CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
NONLINEAR STATIC STRUCTURAL ANALYSIS
OF CYLINDRICAL SHELL ECCENTRIC JOINT
A graduate project submitted in partial
satisfaction of the requirements for the
degree of Master of Science in
Engineering
by
Harry T. Otsuki
January, 1981
The Graduate Project of Harry T. Otsuki is approved:
~essor
G1~sp1e
California State University, Northridge
ii
ACKNOWLEDGMENT
The author is grateful to his advisor,
Or. Tung-Ming Lee, and the committee members,
Professors
E.S. Gillespie and S. Gadomski for
the valuable guidance given during the completion
of this project.
Special thanks to my wife, Kathryn, for
reading the entire report, for making a number
of suggestions toward improving the presentation
of the material, and for assisting in the preparation of the report.
iii
TABLE OF CONTENTS
Page
Acknowledgement.
i i;
Abstract . . . .
xi
I.
INTRODUCTION
1
II. THEORETICAL SIMULATION TECHNIQUES
A.
FINITE ELEMENT METHOD ..
4
B.
APSAC COMPUTER PROGRAM .
6
C.
NONLINEAR ANALYSIS TECHNIQUES
6
1.
Material Nonlinearity with Small Displacement and Small Strain
2.
III.
4
.
7
Geometric Nonlinearity - Large Displacement
16
D.
NONLINEAR EQUILIBRIUM EQUATION . .
E.
TWO DIMENSIONAL SPECIALIZATIONS OF ELASTICITY
F.
FINITE ELEMENT MODELS. .
20
G.
RIGID BODY ANALYSIS
21
.
18
.
19
STRUCTURAL VERIFICATION TEST .
32
A.
TEST PLAN
32
B.
TEST SEQUENCE
. .
33
IV.
RESULTS . .
40
V.
DISCUSSION
69
iv
Page
VI.
VII.
....
SUMMARY AND CONCLUSIONS
REFERENCES.
71
73
VIII. APPENDIXES
75
A.
DESIGN CRITERIA
76
B.
MATERIAL PROPERTIES .
80
v
LIST OF TABLES
Page
Table
I.
II.
III.
Summary of Chamber Pressure Parametric Study .
53
Summary of Critical Cyclic Strain Range
60
Material Properties-Inconel 718 .
81
vi
LIST OF FIGURES
Page
Figure
1.
Kinematic Hardening Behavior Biaxial Stress
State . . . . . . . . . . . . . . . . . . . .
2.
13
Kinematic Hardening Behavior One-Uimensional
Stress State
. . . . . . . . . .
13
3.
Kinematic Hardening under Constant Load
14
4.
Kinematic Hardening under Constant Displacement .
14
5.
Bilinear Stress Strain Curve
15
6.
Normalized Stress Strain Curve
15
7.
Basic Iteration Procedure . . . .
16
8.
Load-Displacement Nonlinear Curve
18
9.
Stress-Strain Nonlinear Curve . .
18
10.
Preliminary Design - Weld Joint
24
11.
FinalDesign-WeldJoint.
25
12.
Test Specimen - Preliminary Design
26
13.
Test Specimen -
27
14.
Finite Element Grid Plot- Preliminary Weld Design
28
15.
Finite Element Grid Plot - Final Weld Design
29
16.
Finite Element Grid Plot - Test Specimen Pre-
Final Design . . .
liminary . . .
30
vii
p •
LIST OF FIGURES
(continued)
Page
Figure
17.
Finite Element Grid Plot- Test Specimen Final
31
18.
Test Loading Setup . . .
34
19.
Test Data Recording System
35
20.
Failure Mode - Test Specimen - Preliminary
36
21.
Strain Gage Plots- Test Specimen Preliminary
37
22.
Test Specimen - Final Design
38
23.
Failure Mode - Test Specimen Final
39
24.
Deformed Shape - Prelininary Design Large Deflection - Plastic .
43
25.
Limit Pressure - Elastic - Sigma-Y Plot
44
26.
Limit Pressure- Elastic - Eff. -Stress Plot
45
27.
Limit Pressure - Elastic- Stress along EB Weld
46
28.
Limit Pressure - Elastic - Strain along EB Weld
47
29.
Limit Pressure - Plastic -
Plot
48
30.
Limit Pressure- Plastic- Eff.-Strain Plot
49
31.
Limit Pressure - Plastic - Stress along EB Weld
50
32.
Limit Pressure - Plastic - Strain along EB Weld
51
33.
Chamber Pressure Parametric Study - Preliminary
34.
Eff.-Stre~s
Design . . . .
52
Limit Pressure - Plastic - Eff.-Stress Plot
54
viii
@
LIST OF FIGURES
(continued)
Page
Figure
35.
Limit Pressure - Plastic - Eff.-Strain Plot . .
55
36.
Limit Pressure - Plastic - Stress along EB Weld
56
37.
Limit Pressure
57
38.
Unloaded - Stress along EB Weld
58
39.
Unloaded - Strain along EB Weld
59
40.
Ultimate Pressure - Plastic - Eff.-Stress Plot
61
41.
Ultimate Pressure - Plastic - Eff.-Strain Plot
62
42.
Equivalent Limit Load - Plastic - Eff.-Stress
Plastic
Strain along EB Weld
Plot Test . . . . . . .
43.
63
Equivalent Limit Load - Plastic - Stress along
....
EB Weld - Test - Preliminary
44.
Equivalent Limit Load
.
.......
. ......
48.
66
Equivalent Limit Load - Plastic - Stress along
EB Weld - Test - Final . . . . . . . . .
47.
65
Plastic - Eff.-Stress
Plot - Test
46.
64
Equivalent Limit Load - Plastic - Strain along
EB Weld - Test - Preliminary
45.
..
67
Equivalent Limit Load- Plastic -Strain along
EB Weld - Test - Final . . . .
68
Inconel 718 Wrought - Stress - Strain Diagram . .
82
xix
'
LIST OF FIGURES
{continued)
Figure
Page
49.
Inconel 718 EB Weld - Stress - Strain Diagram
83
50.
Inconel 718 Wrought - Low Cycle Fatigue
84
51.
Inconel 718 EB Weld- Low Cycle Fatigue .
85
X
ABSTRACT
NONLINEAR STATIC STRUCTURAL ANALYSIS OF CYLINDRICAL
SHELL ECCENTRIC JOINT
by
Harry T. Otsuki
Master of Science in Engineering
This report presents the theory, modeling techniques and
results of nonlinear analysis of cylindrical pressure vessel
shell structure which contains eccentric weld joints.
element was selected for this study.
The finite
The three dimensional
eccentric half shells weld joint structure reduced to two dimensional problem of elasticity was modeled using a two dimensional
structure analysis computer program.
A concise presentation has
been made of the methods and techniques of the nonlinear finite
element analysis.
A discussion of the results has been included
to guide in the plastic or ultimate strength design and analysis
of eccentric shell joint within the design requirements of weight
and space.
The finite element method resulted in an accurate simulation
of the eccentric shell joint structure.
xi
The technique of two
dimensional specialization of elasticity and modeling one quarter
of eccentric shell joint with adjusted boundary conditions at the
planes of symmetry resulted in enormous savings in engineering and
computer solution time.
The analytical solution was verified by the simulated eccentric weld joint static test program.
Good agreements were obtained
between the test and numerical results.
xii
SECTION I.
INTRODUCTION
All phenomena in solid mechanics are nonlinear.
In many
engineering applications, it is practical and adequate to use
linear formulations of problems to obtain engineering solutions.
On the other hand, some problems definitely require nonlinear
analysis if realistic results are to be obtained to meet design
requirements.
The unexplained premature failures of the rocket
engine pressure chamber eccentric welds were presumed to have
resulted from excessive plastic deformations that were not considered in the rigid body elastic analysis.
On this basis, a
more rigorous nonlinear analysis which includes the post yielding
and large deflection behavior of structures are required to confirm
the presumed cause of failure and to insure adequate redesign.
This report presents the nonlinear solution techniques and
methods employed to verify the presumed cause of failure and to
insure adequate engineering redesign and analysis of the pressure
chamber eccentric weld joint structure.
The redesign goal is to
locate the point of yielding and the point of ultimate failure at
locations other than the weld and to accomplish this redesign without weight impact.
The finite element method of solution has been
selected for the detailed structural modeling of the eccentric weld
joint.
With this technique, the overall structural geometry is
1
2
discretized into various elements and an approximation to the
actual solid continuum is made.
The pressure chamber is a long
cylindrical body whose geometry and loading do not vary significantly in the longitudinal direction.
Problem involving a body
whose geometry and loading do not vary in the longitudinal direction are referred to as plane strain problems.
In these problems
the dependent variables can be assumed to be functions of only the
x and y coordinates (when z - axis is taken as longitudinal direction.). Mathematically, this reduces the problem to a two dimensional elasticity problem.
The solution of a three-dimensional
structural analysis problems is impractical because of the large
amount of computer time required.
However, the analysis of two-
dimensional systems can be readily solved with reasonable amount
of computer time.
Therefore, this analysis is confined to the middle
section of the chamber some distance away from the ends that satisfies the plane strain assumption of constant or zero strain in the
longitudinal direction.
A computer program,
11
Finite Element
Axisymmetric and Planar Structural Analysis with Loading and Creep
Duty Cycles,
11
APSAC, (Ref. 1) is chosen for the study.
The tech-
niques for nonlinear analysis used by this program is presented in
detail.
Also, the special considerations of cost and design re-
quirement that must be made prior to and during the evaluation of
3
this type of structural system are discussed in general form.
The results of the finite element analysis and tests and
the calculated low cycle life of the preliminary and final design
of the eccentric shell joint are included in this report.
This
project emphasizes the method employed in the aerospace industry
to insure adequate engineering design and analysis of structures.
It is believed that this report may prove useful to those
who wish to acquaint themselves with some of the various modeling
techniques available.
It may also be useful in the initial phases
of similar engineering problems to suggest a solution procedure
and design philosophy.
SECTION II.
A.
THEORETICAL SIMULATION TECHNIQUES
Finite Element Method
The finite element method is a powerful, numerical solution
tool which may be used when dealing with complex engineering problems.
A problem with an infinite number of degrees of freedom
may be reduced to one with finite degrees of freedom by 11 discretizing11 the continuum and applying the numerical method.
Any
structure may be considered as a series of elements that have
known load-deformation characteristics and are interconnected at
a finite number of nodes.
The conditions of equilibrium and
compatibility are applied to the structure.
The direct stiffness
method is then employed to formulate the element stiffness matrices.
The d)rect stiffness method assumes that a function characterizing the internal displacements of an element at any point
can be uniquely determined by the nodal displacements of the
element.
Then the internal strains, and the internal stresses
using Hooke•s law, are defined in matrix form.
The principle of
virtual work may be applied to equate the internal and external
work done on the element during a virtual displacement.
.
load-deflection relationship for any element,
ness matrix can then obtained.
From the
the element stiff-
The system stiffness matrix
is then assembled using the techniques of matrix structural
4
5
analysis.
The details of the method are illustrated in the
literature (References 2-16).
The modified Iron's quadrilateral element is used for the
finite element approximation of solids.
The continuous structure
is replaced by a system of elements which are interconnected at
nodes.
The element is composed of four corner nodes and a fifth
node at the center.
The element has constant strain at the bound-
aries and a complete linear variation of strain within the quadrilateral.
The shear energy constraint on the flexural response
mode is removed.
This quadrilateral accurately represents
structures that have flexural-type displacements.
As most struc-
tures have zones where flexural-type displacements occur, the use
of Iron's quadrilateral minimizes the chance for gross errors
resulting from not using a fine enough element grid.
The advantage of the finite element method as compared to
other numerical approaches are numerous.
The method is completely
general with respect to geometry and material properties; complex
bodies composed of many different materials are easily represented.
Displacements or stress boundary conditions can be specified at
any point within the finite element system.
Mathematically, it
can be shown that the method converges to the exact solution as
the number of elements is increased; therefore, any desired degree
of accuracy can be obtained.
6
B.
APSAC Computer Program
The APSAC computer program has the capability to analyze non-
linear problems and where plasticity must be considered during a
duty cycle.
The duty cycle is represented by a series of loading
increments.
The incremental load formulation accumulates the
prior plastic strain and yield surface shift.
crement is solved as an inelastic problem.
Each loading in-
The load increment is
a proportional loading with respect to the element yield surface
shifted center.
The proportional load increment is solved with
a secant modulus solution referenced to the shifted yield surface
center.
In this method, the problem is solved by recursive itera-
tion where a series of elastic solutions are made with a secant
modulus representation of the stress-strain law.
The final
solution is obtained wherein the stress and strain results are
consistent with the appropriate secant modulus description.
The inelastic representation of materials is modeled as an
isotropic material with a bilinear stress-strain curve.
The Von
Mises• yield criterion and kinematic strain hardening are used
in the formulation.
C.
Nonlinear Analysis Techniques
In the displacement method of finite element analysis, non-
linearities occur in two different forms.
The first is material
7
or physical nonlinearity, which results from nonlinear constitutive
laws.
The second is geometric nonlinearity, which derives from
finite changes in the geometry of deforming body.
1.
Material Nonlinearity with Small Displacement and Small Strain
In the nonlinear material behavior, use is made of the same
basic conditions of equilibrium and compatibility as those in the
elastic solutions.
The major difference in the two solutions is
the stress-strain relationship.
In the inelastic problem, the strain is no longer proportional to the stress since each strain component contains a linear
elastic component and a plastic component not linearly related.
The method used for the plastic flow problem is the incremental
theory.
The incremental theory relates only the increment of strain
to the increment of stress in a given stress state.
path can be considered.
Hence any load
The analysis technique used in the program
is the total deformation theory using the secant modulus approach.
The load variations is accounted for by dividing the problem into
a series of proportional loading increments.
The strain history
of prior plastic strains and yield surface shift are considered as
initial conditions for each succeeding load increment.
The incremental stiffness method is used by the program to
8
solve inelastic problems by dividing the loading into a series of
increments.
The final stress and strain states are based on the
accumulation of the results of the individual loading increments
where the stiffness of the structure and plastic strains are based
on the current stress and strain state.
In order to utilize this
technique, stiffness formulation that relates the incremental
stresses to the incremental strains have to be developed.
The development of this relationship require a material model that represents a work hardening material.
The material is
described by 1) a yield condition that defines when plastic flow
first starts, 2) a constitutive relationship - flow rule - that
relates the plastic strain increment to the total stress state
and the increment of stress, and 3) a work hardening rule that
specifies how the yield condition is modified during plastic flow.
This analysis uses the Von Mises yield condition, Prandel-Reuss
flow rule, and kinematic strain hardening.
The Von Mises yield
surface relates the components of stress to a yield stress
magnitude that is a function of temperature.
The flow rule states
that the strain increment vector lies on the exterior normal of
the yield surface at the stress point.
The kinematic work hard-
ening rule shifts the yield surface during plastic flow while
maintaining its original size and shape.
The yield surface be-
9
comes temperature dependent by normalizing the yield surface shift
and size to the temperature dependent yield stress.
These criteria
are limited to materials that are initially isotropic.
The kinematic hardening theory predicts an ideal Bauschinger
effect for completely reversed loading conditions since the magnitude of the increased yield strength in one direction results in
a decreased yield strength of the same magnitude in the reverse
direction.
The kinematic hardening behavior assumes that during
plastic deformation, the yield surface translates as a rigid body
in stress space, maintaining its size, shape and orientation.
The
theory with the Von Mises yield curve is shown for a simple example in Figure 1 for a biaxial stress state.
When the load is
increased above the yield point at (1) causing plastic flow (2)
the yield surface translates.
When unloading the completely re-
versed stress state, yielding starts at point 3 and plastic flow
occurs from 3 to 4.
If the loading has been one dimensional (only
one component of stress}, the resulting stress strain conditions
are as depicted in Figure 2.
The importance of the use of the
kinematic hardening is shown in Figure 3 for two common problems
that occur in engineering parts limited by low cycle fatigue.
The
first example is completely reversed loading with cyclic plastic
strains.
10
The second example is constant strain cycling with plastic
strain.
The kinematic theory will properly depict these two
cases (Figure 4).
The kinematic theory does not account for cyclic strain softening or strain hardening effects.
Many materials'stress strain
characteristics change after several cycles of strain-either by
lowering their yield strength (softening) or increasing their
yield strength (hardening).
In general, the solution to a proportional load increment
requires an iterative solution.
The stresses and strains of a
particular element are dependent on adjacent elements and initial
stress state.
The proper value of Esec and Ysec cannot be initial-
ly determined.
This is overcome by iterating the solution - run-
ning the case several times - and adjusting the value of Esec and
Ysec
for all elements.
When values used for generating the stiff-
ness of the element are consistent with the resultant calculated
increments of stress and strain, a converged solution is obtained.
The elastic solution is used as the initial solution.
For the suc-
ceeding increments, a procedure is used to estimate the value of
Esec which is used to calculate Ysec.
The material model used in the program for inelastic behavior
is an isotropic bilinear material.
The material is characterized
11
by the standard elastic properties - E, y, G - yield stress, and
modulus ratio (XN).
The modulus ratio is defined as the ratio of
the slope of the stress - strain curve above the yield stress to
the elastic modulus E (See Figure 5).
The inelastic analysis is handled through a kinematic strain
hardening approach that uses the secant modulus relative to the
shifted yield center.
The secant modulus relationships used in
the analysis are based on the uniaxial stress-strain curve, not
the effective stress-effective strain curve; i.e., XN is based on
the uniaxial curve.
The actual calculations are based on the
normalized stress-strain curve shown in Figure 6.
The effective stress calculation that is compatible with the
normalized form can be written as
=
1
[ (0'1 - 0'2) 2 + (0'2 - 0'3) 2 + (0'3 - 0'1 )2
J~
Where O'y is the yield stress and a 1 , a 2 , a 3 , are the principal
stresses with respect to the shifted, normalized yield surface
center.
The basic iteration procedure used by the program to obtain
a consistent set of effective stresses and secant moduli values for
the finite elements is shown for a typical element in Fig. 7.
only input material property is the modulus ratio XN.
The
The secant
12
modulus parameter is the EPS quantity.
EPS
1
~
For the first iteration
1.0 and an elastic solution is obtained.
During the stress calculations the normalized stress, --&IN is
calculated based on the element yield stress with respect to the
shifted yield surface at the appropriate temperature.
If ""'ltN>l,
then the element has yielded and an estimate of the secant modulus
is made.
This is accomplished by determining the normalized strain
at 1 and calculating the stress-strain curve intercept at 2.
-
The
secant modulus EPS 2 value is then calculated as the ratio oflfN 2
+"EN 2 . Using this EPS 2 value a secant- nu, (Ysec)2, is calculated.
-
With these two quantities for each element the elastic problem can
be resolved for the second iteration.
The basic iteration is shown
through the third iteration of Figure 7.
13
q
2
Yielding first occurs at point 1.
Unloading starts at point 2.
Yielding occurs at point 3 for kinematic hardening and at point 4
for isotropic hardening.
FIGURE 1
FIGURE 2
KINEMATIC - BIAXIAL
KINEMATIC - UNIAXIAL
14
P,
u
P,
u
2
1
6
b., f.
2A
4,8
2A
b., f.
~
5
5
L____ ~~
~
---.- -
...L___
...__
CONSTANT LOAD
MAGNITUDE (:t a )
~
1~:-----1
CONSTANT DISPLACEMENT
MAGNITUDE (:tJ )
Kinematic Strain Hardening Develops a Hysteresis loop in the First
Loops.
1~
Kinematic Hardening Has the Same Stress Range From Tensile Yield to
Compressive Yield. The Range is Equal to Twice the Elasti_c Yield.
FIGURE 3
FIGURE 4
CONSTANT LOAD
CONSTANT DISPLACEMENT
15
a
~- lu
actual u_rliaxial
curve
L
.._ •
represenvavJ.On
~------L_------~~---------------(
str-ain
ly
FIGURE 5
BILINEAR STRESS STRAIN CURVE
(J
al~
"
=(J
y
on
u.Tli ::.xic.l curve
FIGURE 6
NORMALIZED STRESS STRAIN CURVE
16
p '
~
Q)
1.0-
~
+-'
V)
""0
Q)
.....N
r-
ro
E
~
Shifted center of yield
surface
0
z
~-------·--------------·----------------__,.,-lN
0
Initial stress-strain free reference system.
FIGURE 7 BASIC ITERATION PROCEDURE
If this technique were continued, a converged solution would
ultimately be reached where the calculated effective stress would
fall on the stress-strain curve.
2.
Geometric Nonlinearity- Large Displacement
The principal effect of large displacements is that the changes
of geometry brought about by the displacements may no longer be neglected.
The iterative method is used by the program for the large
displacement analysis.
The iterative method for large displacements
consists of applying the total load and using the resulting displacements to revise the locations or coordinates of the nodal points at
17
each cycle of iteration.
stiffness and loads.
The new geometry is used to recompute the
Hence, at every stage the element character-
istics are revised and a linear analysis is performed.
If the
strains are small, the equations for this process may be written
- ll{qi}= {Qi - 1}
J
'
1} = [ Ti
J
1 T { Qe}
lki ,. 1J = rT,. _ 1 : T~keJllT,. _ 1]
L
J l
The process is repeated until the displacements no longer
change significantly.
This procedure is applicable to problems
where relatively small changes in geometry are encounted that
affect the loading.
Good agreement has been obtained from test
of an eccentric pressure vessel weld joint.
stiffness matrix
displacement matrix
load matrix
transformation matrix
D.
Nonlinear Equilibrium Equation
The nonlinear equilibrium equation for a structural element
may be written in matrix form.
18
D.
Nonlinear Equilibrium Equation
The nonlinear equilibrium equation for a structural element
[k]{q}={Q}
where the nonlinearity occur in the stiffness matrix[~, which
is a function of nonlinear material properties C(a).
It can be
indicated that the material parameters in [KJ are no longer constants by writing
The symbolic nonlinear relationship between{Q} and {q} is
shown in Figure 8.
Figure 9 shows the nonlinear stress-strain
curve corresponding to the load, {Q}, and displacement,{q}, in
Figure 8.
It is on the basis of this stress-strain or con-
stitutive law that determines the variable matrix C( a)
for the
nonlinear analysis.
Q
a
FIGURE 8
FIGURE 9
LOAD-DISPLACEMENT CURVE
STRESS-STRAIN CURVE
19
E.
Two Dimensional Specializations of Elasticity
It is costly and time consuming to perform finite element
analyses of three dimensional problems in solid mechanics.
How-
ever, certain classes of three dimensional problems reduce to the
analysis of two dimensional systems by considering their geometry
and loading configurations.
The geometry and loading of the overall pressure vessel body
does not vary significantly in the longitudinal direction.
Prob-
lems involving a body whose geometry and loading do not vary in the
longitudinal direction are referred to as plane strain problems.
In these problems the dependent variables can be assumed to be
functions of only the x andy coordinates.
Mathematically, this
reduces the problem to a two dimensional elasticity problem.
The state of generalized plane strain is characterized by the
strain components
Exx
t 0, Eyyt 0, Ezz=CONSTANT
Yxy t
NowEzz
.1~z
0,
Yyzt 0, Yxz = 0
= constant,
so elastic strain increment~z
= 0.
Since
= 0, the stress .1~z can be expressed in terms of .1~x and
.1~Y as
.10'zz =v(i1aXX + .10'yy )
20
~nd
AUxx, AUyy, and 67Xyare the only dependent stress variables.
The constitutive law for elastic isotropic material reduces to
=
E
1-)' ,..
0
l' 1-)'
0
(1 + y)(l - 2y)
The equations given above results in substantial reductions of
the equations from the general three dimensional stress problem.
F.
Finite Element Models
The finite element technique was selected for the computer
simulation of the nonlinear behavior of the pressure vessel shell
eccentric joint structure.
The computer program used for the in-
vestigation was APSAC (Reference 1).
The simulation models used
here were developed with a nodal geometry, and material properties
typical of the actual structural system.
program assembled the system stiffness.
From these, the computer
The solution technique
was relatively complex and highly automated.
The actual shell structure whose nonlinear behavior was investigated was a rocket engine combustion chamber designed for containment of chamber pressure.
The chamber was constructed of nickel
base precipitation-hardenable inconel 718 alloy.
21
The preliminary and final design concept of the eccentric shell
joint are presented in Figures 10 and 11.
The test specimen
simulating the preliminary and final design are shown in Figures
12 and 13.
Four different finite element models were employed in the
analytical investigation.
Two models were used to simulate the
two designs and the other two models simulated the test specimens.
Figures 14 through 17 presents the finite element geometry and the
generated meshes of the four models.
G.
Rigid Body Analysis:
Weld Joint Design - Figure 10
Geometric Properties:
Weld thickness (h) = .470 in.
Basic shell thickness (t) = .233 in.
Inner surface radius (ri) = 7.7459 in.
Weld offset (k) = .240 in.
Mean radius (r n ) = r.1 + t/2 = 7.8624 in.
C.G. of weld = h/2 = .235 in.
Moment arm (e)= (k + h/2) - (rn-ri) = .3585 in.
Section Properties:
Area (A) = h x 1 = .470 in. 2
Moment of Inertia (I) = l/12 x 1 x h3 = .01038 in. 4
22
= h/2 = .235
Distance to outer surface (c)
in.
Loads:
Chamber pressure (Pc)
= 3000
PSI.
The internal loads on the weld joint are based on the assumption that the hoop load of the basic shell due to Pc
goes through the C.G. of the weld joint.
Hoop load (P) = Pc x rn = 23590 lbs.
Moment @weld C.G. (M)
=P x
e
= 8456
in. - lbs.
Analysis:
= P/A = 50185
{ab) = + Mc/I = ~
Normal stress (aa)
PSI.
Bending stress
191400 PSI.
Combined stress@ inner radius
(~) =~
+u0 = 241585 PSI.
Combined stress@ outer radius (u0 ) =ua _ub
= -141215
PSI.
Peak stress@ inner radius= Kuai = 410690 PSI,
where Ka = 1.70 (geometric stress concentration factor by
R.E. Peterson) (Reference 17).
Material Properties:
Inconel 718
Fty = 70 KSI.
Ftu
= 108
KSI.
Discussion
The combined stresses exceed yield stress.
body is strained into the plastic region.
Therefore, the
23
The bending stress is developed by the self constraint of a
structure which is caused by an imposed strain rather than being
in equilibrium with an external load.
Minor deformation can
satisfy or reduce the discontinuity conditions which cause the
stress to occur.
Also, the local area of high stress will ex-
perience plastic deformation, the result is a redistribution of
stress which relieves stress concentration.
Therefore, the high stresses predicted by the rigid body
elastic analysis will not be experienced by the structure if egometric and plastic deformations are considered.
24
-----------
1-t
I
.470
FIGURE 10
PRELIMINARY DESIGN
WELD JOINT
25
I
f
. 07.5 ::!:. oo-r=-s_ _ l
.020 ~.010
·
-.oooR
I
__j
.1/S
.050:!: .OOS
r-· 6 10.:t.005
I
•
/,305
FIGURE 11
FINAL DESIGN
WELD JOINT
26
-
.
//
. ,4-.."' / ~
•
FIGURE 12
TEST SPECIMEN
PRELIMINARY DESIGN
WELD JOINT
27
I
r<-.t>IO:t.oo5
FIGURE 13
~i~Xls~~~~~~N
:WELD JOINT
28
FIGURE 14
FINITE EtEMENT GRID PLOT
PRELIMINARY WELD DESIGN
29
.
\
,
~.
.
..
~
}1j;Jf(
ffEl
. til
J.
h''f~.·
~:ttt/
:Jffll/1
_i$
--
:: FIGURE 15
FINITE
ELEMW~~TDGDREID PLOT
FINAL
SIGN
30
i : I i
~~,
! :
,
1
H·
I
II
I:
ill
i
I
I lI l'
I
·'
I
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; I
.
II
'
T· i T
·r+H
TITl
d:.
I
I
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Cl:~·
Tt ! T
T
, l I
I
I
·:
I
[T.I1
FIGURE 16
FINITr ELEMENT GRID PLOT
TEST SPECIMEN-PRELIMINARY
31
'
'
t
I I
Il
t I
fTn
tlTI
FIGURE 17
FINITE ELEMENT GRID PLOT
TEST SPECIMEN-FINAL
SECTION III.
A.
STRUCTURAL VERIFICATION TEST
Test Plan
Tensile Test specimen simulating the design weld joint were
tested to verify structural integrity of the weld joint as follows:
1.
Test specimen weld joint including the plastic hing relief
cut and eccentric offset geometry was identical to the design weld joint geometry.
The circular shell section was
replaced by a straight flat section for tensile loading.
The test specimen loading geometry was determined
analytically using the same finite element program and
material properties originally used to determine design
predicted strain level.
2.
The axial tensile test load was applied with a 150 K
capacity hydraulic actuartor strut with pressure regulated
by a Edison load maintainer.
3.
Test specimen were cyclic loaded at room temperature between zero and equivalent limit pressure for 60 cycles with
a hold time of 490 seconds at full load.
4.
Upon successful completion of the cyclic load test, the
test load was increased until failure occurred.
5.
Local strains and lateral deflections were determined
with 20 strain gages and 3 dial indicators as shown in
Figure 19.
32
33
6.
B.
Test set-up is presented in Figure 18.
Test Sequence
1.
Test No. 1
Test specimen (Figure 12) simulating preliminary
weld joint design was tested according to the test plan.
The test specimen failed on the first cycle at 95
percent of the schedule test load.
load is equivalent limit pressure).
(100% scheduled test
The failure occurred
at the weld joint as shown in Figure 20.
The strain gage readings are presented in Figure 21.
The dotted lines are the extension of last test readings
taken to the computer results.
2.
Test No. 2
Test specimen (Figure 22) simulating final design
was tested according to the test plan.
The test specimen successfully sustained 60 cyclic
loading.
This verified the life adequacy of the design.
The testing was continued into failure load test.
The
test specimen failed at 190 percent of scheduled test
load.
This established the ultimate safety factor of
1.90.
The failure occurred at the designed plastic
hinge area of the parent material as shown in Figure 23.
34
FIGURE 18
TEST LOADING SETUP
,~
I
Sl(S2)
S3(S4)
S5(S6)
~S7{S8)
Dl
520
519
518
03
Sl5/l \516
.... -• ._
;~
..
~
"sg
512
_._
..-
-
..L
__... ..._
I
Sl3/
514........_
•
-
'-s1o
,.
-
--••
.,.., Sll
-
I"'"
·I-'
.. r-
FIGURE 19
STRAIN GAGE LOCATIONS
w
(.)1
36
.. • /
.· .
·.
·. ~
.- 1:!- ,
.•
FIGURE 20
-FAILURE MODE
TEST SPECIMEN-PRELIMINARY
:. :,;. :___ ·...
37
=
~r.- !"-
±::~-~~.
f-'-'--~
100
.
·~ ---r- : ·
-
f-- ,.. ~
,
F----i-:-
L
-~i
· - f-
·-- r ---:.....+---'
·+--f-
r-
--+
0
2
4
6
PERCENT - STRAIN
FIGURE 21
STRAIN GAGE PLOTS AT WELD CENTER
TEST SPECIMEN PRELIMINARY
8
10
38
FIGURE 22
TEST SPECIMEN-FINAL DESIGN
39
FIGURE 23
FAILURE MODE
TEST SPECIMEN-FINAL
SECTION IV
RESULTS
The numerical results of the four finite element models used
for the simulation of the structural nonlinear analysis of the
cylindrical shell vessel eccentric joint are presented in this
section.
The finite element analysis generates a vast amount of
numerical informations.
Therefore, for rapid perusal of the
computed quantities, CRT plots were generated by the program.
Since the amount of plots obtained from the program was extensive,
only the results pertinent to this study has been included.
The
types of CRT plots presented for each model are:
1.
Undeformed and deformed element.
2.
Contour plots of stresses and strains at the
critical zone.
3.
Graphical plots of stresses and strains at the weld
joint.
The results of the large deflection computer analysis are
presented in the following sequences:
1.
Preliminary Design Model
The results of the limit pressure analysis (Figures
24 through 28} shows the elastic geometric stress con-
40
41
centration at the inner radius of the weld joint.
The
peak elastic stress of 511 KSI showed good agreement with
the results of rigid body analysis using classical stress
concentration factor in Section II-G.
The results of the limit pressure plastic analysis
(Figures 29 through 32) shows the plastic deformation effect
of strain redistribution in the highly strained areas.
The
peak strain at the inner radius of the weld joint increased
to 9.7 percent.
The results of chamber pressures versus effective
strains parametric study is presented in Figure 33 and
Table I.
2.
Final Design Model
The results of the limit pressure plastic analysis
(Figures 34 through 37) shows the critical strain of 3.37
percent in the designed plastic hinge area of the parent
material.
The maximum strain in the weld joint reduced
to 1.47 percent as compared to 9.7 percent for preliminary
design.
The results of unloaded from limit pressure plastic
analysis (Figures 38 and 39) shows this residual strains
in the body.
The calculated strain range and the corres-
ponding low cycle fatigue life is presented in Table II.
42
The results of the ultimate pressure (1.4 x limit
pressure) plastic analysis (Figures 40 and 41) shows a
maximum peak strain of 6.59 percent which is less than
the minimum tensile elongation of the material.
the result showed ultimate factor of
3.
safety~
Therefore,
1 .40.
Test Specimen Model - Preliminary and Final Design
The results of the equivalent load plastic analysis
(Figures 4Z through 47) show the stresses and strains distribution at the weld joint.
The comparison with the
design models shows good agreements.
43
FIGURE 24
DEFORMED SHAPE-PRELIMINARY DESIGN
LARGE DEFLECTION PLASTIC
44
I ~,·
LtJ ,I
-
I
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r / JIJ
.
IpEE-
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Y-Axis
~
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;:: ->.o?>:to.H
04
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6.50~:10+1) 4
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+il~
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FIGURE 25
LIMIT PRESSURE-ELASTIC
SIGMA-Y PLOT
tr
f_I~05
I
y-,
J /.
45
Y-Axis
r
04
9.83X1(J+0-4
E
·J
K
2:.47:.-<lO+DS
Q
2.25::-.:lU--t-O"':.
3.53:~10 .. 05
L
3.32~:10+1)'5
P.
:3. 74:::;.o+OS
7. 71)(1 o+O'l
s.:::;a o+ c:.5
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p
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H
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+Oc=-
FIGURE 26
LIMIT PRESSURE-ELASTIC
EFF-STRESS PLOT
1 ..
19~:to"'" 05
D
c
3.f.7X10-+04
A
46
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R
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FIGURE 27
LIMIT PRESSURE-ELASTIC
STRESS ALONG WELD
!
-~---~--'
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47
s
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p
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FIGURE 28
LIMIT PRESSURE-ELASTIC
STRAIN ALONG WELD
I
.30
48
Y-Axis
I
£.
i:.-&2~:~:~_:,...'~-.:·
~
:. ! :
:~:::.:
c -,.;~ .!.2:•<! o+ 0 ~
J 4. : ..;~:: ~ D+ o..;
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FIGURE 29
LIMIT PRESSURE-PLASTIC
EFF-STRESS PLOT
i: -.:;.~c.>:l
·M
o- ... -
E -2. 9L~y:ll~-:;..;
-.t,.::..
~~
5.:t::;i:to+
04
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• ,.,t::
49
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I
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~ -3.:.;;,-:lo- 02
r-t
r1
...:. •
-··=·
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L
f .. ~:::::-:1 c-02
FIGURE 30
LIMIT PRESSURE-PLASTIC
EFF-STRAIN PLOT
2.?:?:-<!C:-02
02
;:.• 75:r:1 0-
50
"
s
T
R
P.
I
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:
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t/~-
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ll
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f
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0
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- FIGURE 31
LIMIT PRESSURE-PLASTIC
STRESS ALONG WELD
-'-----~-_l___..L_.J..___.;._______:
..
.;..
_,
51
-. 04 f; >------.-l-,L--T--·'----l--i--+------!
._. (150
~+--+--+--+--+-+---+---+---+---+--+--+--+--+--Ti~-----~~
-. (Jf.O
IT
·I
I,
I l I I
-+--+-----T--~--+-~~~~-+-··~~--+-1
-.ou·\~,1+-j
-.
(I~·QrL--....1~~
1
• j,ll
• 2t.l
FIGURE 32
LIMIT PRESSURE-PLASTIC
STRAIN ALONG WELD
STRESS ALONG WELD
1
~
• ZC•
-+I
1
r
1
52
--
-- :-:..... ~
-
!--
----!
· :.
.-.::~~ =~~ :_~-:~:: !~
-
----
---·· ....
---
~ :..
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- ;-
--- ,.
. . !-
----;-
-
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----=-~.
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--- ..
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--.
6
PERCENT - STRAIN
FIGURE 33
E PARAMETRIC STUDY
CHAMBERP~~~i~~~ARY
DESIGN
---
'---
10
53
TABLE· I
~!mmary
of Chamber Pressure Parametric Study
....----·-···-
Chamber Press. - PSI
3000
2700
2500
2200
1400
Max .lpea k - %
9.7
8.0
6.9
4.6
2.2
Max f~ff -
5.4
4.3
3.6
2.4
1.5
%
Life Cycles
10
Deflections-in~~~f)
.022
.041
Class la - Ftu
= 108
25
62
.021
.038
.020
.036
(Inconel 718) as welded:
EB Weld
'--·-
Material Properties
16
KSI, Fty = 70 KSI
Elongation - 10%
Critical'
Point
"'
220
. 017
.027
.015
.024
54
Y-Axis
//
4.25>:10+04
D
5.12>:10+0 4
E
5.9SI.10+0<\
7.73)<:10+
H
3.-38nO+O<\
4
8.E.•J;-;Jo+0
C
6
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0
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Q
1.'5'5X10+0S
R
l.tS4XlO+OS
A
2.59.X10+0<\
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FIGURE 34
LIMIT PRESSURE-PLASTIC
EFF-STRESS PLOT
55
Y-Axis
A
1.38X10-03
B ·3.12><10- 03
C
5.05:>-:10-0 3
D
6
1.27Al0-0Z
H
1.46>:1o- 02
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LE.E.;~10-02.
K
H
2. 43X1 0-02.
0
2.. E.2n 0-0Z
P
2. 81X1 o- 02
Q
FIGURE 35
LIMIT PRESSURE-PLASTIC
EFF-STRAIN PLOT
6.'?S:·;t~..:ij3
2
1.8:0..:to-0
3. 01X1 0- 02
E
:~.9U<10-0 3
L
2.04X10-
R
:3. 20:•:1 0-02.
02
56
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D==SIGr1F:-Z
FIGURE 36
LIMIT PRESSURE-PLASTIC
FINAL DESIGN
STRESS ALONG WELD
!
E=SIG-I"iR~~
.30
W=EFF-SIG
"'
57
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H=ST~:fi-'(:;v
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FIGURE 37
LIMIT PRESSURE-PLASTIC
FINAL DESIGN
STRESS ALONG WELD
'il~
,~
I
j
lb.__~
.,K
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58
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FIGURE 38
UNLOADED
FINAL DESIGN
STRESS ALONG WELD
I
I
.30
E=SIG-MH:•:
W=EFF-SIG
59
/V
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O=STRri-Z
FIGURE 39
UNLOADED
FINAL DESIGN
STRAIN ALONG WELD
.30
P~STR~H1RX
X=EFF-STRti
60
TABLE II
Summary of Critical cyclic Strain Range
Peak St~in - %
Location
Duty*
Elem.j}{..at '1.
Cycle
Ez
Ex.
Ey
EJcy
Eeff.
1
-.1398
-2.439
3.022
3.528
3.765
-2.047
1.954
2.540
2.736
-2.296
2.893
3.245
3-545
-2.028
1.952
2.517
2.719
-.889
1.402
.046
1.349
-.684
.862
.043
.895
-.864
1.359
.044
1.309
-.684
.863
.043
.895
.o
2
206/P.M.
3
-.1398
I
4
-.1398
1
.o
2
407f\·leld
.o
3
-.1398
.o
4
f-"---
** Strain F..a.nge
206/P.M.
1-2 1 -.1398
-.392
1.068
.988
1.066
2-3
-.1398
-.249
.939
.705
.861
3-4
-.1398
-.268
.941
.728
.874
I
~X* ~ Eeff_._ave
407/Keld
- %
=
.937% - Nr
I
> lo4
l-2 I -.1398
-.205
.540
2-3
-.1398
-.180
.497
.cxn
.439
3-4
-.1398
.496
.001
.438
-:<-::-:<-
I
-.180
I
•
L1 Eeff .ave. = .451% - Nr
oo'"'.:> I
.476
> lo4
E =Strain
* Do.1ty Cycle -(1)
*
Eeff.
~*
Load - (2) Unload - (3) Reload - (4) Unload
="2/3 [<Ez
Nr values -
LCF
-
~) 2
]!
+ (Ex- Ey-) 2 + (Ey- Ez) 2 + 1.5(Exy)2
curve - Appendix B
61
Y-Axis
o• 04
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FIGURE 40
ULTIMATE PRESSURE-PLASTIC
EFF-STRESS PLOT
E
62
Y-Axis
. . . .......-"'T.,J
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... ,
~
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·-
t-.25'-:1(:- 02
r r
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:: FIGURE 41
ULTIMATE PRESSURE-PLASTIC
EFF-STRAIN PLOT
63
Y-Axis
l:,_
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:
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2. 01::1 0-tDS
~-1~:1
FIGURE 42
EQUIVALENT LIMIT LOAD-PLASTIC
EFF-STRESS PLOT-TEST
(•
..
L
64
~­
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R
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s
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s
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-
FIGURE 43
EQUIVALENT LIMIT LOAD-PLASTIC
TEST-PRELIMINARY
STRESS ALONG WELD
. I
65
•
'
. -~
'-~
;~
r_;
' - -'
FIGURE 44
EQUIVALENT LIMIT LOAD-PLASTIC
TEST-PRELIMINARY
STRESS ALONG WELD
66
~
\
\
Y-Axis
H
::· • .! •:··.,.:: (r- 0 ~
!.';~ ...:;,(:-(! 2
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.: . 0::·:.:; L-r.r;:
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FIGURE 45
EQUIVAL~NT
02
LIMIT LOAD-PLASTIC
EFF-STRESS PLOT-TEST-FINAL
\
1.J4i·:!O-o:::
7!:·>~1 O-D 2
R 4. 35>~! D- 02
2.
67
STRESS ALOtiG EB LJELD
l NP UT
EL
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FIGURE 46
EQUIVALENT LIMIT LOAD-PLASTIC
TEST FINAL
l
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68
9
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FIGURE 47
EQUIVALENT LIMIT LOAD-PLASTIC
TEST FINAL
STRAIN ALONG WELD
.
SECTION V.
DISCUSSION
The initial stress analysis presented in Section II-G was
performed using hand calculations based on rigid body elastic
analysis.
These hand calculations with simplified considerations
indicated structural inadequacy of the design and provided the
basis to define the approach and scope of further analysis.
An
adequate analysis and tests are both necessary to confirm the
structural integrity of the design.
The substitution of the two dimensional system analysis for
the three dimensional structure was primarily for economy consideration through reduction of the finite element computer solution
time.
Because of symmetry, the computer model can be reduced to
simulate one-quarter of the shell structure.
Appreciable expense
was saved by using these models, with adjusted boundary conditions,
over a full vessel shell simulation.
The computer
progra~
run
costs for a particular problem are dependent on the total number
of element used, the stiffness matrix bandwidth, and the number of
iterations and loading increment required--the increase in any one
of these variables, the higher the cost.
The finite element models required a rather fine simulation
mesh to yield accurate results at the geometric discontinuities.
One must recognize that the choice of element size, geometry and
69
70
stress-strain curve will affect the calculated magnitude of the
stress and strain concentration.
A direct comparison of the results
with a classical stress concentration factor by R.E. Peterson
(Reference 17) would serve as a tool to evaluate the validity of
the results.
Many structural considerations were given during the iterative
redesign of the weld joint.
These considerations would apply in
general to any evaluation of a weld joint.
The weld criteria--weld
joints shall be designed such that yielding initiates in parent
material rather than in the weld joint--was the prime consideration
given for the redesign.
The unknown factors that must be con-
sidered for analytical evaluations of the weld joint are as
follows:
1.
Include effects of geometric discontinuities.
2.
Include effects of allowable weld specification mismatch.
3.
Include consideration of residual stresses due to
weld shrinkage.
Hereupon, the numerical results become dependent on the
assumed numerical values assigned to the unknown factors.
The objectives of the test program were to verify the
methods and techniques that were employed for the numerical
solutions and determine the ultimate strength capability of the
structure.
SECTION VI
SUMMARY AND CONCLUSIONS
The two-dimensional system analysis of the three-dimensional
chamber eccentric weld joint structure by the plane strain finite
element models are restricted to the middle section of the chamber
some distance away from the ends that satisfies the plane strain
assumption of constant or zero strain in the longitudinal direction.
The discretization of the structural body into a series of quadrilateral planar solid elements resulted ina model which yielded
reasonable results for combined bending and stretching structural
effects.
The finite element model did require a very fine mesh at
the weld joint section to obtain both the stress and strain concentration effects.
The plane strain chamber joint finite element model employed
repres.ent one-fourth of the shell about the two plane of symmetries.
Modeling technique using the smallest representative portion of the
structure that will define the entire structure is one important key
to reducing the expensive computer solution time.
The agreement
with the test data deflections and strains at the weld joint was
fairly close, considering the complexity of the system involved.
Computer simulation resulted in a maximum peak strain of 9.7 percent
at limit load versus a test strain measurement of 10.5 percent at
limit load {allowable material elongation is 10 percent).
71
72
This lent support to the technique of modeling only onequarter of the shell structure with an adjusted boundary conditions
at the plane of symmetries.
By the simplifications of the
analytical model, the finite element method becomes economical for
analyzing complex problems.
Iterations through the application of
the finite element method, the optimum design was achieved quickly
with specific goals for weight saving and part rigidity achieved.
The critical strained area for the final iterated design was
in the region of the designed plastic hinge parent material zone.
The numerical results satisfied the strength and design requirements
as defined in the design criteria for the component.
The tests verified the structural integrity and added support
to the methods and techniques employed in the numerical analysis.
The ultimate strength test determined the ultimate strength capability and the mode of failure of the design.
SECTION VII
REFERENCES
1.
Newell, J.F., and Persselin, S.F., 11 APSAC- Finite Element
Axisymmetric and Planar Structural Analysis with Loading and
Creep Duty Cycles, .. Rocketdyne's Implementation and
Modification of Ref. 2, Nov. 1968.
2.
Wilson, E.L., and Jones, R.M., 11 Finite Element Stress Analysis
of Axisymmetric Solids with Orthotropic, Temperature-Dependent
Material Properties, .. TR-0158(s3816-22)-1, The Aerospace Corporation, San Bernardino, California (September 1967).
(Available only from the Defense Documentation Center.)
3.
Jones, R.M., and Crose, J.G., 11 SAAS II, Finite Element Stress
Analysis of Axisymmetric Solids with Orthotropic, TemperatureDependent Material Properties, .. TR-0200(S4980)-1, The Aerospace Corporation, San Bernardino, California (September 1968).
(Available from the Defense Documentation Center.)
4.
Doherty, W., 11 Stress Analysis of Axisymmetric Solids Utilizing
Higher-Order Quadril at era 1 Finite Elements, 11 University of
California, Berkeley, California, January 1969.
5.
Crose, J.G., and Jones, R.M., 11 SAAS III: Finite Element Stress
Analysis of Axisymmetric and Plane Solids with Different Orthotropic, Temperature-Dependent Material Properties in Tension
and Compression, .. The Aerospace Corporation, San Bernardino,
California (June 1971).
6.
Armen, H., Isakson, G., Pifko, A., 11 Discrete Element Methods
for the Plastic Analysis of Structures Subjected to Cyclic
Loading, .. AIAA/ASME 8th Structures Structural Dynamics and
Material Conference, Palm Springs, California, March 29-31,
1967.
7.
Bathe, K.J., Wilson, E.L., and Peterson, F.E., 11 SAP IV- A
Structural Analysis Program for Static and Dynamic Response of
Linear Systems, .. Earthquake Engineering Research Center,
College of Engineering, University of California, Berkeley,
1973.
8.
Desai, C.S., and Abel, J.F., 11 Introduction to the Finite
Element Method- A Numerical Method for Engineering Analysis, ..
~n Nostrand Reinhold Company, 1972.
73
74
9.
Cook, R.D., 11 Concepts and Applications of Finite Element Analysis - A Treatment of the Finite Element Method as Used for
the Analysis of Displacement, Strain, and Stress, .. Wiley and
Sons, New York, 1974.
10.
Zienkiewicz, O.C., 11 The Finite Element Method in Engineering
Science, 11 McGraw-Hi 11 , London, 1971 .
11.
Manson, S.S.,
Hi 11, 1966.
12.
Nadhai, A., 11 Theory of Flow and Fracture of Solids, .. McGrawHill, 1950, Chapt. 24.
13.
Smith, J.O., and Sidebottom, O.M., 11 Inelastic Behavior of
Load-Carrying Members, .. Wiley, 1965, Chapt. 3.
14.
Timoshenko, S. and Woinowsky-Krieger, S., 11 Theory of Plates
and She 11 s, .. McGraw-Hi 11 , New York, 1968.
15.
Armen, H., Pifko, A.B., Levine, H.S., Isakson, G., 11 Plastic
Analysis of Structures 11 Grumman Research Department Report
RE-380J, April 1970.
16.
Mendelson, A., 11 Plasticity, Theory and Application, 11
MacMillan Company, New York, 1968.
17.
Peterson, R.E.,
New York, 1953.
11
Therma1 Stress and Low-Cycle Fatigue, .. McGraw-
11
Stress Concentration Design Factors, 11 Wiley,
APPENDICES
SECTION VII.
75
76
APPENDIX A
Design Criteria - Rocket Engine Component
The design should reflect structural considerations from the
standpoints of optimum structural configuration, satisfying the
design criteria, and minimizing system weight.
The structural analysis requirements are satisfied by performing
load analyses to establish the structural loads criteria
and by performing strength analyses, including, as appropriate,
fatigue life analyses to fully substantiate the capability of the
component analyzed to meet the design criteria.
The strength analyses will be accomplished, when applicable
by state-of-the-art techniques such as the use of elastic-plastic
finite element analysis computer programs to achieve optimum,
lightweight designs.
A service life evaluation will be performed
and included as part of the strength analyses to verify a component's capabiiity to sustain any cyclic loads; such standard methods
as Miner's method shall be used to determine the combinoodamage.
1.
Structural Criteria
Each structural component shall be designed to the following
basic structural criteria:
Minimum Yield Factor of Safety>l .1
Minimum Ultimate Factor of Safety>1.4
77
These safety factors govern the stresses induced by all
limit loads and shall be based upon minimum guaranteed material
properties that include the effects of the component environment.
All other criteria, such as the fatigue criteria, are special
additions to the basic criteria.
The yield safety factor is the ratio of the material minimum
guaranteed yield strength at the design temperature to the maximum
principal stress.
Yielding due to secondary stresses is permitted
provided there are no deformations adversely affecting the function
of the structural elements.
Yielding due to secondary stresses is
controlled by the ultimate safety factor or fatigue criteria.
The minimum ultimate safety factor shall be maintained on the
stresses, strains, or load that would cause failure whether the
failure mode is tensile ultimate or buckling.
The ultimate factor
of safety is the ratio of the allowable load to the limit load.
Primary stress is stress developed by the imposed loading
which is necessary to satisfy the law of equilibrium between external and internal forces and moments.
The basic characteristic
of a primary stress is that it is not self-limiting.
Secondary stress is stress developed by the self-constraint
of a structure which is caused by an imposed strain rather than
being in equilibrium with an external load.
The basic character-
istic of a secondary stress is that it is self-limiting since minor
78
distortions can satisfy the discontinuity conditions which cause
the stress to occur.
2.
Fatigue Criteria
Each structural component that experiences cyclic loading dur-
ing operation, excluding cyclic wear, shall be designed to the following fatigue criteria:
Fatigue
Life~4
x Service life Operational Cycles
Fatigue damage shall be evaluated by a linear damage accumulation,
Where ni is the actual number of cycles at a particular stress or
,
strain amplitude and Nf. is the cycles to failure at the same amplitude(Minor's Rule).
In low cycle fatigue analysis, Kf' empirical factor that reflects the actual effects of a discontinuity, values apply to
strains, and are applied to the strain component normal to the
discontinuity only.
Finite element analysis can be used to obtain both the stress
and strain concentration effects in a structure.
additional
11
K11 values are not needed.
In this case,
One must recognize that the
choice of element size, geometry and a-E curve will affect the calculated magnitude of the stress and strain concentration, and re-
79
sults must be used carefully.
3.
Weld Joint Criteria
Weld joints shall be designed such that yielding initiates in
parent material rather than in the weld joint independent of
analytically predicted operating stresses-strains.
Weld joint efficiency factors shall be included in the design
of all weld joints.
The weld joint efficiency factor is a function
of the weld quality requirements, designated by the weld classification(Class I, II, or III), and the weld inspection requirements.
These factors are applied to the weldment minimum guaranteed
material properties.
80
APPENDIX B
Material Properties
Inconel-718 is a nickel-base precipitation-hardenable alloy.
It is used where good corrosion and oxidation resistance are
required over a wide temperature range.
Mechanical properties
permit it to be used as a structural material from -420 to 1200F.
Short time usage up to 1400F is possible but not recommended.
The material is weldable and brazeable but presents some
difficulties in machining.
Weldability is judged by its ability
to resist post-weld strain-age cracking during heat treatment, a
problem common to many of the precipitation-hardenable nickel-base
alloys.
In general, the alloy should be plated for brazing.
The alloy has wide usage and application in gas turbines,
rocket engines and aerospace structures.
Typical products are
fasteners, turbine housings and components, ducts, bellows, injectors, valves and pressure vessels.
Specific heat-treatment procedures for Inconel-718 will depend
upon the application.
Materials and Processes should be consulted
for heat-treating details.
The Inconel-718 properties used in this report are presented
in Table III and Figures 48 through 51.
81
TABLE II I
INCONEL - 718 PROPERTIES AT 70F
(AIR ENVIRONMENT) (STA- 1)
Minimum Tensile Ultimate Strength (KSI)
180.0
Minimum Tensile Yield Strength (KSI)
150.0
Minimum Tensile Elongation (percent)
Young's Modulus (10 6 psi)
10.0
Torsional Modulus (10 6 psi)
11.2
29.6
Poisson's Ratio
0.29
Thermal Conductivity (BTU-it/hr-it 2-F)
6.4
Thermal 6oefficient of Expansion:
(10- in./F)
7.5
Density, (lb/in.3)
Hertz Stress (KSI)
70-40GF
0.297
400
82
1~<CJ:U 11
e
STR£55-5TRAIN DIAGRAM
STA
0
2
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FIGURE 48
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FIGURE 49
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FIGURE 50
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FIGURE 51
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