of the
time intervals i and i+1. The above expressions are substituted in
the general equations of motion amd a new estimate for the acceler
ation at the end of the time step i+1 is determined. The process is
repeated until successive values of the acceleration agree within a
specified tolerance.
103
Thci parameter B in equation (6.1'J) cjoverns the influence of
the acceleration at the end of the time interval i+1 (i.e. ii ) on i+1
tlie displacement at that instant. Furthermore, the value selected for
P determines the variation of the acceleration during the interval At
from i to i+1. For B = 1/6, the method becomes a linear acceleration
assumption. A B value of 1/4 represents a constant acceleration
throughout the interval and i. = 1/8 may be interpreted as a step • •
function having an acceleration u over the first half of the time 1
• •
interval and u. , through the last half. 1+1
The iterative technicjue lends itself easily to nonlinear analy
sis. However, for linear analysis, a direct solution is possible for
the deflections at the new time station. Using the 3 parameter method
Chan, Cox, and Benfield (83) have developed a recurrence relation
which eliminates both velocities and accelerations from the equations
of motion. This recurrence method is used in the present analysis.
The derivation of the recurrence relation is given in reference
(84). The final form of the expression is
[ ^ ^ 2At W J {u.^,} =
{ 3 {F^^^} + ( 1 - 23 ) {F^} + 3 {F^_^} I
- r - 2tMl + ( 1 - 23 ) [Kl I {u.} L At2 J ^
- r im. - M . + [Kl 1 (u. ,} (6.20) L At2 2At J 1-1
where {u^^^}, {u^}, and in^^j^} are the displacements at the end of
104
the time intervals i+1, i and i-1 respectively; {F }, {F.}, and
{F. j } are the external forces at the corresponding instances; [M],
[Cl, and [Kl being the mass, damping, and stiffness matrices respect
ively for the entire structure. If the displacements at the end of
the previous two time intervals are known, the above equation yields
the displaconents at the end of amy time interval. Equation (6.20)
may be written symbolically as
[Al {u} = {F} (6.21)
where [Al = f " ^ •»• ^ + ^ fKll (6.22)
L At^ 2At J
{u} = {u .i> (6.23)
and {F} = < 3 (F. ,} + ( 1 - 23 ) {F.} + 3 (F } I 1+1 1 1-1
- [A.l {u.} - [A. 1 {u. } (6.24) 1 1 1-1 1-1
where [A.l = f ^ - ^ ^ + ( 1 - 23 ) [Kl J ^ ^ At
*- At
(6.25)
(6.26)
For programming in the computer, the matrices [Al , [A. ] , amd
[A. 1 aure assembled from the stiffness, mass and damping matrices of
the elements. The matrix [Al is triangular!zed and kept in storage for
back substitution. The equivalent force vector {F} is calculated for
each time interval as per equation (6.24) amd by back substitution
the displacement vector {u} is calculated. For starting values {u }
and {u_^} can be taken as {0}, which imply that there is no force
at the instant t = 0. Even if there is a suddenly applied force at t = 0,
105
it can be considered as a force at the end of the first interval of
time by making the time interval At small enough.
6.6 Selection of the Time Interval At
The recurrence formula given by equation (6.20) is obviously
approximate, but it gives sufficiently accurate results provided that
the time interval At is chosen small in relation to the variation in
the acceleration. In general, it has been found that results sufficiently
accurate for practical purposes can be obtained if the time interval
is chosen no larger than one-tenth of the natural period of the system
(79). However, a second criterion must also be considered while select
ing the time interval. The interval should be small enough to represent
properly the vauriation of the load with time.
The size of the time interval also influences the stability of
the solution. Small time steps are required for stable solutions to
many initial value problems. Determination of the sta±>ility of a
nxanerical procedure is based on the investigation of the propagation
of errors introduced at any time step. If after a number of time steps
the errors are unbounded, the solution is said to be unstaQjle. A
stability analysis for the 3 parameter method for a value of 3 = 1/6
is given in reference (84), and it is reported that a satisfactory
estimate for the maximum time increment is 1/4 of the smallest pericxi
of vibration. In reference (85) it is reported that, if the damping
is uniform amd positive (or zero) and if there are no nonlinear terms
on the righthand side of equation (6.20), the solution will be stable
for any time step provided 3 is greater than or equal to 1/4. A value
106
of 3 ecjual to 1/4 is used for all problems in the present investiga
tion and no instability was noticed.
6.7 Results of Dynamic Analysis - Simply Supported Rectangular
Plate under Dynamic Loading
The response of a simply supported rectangular plate under
dynamic loading was studied. The dimensions and material properties
are as shown in Figure 6.1. Because of the symmetry about both x and
y axes, it was possible to amalyze only one quarter of the plate. Two
types of dynamic load were applied. One is a suddenly applied vertical
pressure which was kept constant, amd the other was a suddenly applied
vertical pressure which reduced with time. The details of the forcing
functions are also shown in Figure 6.1.
The 3 value used is 1/4. The analysis of the plate under the
secx>nd case of loading was done by Biggs (79). The natural frequency
was calculated as 0.0117 sec. The time step used in the analysis was
0.00125 sec, which is about 1/9.4 of the natural period. It was also
possible to take into account the discontinuities in the forcing
function of case 2 with this time step. The values of the central
displacement for each time step are tabulated in Tables 6.1 and 6.2
for the two cases of loading. Figures 6.2 and 6.3 show the plot of
these values. The values of the maximum deflections and the natural
pericxi from reference (79) are also shown in these figures for
cxjmparison.
107
TABLE 6.1 SIMPLY SUPPORTED RECTANGULAR PLATE UNDER DYNAMIC LOADING.
CASE 1. CONSTANT FORCING FUNCTION.
Time - Secx^nds Central Displacement - inches
Time - Seconds Central Displacement - inches
0.0
0.00125
0.0025
0.00375
0.005
0.00625
0.0075
0.00875
0.01
0.01125
0.0125
0.01375
0.015
0.01625
0.0
0.025
0.126
0.305
0.479
0.568
0.543
0.424
0.248
0.082
0.004
0.052
0.189
0.362
0.0175
0.01875
0.02
0.02125
0.0225
0.02375
0.025
0.02625
0.0275
0.02875
0.03
0.03125
0.0325
0.518
0.579
0.514
0.360
0.185
0.050
0.005
0.082
0.253
0.427
0.544
0.568
0.476
108
TABLE 6.2 SIMPLY SUPPORTED RECTANGULAR PLATE UNDER DYNAMIC LOADING
CASE 2 - TRIANGULAR FORCING FUNCTION
Time - Seconds
0.0
0.00125
0.0025
0.00375
0.005
0.00625
0.0075
0.00875
0.01
0.01125
0.0125
0.01375
0.015
0.01625
0.0175
Central Displacement - inches
0.0
0.025
0.126
0.301
0.468
0.544
0.506
0.372
0.186
0.014
-0.006
-0.018
0.118
0.286
0.432
F" '
109
o
60 i n .
ujq::i:j O- C - • • -tv- •— - o ;.
MM
h = 1 i n .
6 E = 30 X 10 p . s . i .
y V = 0 .25
P
I . . . i—i
2 3 0 .00073 l b - s e c / i n
DIMENSIONS AND FINITE ELEMENT SUBDIVISION
m
m
04
40
0 J . I .
0.01 TIME t seconds
0.02
PRESSURE TIME FUNCTION - CASE 1
0.05 TIME t secorids
PRESSURE TIME FUNCTION - CASE 2
FIGURE 6.1 SIMPLY SUPPORTED PLATE UNDER DYNAMIC LOADING
110
pq
§ •J
i Cu (C a D o
u g Q pq
O
D CO
H CO
O
pq CO z o 0* CO
g
D O H (14
saHDNi - HajiNSD aHi. J.V JJ^awaov^dSIa
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a 04
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o 04 04 D CO
>i i-q 04
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PL4 O
pq CO 2 O 04 CO
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D
in
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saHDNi - Haj.N3D aHi. J.V i,N3waD\ndsia
CHAPTER VII
RESULTS OF PRESSURE VESSEL ANALYSIS
The results of the analysis of a pressure vessel junction by
the finite element methcxi are presented in this chapter. The pressure
vessel was amalyzed for a static load case amd also a dynamic load.
The static analysis was done in two steps. The first was an amalysis
using the cjuadrilateral shell element. A portion of the structure
was then analyzed using the three dimensional octahedral element in
order to determine the stress concentrations. The dynamic amalysis
was done for an impulse load on the manhole cover. The results of
the static analysis are cxxnpared with experimental results.
7.1 Static Analysis using the Shell Element
A schematic diagram of the junction between a mamhole amd the
main body of a pressure vessel is shown in Figure 7.1. The dimensions
used in the analysis of such a junction are also shown in the above
figure. These dimensions are of a pressure vessel which had been
instrumented and tested experimentally by Chicago Bridge and Iron
Ccxnpany (24). One of the big tasks in solving this problem by the
finite element method is the generation of the coordinates of the
ncxial points. The schonatic diagram of the finite element idealiza
tion is sho%m in Figure 7.2. The results of the experimental inves
tigation indicated a symmetry in the stress distribution across the
four cjuarters of the manhole and the coverplate. Hence, only one
cjuarter of the junction was amalyzed.
The loading on the structure was an internal pressure of 40 p.s.i
112
113
This load war. choson for the sake of comparison with the experimental
results. TI»o uatuio o! the oxpot imont was as follows. After the strain
gages were installed in position, the regenerator (pressure vessel)
was filled with water. The strain gage readings were recorded. These
readings were tadcen as the initial readings for the test. The regene
rator was then pressurized to the 40 p.s.i. pressure and returned to
the zero pressure. The strain gage readings were recorded once again
and checked with the initial readings for errors and also to check
for proper functioning of the strain gages. Then the pressure was
increased frcxn 0 to 40 p.s.i. in increments and the strain gage
readings were noted at each stage. The pressure was then decreased
in increments and the readings were once again recorded. This c:ycle
was then repeated and another set of readings was taken. The stresses
were calculated for each set of readings and the average values for
the stresses were taken.
Table 7.1 gives the results obtained by the finite element
analysis for the stress distribution along the vertical section
through the center of the manhole (line connecting nodes 28 and 100
in Figure 7.2). From the membrane stresses and bending moments obtained
by the analysis the stresses at the inside and outside surface of the
coverplate are calculated. These values are also given in Table 7.1.
Figure 7.3 shows the plots of these results. The plots of the experi
mental results are also shown on the same figure.
Table 7.2 gives the stress values at various points in the
coverplate. Stresses in the coverplate along the radial and tangential
114
directions to the manhole are calculated from the hoop and longitudi
nal stresses obtained by the finite element analysis. The stresses
are calculated for the nodes along five radial lines from the center
of the mamhole. Figure 7.4 shows the plot of the tangential stresses,
and Figure 7.5 shows the plot of the radial stresses. The results of
the experimental investigation are also plotted in these figures for
comparison.
It may be noted that the experimental and finite element results
agree very closely in many regions. More detailed stress analysis can
be accxMi^lished by further subdivision of the plates in the regions
of interest thus getting more accurate results for the stresses in
the plates.
115
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9 9
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117
z RESS IN THE
lAL DIRECTIO
THE MANHOLE
p.s.i.
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DIRE
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S.i.
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118
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119
^ . L ."C*
1.25 in.'\
V
-_L-
COVER PLATE 3 in. THICK
V ^L MANHOLE
I \"-Vl,:l;/
•H
in
in
\ ^- \ 2 in.
FLANGE
\ \-\ '3.125 in.
\
T--
/
„*.. 30DY OF THE [PRESSURE VESSEL
\ .
FIGURE 7.1 SCHEMATIC DIAGRAM OF A JUNCTION AT A MANHOLE IN A PRESSURE VESSEL
120
124
COVER PLATE
\y^ r7?r|36 45 54 63 I
^ ^— COVER *< COVER PLATE
FIGURE 7.2 STRESSES AROUMD A MANHOLE IN A PRESSURE VESSEL SCHEMATIC DIAGRAM OF FINITE ELEMENT IDEALIZATION
121
7J
^ SHELL
— t - 1.213"
^ \ A O Q EXPERIMENTAL RESULTS
2 4 6 8 10 12
HOOP STRESS KSI ( AT 40 PSI PRESSURE )
FIGURE 7.3 STRESS DISTRIBUTION IN THE COVER PLATE ON A VERl'ICAL SECTION THROUGH THE CENTER OF THE MANHOLE
122
.r_67.5<
• F.E.M. Analysis Results
O Experimental Results
SCALE 1" = 10 k.s.i
1" = 20"
FIGURE 7.4 PRESSURE VESSEL ANALYSIS - DIRECT STRESS TANGENT TO THE MANHOLE AT 40 PSI PRESSURE
123
67.5"
Q F.E.M. Analysis Results SCALE 1" = 10 k.s.i
O Experimental Results 1" = 2G"
FIGURE 7.5 PRESSURE VESSEL ANALYSIS - DIRECT STRESS RADIAL TO THE MANHOLE AT 40 PSI PRESSURE
124
7.2 Analysis for Local Stress Concentrations
The analysis using the shell ele:nent gave a clear picture of
the stress conditions around the opening. From the membrane stresses
and bending moments obtained by the analysis, the stress variation
across the thickness of the cover plate and other parts of the struc
ture were calculated. However, along the line of intersection of the
pressure vessel wall and the manhole the stresses and moments obtained
by the shell analysis represented the average conditions only. This
was because of the cibrupt discontinuity in the shell geometry along
this line. To determine the precise nature of the stress variation
across the thickness of the plates at the intersection and also the
stress concentration at the corners and fillets, a detailed analysis
of a small portion of the structure was done by the three dimensional
finite element analysis. Two shell elements, one of which is part of
the main body of the pressure vessel and the other which is part of
the manhole, were analyzed using the octahedral elements explained
in sections 4.8 and 4.9. The shell elements analyzed are those
marked El and E2 in Figure 7.2. The scheme used for discretization
into finite elements is as shown in Figure 7.6. The boundary forces
and displacements for this analysis were those obtained from the
shell analysis. Figures 7.7, 7.8, 7.9, and 7.10 show the results of
this analysis.
Figure 7.7 shows the hoop stress distribution at the inside
and outside surface of the coverplate at the junction along the verti
cal section. The experimental result for the stress at the inside
125
corner of the coverplate and manhole intersection is also marked in
the figure. It can be seen that the analytical result is very close
to the experimental result. Figure 7.8 shows the hoop stress distri
bution across the thickness of the coverplate along the lines join
ing the nodal points. These lines are drawn parallel to those joining
the ncxies in Figure 7.6 which shows the element subdivision and hence
are inclined to the horizontal. The stress variation is almost linear,
Figure 7.9 shows similar plots for the stresses across the thickness
of the manhole plate. Here again the stress distribution is almost
linear along the inclined lines joining the nodal points. Figure 7.10
shows the plot of the radial and tangential stresses along the inside
edge of the cxjverplate and manhole intersection.
The above results give a detailed picture of the stresses at
the corners of the intersection and also the distribution of stresses
across the thickness of the plates. Similar results ccin be obtained
for all regions of interest by the three dimensional analysis.
126
"GCmE 7.6 PI«x^ , ^ ^ IDEALIZATION FOR THREE DIMENSIONAL ANALYSIS
127
141 157
vo
^il21^
«IH
& 101
pq A
Q O Z
81^
137
.. OUTSIDE SURFACE
117
STRESS AT THE OUTSIDE SURFACE
97
INSIDE SURFACE STRESS AT THE INSIDE SURFACE
EXPERIMENTAL RESULT
8 10
HOOP STRESS k.s.i.
12 14
FIGURE 7.7 HOOP STRESS DISTRIBUTION ON THE COVERPLATE AT THE JUNCTION
128
t •145 '149
—o~—
153 IT 157
137
117
NODAL NUMBERS ( Refer Figure 7. 6 )
3 i n . THICK
SCALE 1" = 20 k . s . i .
r c ^ X ^ - - B ^ I 0 N ^ ^ e ^ 3 S THE SICKNESS OP
129
77 NODAL NUMBERS ( Refer FIGURE 7.6)
o H X in
G -H
in
CM
X
SCALE 1" = 20 k.s.i,
FIGURE 7.9 HOOP STRESS DISTRIBUTION ACROSS THE THICKNESS OF THE MANHOLE PLATE AT THE JUNCTION
130
NODAL NUMBERS ( Refer Figure 7.6 )
TANGENTIAL STRESS
RADIAL STRESS
SCALE 1" = 10 k.s.i.
FIGURE 7.10 STRESS DISTRIBUTION AT THE INSIDE EDGE OF THE JUNCTION
131
7.3 Results of Dynamic Analysis
The pressure vessel junction was analyzed for an impact load.
The nature of the load was a pressure of 100 p.s.i. on the flange of
the manhole for a duration of 0.002 seconds. The time interval for
the analysis was also chosen as 0.002 seconds. A 3 value of 1/4 and
a damping coefficient of 0.01 was used in the analysis. The boundary
conditions prescribed were the same as for the static analysis.
Table 7.3 shows the results for the displacement response at two
typical points in the pressure vessel - nodes 28 and 64 in Figure 7.2.
Node 28 is at the junction between the manhole and the coverplate,
and node 64 is at the junction between the coverplate and the bcxiy of
the pressure vessel. Figures 7.11 and 7.12 show the plots of these
values. Figure 7.13 shows the propagation of the displacements with
time. The section chosen is the horizontal semicircle joining the
ncxies 36, 118, and 128 in Figure 7.2. The u displacements at the ncxies
£Llong this section at three instants of time - 4, 10, and 16 times
the time interval of 0.002 seconds - are plotted. Figure 7.14 shows
the vauriation of the hcx p stress at node 28 with time.
Since the dynamic analysis required a considerable amount of
computer time, the analysis was limited to a few time steps. The
increase in computer time was mainly ciue to the use of auxiliary
storage units which became necessary due to limited core storage.
However, the results indicate the applicability of the finite element
methcxi for solving pressure vessel problems under dynamic loading
conditions using the step by step integration technique.
TABLE 7.3
DYNAMIC ANALYSIS OF A PRESSURE VESSEL
132
TIME STEP
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
NODE
3 u X 10 inches
1.5537
5.0522
8.1163
10.3343
12.3576
14.1714
15.8528
17.4009
18.4160
20.1772
21.4238
22.5856
23.6748
24.6964
25.6604
26.5726
28
5 w X 10
inches
-1.0849
-1.9814
-2.6117
-6.0527
-10.4343
-14.1324
-17.4348
-20.0128
-22.0695
-23.5578
-24.6016
-25.2245
-25.5078
-25.4900
-25.2288
-24.7621
NODE
3 u X 10
inches
1.3950
4.7867
7.9741
10.2920
12.3949
14.2776
16.0103
17.6007
19.0712
20.4301
21.6912
22.8630
23.9561
24.9786
25.9395
26.8467
64
5 w X 10
inches
0.5140
2.0456
1.6588
-2.5975
-7.5949
-11.9086
-15.6887
-18.6707
-21.0381
-22.7780
-24.0126
-24.7862
-25.1813
-25.2494
-25.0495
-24.6276
133
Ui
(C
tu o Pi a m
H1
Vi Vi
D cn CO
s Cu
Vi yi Vi
o
Q
g D CJ H
I O (saqouT) •"•
{ Z'L ajnfix^ aa jan ) 8? 3aON iV aSNOdSaa
134
^q a Vi Vi
Vi Vi
cu
O
cn H
cn
u Q
CM
r-i •
H EL4
(saqouT) ( 2 * / , a jn f i j j j a j a ^ ) i g aaON iV aSNOdSETa
cn
135
CM
0) U
•H
U
pq CQ
i o z
§ H i I P4
cn cn
cn cn
g Pt
o cn H cn
u
Q
cn
8
(saqouT) 01 X n
CM
136
00
vo
H
tu O
a m
CM
o CM
o
T - s - d SSaHJiS dOOH
D
CHAPTER VIII
CONCLUSIONS AND RECOMMENDATIONS
8.1 Conclusions
A rational methcxi is presented for the analysis of stresses
around openings in pressure vessels using the finite element method.
To select a suitable element for the pressure vessel analysis
a detailed study was made of the different elements available. Taking
into consideration the capacity of the available computer and the time
required for an analysis, the quadrilateral shell element developed
by Johnson (23) was found to be suitable.
A computer progrcim was developed specifically for the pressure
vessel amalysis. The accuracy of the program as well as that of the
element used was checked by solving some example problems on plates
amd shells and comparing the results with those of other investigators.
The results obtained using the quadrilateral shell element ccxnpare
very well with those using higher order finite elements and also
other methcxis of analysis such as the finite strip method (76, 77).
A comprehensive stress analysis of a pressure vessel junction
was done in two steps. The first used a shell element which gave the
general nature of the stress distribution at the junction. The second
was an amalysis for the local stress concentrations using three dimen
sional elements and the results of the shell analysis for boundary
conditions.
The results of the finite element analysis were compared with
137
138
those of an experimental investigation. The finite element results
were slightly higher than those of the experiment. The percentage
difference was less than 5% for 80% of the results. There were no
factors which could cause significant local errors in the finite
element results. However, the above statement may not be true for
the experimental results since each strain gage operates independently.
The close agreement between a gocxi percentage of the finite element
results and the experimental results indicates that the finite ele
ment methcxi can be applied confidently for the analysis of pressure
vessel problems.
The finite element method was applied to the analysis for
dynamic loads CLISO. Nevmiark's 3 parameter methcxi was used for the
step by step integration of the equations of motion.
An example problem of a rectangular plate subjected to a
dynamic load was analyzed. Very gocxi agreement was obtained between
the results of the finite element analysis and the availaJole solution
for the example problem. The percentage error between the results
was less tham 4%.
The pressure vessel junction was analyzed for an impact load.
Preliminary results obtained indicate that the finite element method
can be applied to dynamic load cases also. Detailed dynamic analysis
of the pressure vessel junction was not possible due to restrictions
of the computer facilities.
139
Even though the example chosen for the analysis was a circular
opening, there is no restriction for the analysis by the finite element
methcxi as regards goemetry or material properties. Any other shape
of opening amd also oblique junctions can be amalyzed with equal ease.
The methcxi is equally appliceO^le to other pressure vessel
components such as closure plates, conical heads and reducers, tapered
tramsition joints, etc. The local stress concentrations can be deter
mined by the three dimensional analysis.
Finally, dynamic problems, thermal elasticity problems, elastic
plastic problems and creep problems of pressure vessels, for which
other types of solutions may be difficult, if not impossible, can be
solved by the finite element method with relative ease.
8.2 Recommendations for Further Research
The dynamic analysis in the present investigation was done
using Newmaurk's 6 parameter methcxi. The dynamic analysis can be per
formed by using the conventional modal analysis, by computing all the
normal modes of vibration and the corresponding natural frequencies.
The finite element method using the quadrilateral shell element
may be extended to thermal problems in pressure vessels.
LIST OF REFERENCES
1. Nichols, R. W., Pressure Vessel Engineering Technology , Elsevier Publishing Company Limited, New York, 1971.
2. Leckie, F. A., and Penny, R. K., " Stress Concentration Factors for the Stresses at Nozzle Intersections in Pressure Vessels ", Welding Research Council Bulletin, No. 90, 1963.
3. Bickel, M. B., and Ruiz, C , Pressure Vessel Design and Analysis , Macmillan Publishing Company, New York, 1967.
4. Waters, E. O., " Stresses near Cylindrical Outlet in Spherical Vessel ", Welding Research Council Bulletin, May, 1964.
5. Palmer, A. C , " Direct Design-Technicjue for Pressure Vessel Intersections ", ASME Prcx:eedings, First International Conference on Pressure Vessel Technology, Netherlands, SeptemberHDctober, 1969.
6. Pan, K. C , " Stress amd Displacement Analysis of a Shell Intersection ", Journal of Engineering for Industry, ASME, May, 1970.
7. Lind, N. C , " Elastic Shell Analysis of the Stress Concentration of a Pressurized Branchpipe Connection ", ASME Proceedings, First International Conference on Pressure Vessel Technology, Netherlands, September-October, 1969.
8. Maye, R. F., and Eringen, A. C., " Further Analysis of Two Normally Intersecting Cylindrical Shells Subjected to Internal Pressure ", Nuclear Engineering Design, Volume 12, No. 3, 1970.
9. Jones, N., and Hansberry, J. W., " Theoretical Study of Elastic Behaviour of Two Normally Intersecting Cylindrical Shells ", Journal of Engineering for Industry, ASME, August, 1969.
10. D. H. van Campen, " Mechanical and Thermal Stresses in Cylinder to Cylinder Intersections of Equal or Nearly Ecjual Diameters ", ASME Proceedings, First International Conference on Pressure Vessel Technology, Netherlands, September-October, 1969.
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