Optimization techniques for contact stress analysis. McDonald, Eric S. 1992

Optimization techniques for contact stress analysis. McDonald, Eric S. 1992
Calhoun: The NPS Institutional Archive
Theses and Dissertations
Thesis Collection
1992
Optimization techniques for contact stress analysis.
McDonald, Eric S.
Monterey, California. Naval Postgraduate School
http://hdl.handle.net/10945/23999
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Optimization Techniques
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Contact Stress Analysis
by
Eric S.
McDonald
Lieutenant, United States
Navy
Merchant Marine Academy, 1986
B.S., U.S.
Submitted
in partial fulfillment
of the
requirements for the degrees of
MASTER OF SCIENCE
IN
MECHANICAL ENGINEERING
and
MECHANICAL ENGINEER
from the
NAVAL POSTGRADUATE SCHOOL
December, 1992
^
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Richard S. Elster
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OPTIMIZATION TECHNIQUES FOR CONTACT STRESS ANALYSIS
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Stress, Optimization.
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The analysis of stresses induced by contact between two bodies is inherently difficult because the size of the contact z
is unknown and constantly changing throughout loading.
To overcome these difficulties, two approximation methods h
been developed to determine the magnitude of contact stresses using the Rayleigh-Ritz method and the finite element met!
Numerical optimization methods are employed to solve the contact problem. The solution techniques are comparec
known analytical solutions and shown to yield accurate results. An application oi' this approach to solving the con
problem is illustrated by examining the response of a clamped sandwich composite beam to low velocity impact. It was fcv
maximum shear stress is insensitive to lamina thickness, however an increase in the contact layer thickness resu
reduction in interfaeial shear stress. In addition, it was noted that a nonlinear bending stress distribution in the con
layer intensified as the thickness of this layer increased. This phenomenon was found to be localized to the region of cont
Finally, it was found that the compressive transverse normal stresses increased as the thickness of the contact lamina increa
that the
in a
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ABSTRACT
The
because the
to
size
of the contact zone
To overcome
loading.
oped
and shown
The solution techniques
beam
to
is
illustrated
constantly changing throughout
methods have been
devel-
An
are
compared
to
known
analytical solutions
application of this approach to solving the
It
was found that the maximum shear
however an increase
reduction in interfacial shear
stress.
in the
In addition,
This
phenomenon was found
stress
it
was noted
of the contact lamina increased.
in
in-
that a nonlinear
of
this layer
to be localized to the region of contact.
was found that the compressive transverse normal
is
contact layer thickness resulted
stress distribution in the contact layer intensified as the thickness
increased.
to solve the
by examining the response of a clamped sandwich com-
low velocity impact.
sensitive to lamina thickness,
it
inherently difficult
Numerical optimization methods are employed
to yield accurate results.
contact problem
bending
unknown and
is
determine the magnitude of contact stresses using the Rayleigh-Ritz method and
contact problem.
in a
is
these difficulties, two approximation
the finite element method.
posite
two bodies
analysis of stresses induced by contact between
Finally,
stresses increased as the thickness
TABLE OF CONTENTS
INTRODUCTION
MOTIVATION
A.
B.
LITERATURE SURVEY
I.
II.
1
I
1
FORMULATION OF THE CONTACT PROBLEM
A.
PRINCIPLE OF MINIMUM TOTAL POTENTIAL ENERGY
B.
CONTACT PROBLEM DESCRIPTION
C.
NUMERICAL OPTIMIZATION
V.
8
2.
Augmented Lagrange
3.
Optimizer and One-Dimensional Search
14
4.
Convergence
15
8
Multiplier
Method
13
16
1.
Background
2.
Application of the Rayleigh-Ritz
3.
Trial Function Selection
FINITE
16
16
Method
to the
Contact Problem
17
17
ELEMENT APPROACH
22
1.
Total Potential Derivation
22
2.
Optimization and Static Condensation
26
RESULTS AND DISCUSSION
A.
PROCEDURE VALIDATIONS
B.
6
Optimization Fundamentals
APPROXIMATE SOLUTION TECHNIQUES
RAYLEIGH-RITZ APPROACH
A.
IV.
5
1.
III.
B.
5
1.
Rayleigh-Ritz
2.
Finite
Method
30
30
30
Results
Element Method Results
a.
Two
b.
Roller-Foundation Contact Problem
Thin Plates
in
Contact
APPLICATION
35
35
39
43
CONCLUSIONS AND RECOMMENDATIONS
IV
76
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NAVAL POSTGRADUATE SCHOOL
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A.
B.
C.
D.
E.
RAYLEIGH-RITZ APPROACH
76
ELEMENT APPROACH
COMMENTS ON OPTIMIZATION
SANDWICH COMPOSITE MATERIAL STUDY
RECOMMENDATIONS FOR FURTHER STUDY
77
FINITE
77
78
78
REFERENCES
79
INITIAL DISTRIBUTION LIST
81
LIST
Table
1.
Table
2.
Table
3.
OF TABLES
RAYLEIGH-RITZ RESULTS AT CONTACT POINT
FINITE ELEMENT RESULTS: TWO PLATES IN CONTACT
COMPARISON OF STRESSES NEAR THE POINT OF CONTACT
VI
32
39
.
43
LIST OF FIGURES
Figure
1.
Roller-foundation assembly
7
Figure
2.
Contact zone cross section
7
Figure
3.
Analytical solutions for a y and a x
9
Figure
4.
Deformed contact zone
10
Figure
5.
Unconstrained minimization
11
Figure
6.
Constrained minimization
12
Figure
7.
Rayleigh-Ritz method applied to contact problem
IS
Figure
S.
Contact zone
19
Figure
9.
Finite element
Figure
10.
Figure
1
Figure
12.
Stress
contour of a y from Rayleigh-Ritz method
35
Figure
13.
Stress
contour of a from Rayleigh-Ritz method
36
Figure
14.
Strain contour of
from Rayleigh-Ritz method
37
Figure
15.
Strain
Figure
16.
Two
Figure
17.
Finite element
mesh
for
Figure
18.
Finite element
mesh
for roller-foundation
Figure
19.
Deformed
1.
method applied
to contact
problem
29
Comparison of a y
in a roller
contact problem
33
Comparison of a
in a roller
contact problem
34
x
x
Contour
e,
c
y
from Rayleigh-Ritz method
38
thin plates in contact
finite
39
two plates
40
in contact
problem
41
element solution: deformation magnified 100 times
... 44
Figure 20. Analytical solution
vs.
approximate a y from
finite
element results
45
Figure 21. Analytical solution
vs.
approximate o from
finite
element results
46
x
Figure 22. Stress contour ay from
finite
element results (increment 0.01 GPa)
.... 47
Figure 23. Stress contour a x from
finite
element results (increment 0.02 GPa)
.... 48
from
finite
element results (increment 0.000035)
49
from
finite
element results (increment 0.00001)
50
Figure 24. Strain contour
t
Figure 25. Strain contour
zx
Figure 26.
y
Clamped composite beam
51
Figure 27. Finite element mesh for clamped
beam model
52
Figure 28. Determination of nodal contact
55
Figure 29. Loaded beam: deformation magnified 100 times
56
Figure 30. Stress contour: j xy for clamped
beam model
57
Figure 31. Stress contour: a x for clamped
beam model
58
vu
Figure 32. Stress contour: a y for clamped
beam model
(In region of contact)
59
Figure 33. Stress distribution: r xy for 0(3)-ISO(10)-0(3) laminate
60
Figure 34. Stress distribution: r xy for 0(4)-ISO(8)-0(4) laminate
61
Figure 35. Stress distribution: r ty for 0(5)-ISO(6)-0(5) laminate
62
Figure 36. Stress distribution: o y for 0(3)-ISO( 10)-0(3) laminate
63
Figure 37. Stress distribution: a y for 0(4)-ISO(8)-0(4) laminate
64
Figure 38. Stress distribution: a y for 0(5)-ISO(6)-0(5) laminate
65
Figure 39. Stress distribution: a, for 0(3)-ISO( 10)-0(3) laminate
66
Figure 40. Stress distribution: a
for 0(4)-ISO(8)-0(4) laminate
67
Figure 41. Stress distribution: a for 0(5)-ISO(6)-0(5) laminate
68
x
x
Figure 42. Stress distribution: a x for 0(5)-ISO(6)-0(5) laminate at cross section
C
.
69
Figure 43. Stress distribution: z xy for 0(3)-ISO(6)-0(7) laminate
70
0(8)-ISO(6)-0(2) laminate
71
Figure 45. Stress distribution: a y for 0(3)-ISO(6)-0(7) laminate
72
Figure 46. Stress distribution: a y for 0(8)-ISO(6)-0(2) laminate
73
Figure 47. Stress distribution: a for 0(3)-ISO(6)-0(7) laminate
74
Figure 4S. Stress distribution: a for 0(8)-ISO(6)-0(2) laminate
75
Figure 44. Stress distribution:
r lv for
t
t
vui
INTRODUCTION
I.
MOTIVATION
A.
Contact stresses occur when two bodies exert forces over limited contact regions.
Examples of contact
rollers in plate
meshing gear
stress include
forming operations,
and
roller
teeth,
cam
shaft
and pushrod contact,
ball bearings in contact with races, shaft
and journal bearing contact, and plate-pin connections.
Contact zones can be point,
line,
or surface in geometry.
Because of the limited
contact zone, the local stresses can be sufficiently high to be of major concern to the
Consequently, a thorough understanding of this phenomenon
designer.
successful analytical solution to the contact
first
solved in the late 19th century by Hertz.
is
essential.
The
problem between two spheres was
His solution can only be applied to simple
geometries such as spheres, cylinders, and
flat plates.
ternate solution techniques were needed to
accommodate more complex geometries and
Because of these limitations,
al-
boundary conditions.
Unfortunately, the contact problem
culty
is
that the size of the contact zone
loading.
is
difficult to study.
unknown and
its
behavior.
It
is
and numerical methods are preferred
two general approaches used
One
class applies
in the
specific
because of these
in the
significant diffi-
Consequently, the problem
difficulties that
is
approximation
solution of the contact problem.
There are
approximate techniques to solve the contact prob-
iterative
equations that represent the contact
The most
constantly changing throughout
Solutions are obtained by an iterative process.
highly nonlinear in
lem.
is
state.
procedures to solve a nonlinear system of
For example, the contact condition can be
simulated by the introduction of additional coupling terms into the system of equations.
The other
functional
is
body
constraint.
The
then minimized using specific numerical programming techniques.
Al-
class constructs a functional that includes the contact
though these two
classes use different procedures, there are several
porate the contact condition into the problem formulation that are
classes.
methods
to incor-
common
to both
Examples of these include the penalty method and the augmented Lagrange
multiplier method.
These methods specify the manner
in
which the contact boundary
conditions are treated.
The
objective of this study has two parts.
First,
two approximate solution
niques will be developed to obtain the solution of the general contact problem.
tech-
These
techniques belong to the general class of methods that calculate a specific functional and
then applies numerical programming methods to solve the contact problem.
these solution techniques will be verified by comparison with
With the solution techniques
methods
verified, the
known
Second,
analytical solutions.
be applied to study an actual
will
contact problem.
In order to accomplish the
method
utilizes the
deformation
field
first
objective,
two models
will
be developed.
Rayleigh-Ritz method to solve the contact problem.
that satisfies the
boundary conditions
is
an expression for the system's
enabling calculation of the total potential.
The minimization of the
first
An assumed
carefully selected.
elasticity relationships are applied to obtain
The
Theory of
strain energy
total potential en-
ergy enables the calculation of the contact stresses at any point in the body. The second
model developed uses
element analysis to accomplish the same objectives.
finite
numerical minimization technique used
plier
in
both cases
is
the
augmented Lagrange
The
multi-
method.
To accomplish
the second objective, both
methods
will
be verified by comparison
With
with the Hertz solution of an infinitely long cylinder
in
verification completed, the contact stresses resulting
from low velocity impact between
objects
B.
and composite sandwich materials
will
contact with a
plane.
be studied.
LITERATURE SURVEY
As discussed
previously, there are
One approach
problem.
The other
equations.
Although
straints.
uses
two general approaches
special
procedures to
solve
are the Lagrange
1].
2] to solve the
proach
is
a
nonlinear
system of
boundary con-
both approaches use similar mathematical methods to
different,
method and
the penalty method.
of employing the Lagrange method
Pian [Ref.
to solving the contact
creates a functional that includes the contact
corporate the contact condition into the problem formulation.
is
flat
in
nonlinear
A
multibody contact problem.
of these methods
detailed explanation of the
finite
Alternately, the penalty finite element
Two
method
element analysis was discussed by
method was used by Cheng
The fundamental concept of
(i.e.,
Although both methods are
of both methods.
[Ref.
the latter ap-
the transformation of a constrained problem into an unconstrained one.
done by penalizing
in-
This
increasing) the objective function for constraint violations.
effective, there
Nour-Omid
is
substantial discussion
[Ref. 3] described the positive
on the
limitations
and negative aspects of
both methods. The Lagrange method has been shown to be the more accurate method.
However,
its
usage requires the introduction of additional unknowns thus increasing the
II.
FORMULATION OF THE CONTACT PROBLEM
PRINCIPLE OF
A.
As
MINIMUM TOTAL POTENTIAL ENERGY
discussed in the introduction, the limited
cessitated the
development of solution techniques capable of handling the nonlinear be-
havior of the contact
problem with complicated geometry and complex boundary
This study intends to develop two numerical procedures to solve the contact
conditions.
In short, the procedures will use different
problem.
of the analytical solutions ne-
utility
methods
to obtain a functional, the
system's total potential energy, and then use similar methods to obtain the equilibrium
condition.
Determination of the equilibrium position
principle of
minimum
work and the
Given a body
principle of
in
minimum
equilibrium,
infinitesimal displacements
and the
it
The
The
and
is
since the
fact that this
virtual
virtual
is
from a system of forces.
However,
and work
work vanishes
work discussed thus
SU, that
results
far
is
If
each particle
body
in the
to
body
if
the particle
forces in the
in this
and
x, y,
in equilibrium, the
z directions
example are referred
is
zero.
to as virtual in
referred to as the principle of virtual work.
From
the definition of strain energy, virtual
virtual displacements
can be calculated.
viewed as energy against the bonds between elements,
forces
is
can be subcategorized as virtual strain energy
forces.
from
The work done by external
tity.
must be discussed.
desired to describe the response of that
summation of
work done by external
strain energy,
energy
potential energy
simply the product of the generalized forces acting on each
infinitesimal displacements
nature.
is
particle's displacement.
work must be zero
The
stresses
by some generalized coordinates, then the work resulting from these
described
particle
With equilibrium established, contact
potential energy.
infinitesimal displacements resulting
is
application of the
In order to understand the details of this approach, the principle of
can be quantified.
virtual
made by
is
is
designated S XV and
is
SU
is
Since this
a negative quan-
simply the summation
of the product of the external forces and the displacements of the generalized coordinates.
Since the principle of virtual
displacements
is
work
states that the
zero,
SlV-SU =
work done
as a result of virtual
Alternately, this can be expressed as,
n =d(U-
n
where
represents the system's total potential.
The above equation
This
330-331]
This principle states that
total potential
is
stationary.
minimum
This
is
The
12:
pp.
Hence, determination of a system's total potential energy
enable the calculation of the equilibrium po-
will
the basis for the numerical techniques developed in this study.
objective of the solution techniques to be developed
calculating the total potential energy and minimize
To accomplish
dition.
this,
two
isotropic case, the solution of
dated for use
in
different
which
is
models
known.
equal a distance of 2a.
An
work done by Hertz.
normal
in the
is
is
shown
Figure
in
half zones.
solve a simple
first
manner, our models can be
vali-
plane as shown
in
Figure
1.
A
problem
is
available as a result of
In developing the solution methods, only a very limited region
from
this
Figure
The reason
than one half-contact zone
Because of
this,
it
is
[Ref. 13]
The diagram shows
(a)
for this
is
that the contact
represents the analytical solution of the
3
contact problem.
foundation diminish very rapidly.
less
means of
Let the width of the contact zone
2.
analytical solution of this
a very local one.
stresses resulting
depths
be created to
will
infinite
adjacent to the contact zone will be examined.
phenomenon
to obtain a
more complicated arrangements.
cross section of the contact zone
the
is
to determine the equilibrium con-
it
In this
Consider contact between a cylinder and an
3
potential energy. [Ref.
CONTACT PROBLEM DESCRIPTION
B.
at
total potential of a system.
condition of stable equilibrium, the system's
in a
and the minimization of that quantity
sition.
minimum
condition of
illustrates the
the foundation of the principle of
is
=
ll')
As shown,
that a z
and a y diminishes
the stresses
becomes
negligible
significantly in less than
reasonable to assume that displacements beyond a
very limited region are negligible in the strain energy calculation of the contact problem.
A number
1.
of simplifying assumptions
will
be
As discussed above, displacements beyond
made
for this study:
a limited region are negligible in strain
energy calculations.
2.
The foundation
is
an
elastic isotropic material.
The
cylinder (roller)
is rigid.
system's degrees-of-freedom and computational time.
quire additional computational time, but
conditions are exactly satisfied only
is
has been shown to result in
boundary conditions.
solutions in satisfying the contact
choice of the penalty parameter
it
The penalty method does not
when
less
re-
accurate
Since the contact boundary
the penalty term goes to infinity, the correct
the key to an acceptable solution.
Guerra [Ref. 4]
supported these claims.
Because of the limitations o[ the Lagrange and penalty methods, Bishoff [Ref. 5]
advocated the use of the augmented Lagrange multiplier method
contact problems.
This method
is
favorable since
methods discussed above. The augmented Lagrange
penalty
method
in
that the objective function
However, an additional multiplier term
As
combination of both terms.
of
this
method
is
is
it
is
element
avoids the limitations of both
multiplier
method
is
similar to the
penalized for constraint violations.
is
added so the optimum can be achieved by
stated by Vanderplaats [Ref.
that the penalty term
to solve finite
6: p.
a
advantage
141], the
not required to grow to infinity to achieve exact
constraint satisfaction.
The augmented Lagrange
techniques to find the
optimum of any
detailed analysis of the
functional.
method
in
6:
numerical techniques.
this
method
and Lowe [Ref.
in
7] provide a
applying this method.
Rothert
et al.
[Ref. 8] used a
to solve a nonlinear contact prob-
will
obtain the same functional.
The augmented Lagrange
in a similar fashion to solve
be used to determine a
set
augmented
be used as an integral part of two solution methods
developed to solve the contact problem. These methods
will
numerical programming
In this study, an existing numerical optimization routine utilizing the
Lagrange multiplier method
used
in
pp. 140-147] provided an excellent discussion on the
numerical programming code based on
lem.
Pierre
programming techniques necessary
Additionally, Vanderplaats [Ref.
practical usage of this
method can be used
multiplier
each problem.
will use different
multiplier
techniques to
method
will
In each case, the optimization routine
of design variables that describes the contact
state.
Following the development and verification of the numerical procedures,
will investigate the
One of the common
correlation
maximum
this
study
response of composite sandwich materials to low velocity impact.
failure
mode of low
velocity impact
is
delamination. Joshi and Sun
[Ref. 9] studied the impact response of a three layer cross-ply graphite
A
then be
was obtained between delamination cracks
shear stress points determined numerically.
epoxy laminate.
initiated experimentally
Sun and Rechak
and
[Ref. 10] fol-
lowed up these findings and found that the introduction of adhesive layers between
laminae reduced the shear
stress distribution thus
reducing delamination.
Choi, Wang,
and Chang [Ref.
11]
studied the effects of laminae orientation, ply thickness, and
stacking sequence on impact
damage of graphite epoxy composites.
that stacking sequence affects impact
Much
damage more than laminae
It
was determined
thickness variations.
of the previous work has focused on the behavior of the graphite epoxy laminate.
However, there
is
currently interest in the development of turbine blades constructed of
sandwich composites.
It is
therefore beneficial to investigate the response of composite
sandwich materials to low velocity impact.
Figure
1.
Roller- foundation assembly
Figure
2.
Contact zone cross section
Deformations normal to the cross-section are
3.
negligible,
hence a condition of plane
strain exists.
4.
The
5.
The roller-foundation contact
An
roller
is
subjected to a vertical distributed load.
is
frictionless.
important restriction on the minimization problem
prohibited from penetrating into the other body.
however
striction,
a
method of mathematically
This
will
be that one body
may seem
stating this restriction
Figure 4 shows the deformed and undeformed contact zone.
flection of the roller
due to an external force and
of the foundation at any point
like
v(jCjy)
Let
be
will
an obvious
re-
must be discussed.
represent the de-
<5
represent the vertical deflection
At the point of Contact A, the condition of no
(xj,-).
interference can be expressed as.
v(0.0)
At point
condition can be stated
B, this
\{r sin 0,0)
The
latter condition
>
c)
as.
>d -
r{\
many
can be specified at as
-
cos 6)
points as necessary to define this re-
striction.
C.
NUMERICAL OPTIMIZATION
1.
Optimization Fundamentals
Before developing the models to be used in this study, one final area must be
discussed.
it
Once
the total potential energy has been calculated, a
to find the equilibrium position
mization technique to be used
is
A
study of the numerical opti-
required.
The technique being used
optimization.
must be employed.
method of minimizing
in this
Design optimization
is
study
relies
heavily on the methods of design
the utilization of mathematical techniques to
minimize or maximize a particular value to obtain an optimum solution.
is
ideally suited for design.
utions.
may
is
may have an
number of
sol-
a matter of the designer's experience
and
given design task
However, finding the best solution
intuition.
ution
A
The method
infinite
In the absence of significant experience in a particular
field,
reduce to examining a range of possible solutions by
trial
finding this sol-
and
error.
Opti-
mization routines can be utilized to find this solution mathematically.
Optimization problems can be constrained or unconstrained.
may be
desired to determine the
minimum
of the parabola, F{x)
=
(x
-
For example,
5)
2
+
2.
As
it
seen
—
"
<
n
a
LEGEND
to
U
C/1
a-
* -
SIGMA X
• -
SIGMA Y
M
in
a
u
ISJ
§
z
o"
a
—
'
0.0
.5
1.0
l.S
«
3.0
NORMALIZED DEPTH BELOW CONTACT SURFACE
Figure
3.
Analytical solutions for a y and
<x,
M
2.S
1
3 .0
Figure
in
5,
the
minimum
A
constrained problem.
clearly identified at point A.
is
constrained counterpart of
Minimize:
Subject
As
seen
Figure
in
6,
the
minimum
this
This
is
problem
an example of an unis:
F{x)
to: ^(.v)
>
.5jc
+
4
of the constrained problem
simple illustration of constrained minimization.
is
at point B.
The contact problem
is
This
is
a
a constrained
minimization problem.
Figure
4.
Deformed contact zone
The value
The parameters
process
is
an
to be
minimized or maximized
is
referred to as the objective function.
to be determined are referred to as design variables.
iterative one.
are varied thus obtaining a
The
new
objective function
objective function.
tolerance or meets a certain convergence criteria, the
is
out of tolerance, the cycle repeats.
10
is
evaluated.
The optimization
The design
If the difference
variables
within a certain
is
optimum has been
obtained.
If
it
Figure
5.
Unconstrained minimization
11
o
\
a
/
/
\
a
X
U.
a
^-^\B
a
o
1
9.0
2.0
4
B .0
.0
X
Figure
6.
Constrained minimization
12
8 .0
IE 1.0
Augmented Lagrange Multiplier Method
2.
The contact problem belongs
in the class
The technique used
several techniques of solving constrained problems.
belongs to a class of solution techniques
tion techniques
this
is
approach
using an unconstrained minimization technique.
if
the design variables are varied in such a
is
violated
(i.e.,
(i.e.,
way
is
that a constrained
This transformation
Thus,
in
as to enter the region
6:
is
problem
minimized
accomplished
For example,
where a constraint
would be assessed
a penalty
order to minimize the objective function, the design variables
remain within the region where no constraints are violated
[Ref.
is
for constraint violations.
the infeasible region), the objective function
increased).
study
designed for the general nonlinear
transformed into an unconstrained problem and the objective function
by assessing a penalty to the objective function
in this
as sequential unconstrained minimiza-
class of techniques
The fundamental concept behind
problem.
is
(SUMT). This
known
There are
of constrained problems.
(i.e..
the feasible region).
pp. 121-123]
There are a number of methods within
differ in the
way
in
this class
of techniques. They essentially
which penalties are assessed. The technique used
augmented Lagrange
multiplier
(ALM)
in this
study
is
the
method.
Given the constrained inequality optimization problem;
Minimize:
Subject
The Augmented Lagrangian
is
to:
F{X)
g (X)
t
<
0,
/
=
1
.2
n
defined as,
m
A(X, k,p)
=
F[X)
+
£{/M +
2
s,
]
+ p[g/LX) +
sff}
where,
X=
).
t
vector containing the design variables
= Lagrange
multipliers
p =penalty parameter
5,
=slack variables which convert inequality constraints to equality con-
straints
13
The
first
two terms of A(X,
Lagrange multipliers,
mality.
known
X
unknown. Hence,
is
used to select and modify
/
method of
the
that minimization of the Lagrangian represents opti-
Like simple problems where X
AL.M method
is
is
it
From
Lagrangian.
represent the
X,p)
is
simply an additional
unknown
a mathematical routine based
for successive iterations.
[Ref.
6:
to obtain, in the
on constraint values
pp. 140-147]
Initial
selection of this term can have a significant impact of the problem's convergence.
As discussed above,
violations.
This feature
is
a penalty
is
assessed to the objective function for constraint
apparent by examining the
results in a positive value for g(X) thus resulting in
which
a scaling term
that there
small, convergence
A
term.
an increase
constraint violation
in A.
The value
sequentially increased throughout optimization.
is
between convergence and numerical conditioning.
a balance
is
last
may
occur with major constraint violations.
If
of/?
This ensures
I
f
/?
remains
p remains large,
constraints will be satisfied at the expense of an ill-conditioned problem [Ref.
169].
As with
is
7:
p.
the Lagrange term, selection of this term has a significant impact on the
outcome of the problem.
3.
Optimizer and One-Dimensional Search
Thus
far.
a procedure has
been defined which has transformed
minimization problem into an unconstrained one. This
is
of the optimization process
The formulation of
referred to as the optimization strategy.
function via the augmented Lagrange multiplier
optimization process.
level
a constrained
the modified objective
method represents
However, there are additional parts of
a key portion of the
this process that require
comment.
With
the modified objective function
and
a procedure for assessing penalties in
place, a procedure for minimizing the objective function
of the process
is
carried out by the 'optimizer.'
atically alter the design variables in a
If
X represents
manner
The
must be defined. This portion
optimizer's function
X
X —
l
<*(/_!)
+
*<-><
where,
=
current iteration
number
5 = search vector
k
=
scalar representing distance traveled in direction
14
to system-
that reduces the objective function rapidly.
a vector containing the design variables, the following process
the optimizer to alter
/
is
S
is
used by
In general,
two processes must be accomplished
S must be determined by
direction
steepest descent
method where
An example
a systematic process.
the direction of steepest gradient
must be determined such that the objective function
scalar k
The
possible in the search direction of the current iteration.
as one-dimensional search. [Ref.
The optimizer used
complexity of
count
optimum.
to find the
this
method,
this
study
a discussion
routine used in this study
method of
count of
this
finding the
approach
is
chosen.
is
the
Second, the
minimized as much as
is
latter
phase
referred to
is
6:
Due
the variable metric method.
pp. 92-93].
the golden section
is
The one-dimensional search portion of
efficient
is
of their formulation
available in Vanderplaats [Ref.
is
of this phase
pp. 10-12]
6:
in
is
search
First, the
is
A
omitted.
to the
detailed ac-
The one-dimensional search
method with polynomial
interpolation.
the process merely represents a systematic and
minimum
in the
A
chosen search direction.
again available in Vanderplaats [Ref.
6:
detailed ac-
pp. 26-49].
Convergence
4.
The
convergence.
final point to
be discussed relevant to optimization fundamentals
Convergence
There are a number of convergence
terminate calculations.
The most obvious
is
criteria are utilized to identify the
optimum
is
solution and
can be
criteria that
that of
utilized.
absolute convergence where the objective functions from two suc-
cessive iterations are compared.
prescribed limit, optimization
is
If the difference
terminated.
A
between the values
is
within
some
second method signifying optimality for
unconstrained problems
is
vector
approximately zero, optimality has been achieved. This method
is
A'.
If this
called the
value
is
calculation of the gradient with respect to the design variable
Kuhn-Tucker conditions
for unconstrained minimization.
conditions are more involved for constrained problems. [Ref.
6:
The Automated Design Synthesis (ADS) System used
iterations
is
100-101]
in this
study uses both
Relative convergence
these termination criteria as well as relative convergence.
to absolute
pp.
Kuhn-Tucker
is
similar
convergence except normalized versions of the difference between successive
calculated.
Again,
if
a specified tolerance
minated.
15
is
achieved, optimization
is
ter-
APPROXIMATE SOLUTION TECHNIQUES
III.
RAYLEIGH-RITZ APPROACH
A.
Background
1.
The Rayleigh-Ritz method
tential
is
utilizing the theory
the
Since the total potential
function.
trial
enables determination of the
unknown
differentiating with respect to the
n'
is
Second, the system's total potential energy
constants.
equations and
'n'
is
a
minimum
The
constants.
unknown
unknowns, the
trial
A
total potential
L with the origin
at the left
is
terms oC un-
in
is
deflection
A
end of the beam.
12:
is
and
minimized by
The
result
pp. 335-336
one that
slope).
a simply supported
ofl
minimization
that the trial function
kinematically admissible solution
(i.e.,
method
calculated in terms
function constants. [Ref.
As an example, consider
satisfied.
is
this
constants and equating to zero.
geometric boundary conditions of the system
ments need not be
chosen
at equilibrium,
The only requirement of the Rayleigh-Ritz method
kinematically admissible.
minimum po
of
The fundamental concept behind
energy to solve a given problem.
that a trial function that represents the deformation field
known
is
method of
a
is
]
is
satisfies the
Other require-
beam of length
kinematically admissible solution to
describe the beam's one dimensional deformation from vertical loading in the y direction
is,
y(x)
where a represents the
n
A
a n sin(
~-
)
coefficient to be determined.
Deflection boundary conditions at x
increased
=
number of terms
=
in the trial
and
jc
= L have been
satisfied.
Obviously, an
function will yield a far more accurate solution.
Fourier sine series would be a reasonable selection in this case.
Although the Rayleigh-Ritz method does not
on the
trial
significantly.
now
function, sensible choices of
stipulate
numerous requirements
functions will increase solution accuracy
trial
For example, consider the same simply supported beam with the origin
at the center.
A
kinematically admissible function
/
\
y[x)
=
•
/
an sin(
16
——
27T.V
,
).
is,
Naturally however, due to the placement of the origin in this problem, an even function
a
is
much
A more
for sensible selection for a trial function.
appropriate selection would
be,
y(x)
The
throughout
A
-^-
).
trial
function will be discussed
method
brief overview of the
Method
to the
Contact Problem.
to be developed
in order.
is
Rayleigh-Ritz method necessitates the selection of the appropriate
of
unknown
in
this study.
Application of the Rayleigh-Ritz
2.
an cos(
concerning sensible selection of the
latter point
detail
=
A
coefficients.
the desired features of the
The theory of
Application of the
trial
function in terms
discussion of the physical nature of this problem as well as
trial
solution
is
required.
elasticity relationships will
obtain the system's strain energy
in
terms of
be applied using the
unknown
coefficients.
trial
function to
The system's
total
potential energy will then be minimized utilizing the optimization techniques discussed
in
Chapter
II.
The design
variables are the
the coefficients determined, the displacement
to be
determined throughout the body.
unknown
is
Figure 7
utilized.
The post-processing procedure shown
using the
now determined
3.
known
is
is
trial
for
function coefficients.
all
With
points enabling the stress
a flow chart of the
procedure to be
simply the calculation of the stresses
coefficients.
Trial Function Selection
In order to
choose an appropriate
trial
function,
it
is
necessary to have an
understanding of the physical phenomena to be modeled. Consider the roller-foundation
system shown
tical
in
1.
Two
trial
functions are needed to model horizontal and ver-
There are a number of important characteristics that should be
deformation.
herent within the
1.
Figure
trial
functions.
in-
These are outlined below:
As seen
in Figure 3, a t and a y are equal and compressive at the point of contact.
Additionally, a t decays more rapidly than a y as distance from the contact point in-
creases.
2.
Taking the origin at the point of contact as shown
takes the form of an odd function.
3.
The v-deformation takes the form of an even function
origin
4.
To
and greatest
satisfy the
be satisfied.
boundaries.
in
Figure
(i.e.
2,
the u-deformation
symmetric about the
at the origin).
Rayleigh-Ritz method requirements, the boundary conditions must
and v deformation to be zero at the far
In this case, this requires u
17
f
Initiialize Program
2>
Define Optimization Design Variables
Define Assumed Solution
Compute Strain Energy
, f
Calculate Total Potential
Alter Design
Variables
•
Define Constraints
1
i
Call Optimizer to Minimize
Total Potential
Figure
7.
Rayleigh-Ritz method applied to contact problem
18
The characteristics outlined in item 4 satisfy the requirements of this method.
However, since this study will be a stress analysis, it is also desired that the stresses
also reflect the physical phenomena. Since a and a y are functions of c and c c,
and c y should also exhibit certain characteristics. Referring to Figure 8 for dimensions and the coordinate system, it is desired that c and c y equal zero at x= L 2 and
y= H. This will ensure that stresses are zero at the boundaries. To satisfy the requirements of item
above, c, and c y should be equal and negative at the point of
contact and decrease in magnitude as the distance from the point of contact in.
t
s
t
x
1
creases.
Figure
8.
With
Contact zone
the
above guidelines
functions chosen for this study are
in
mind, the
trial
composed of a
=
an {H-y)
a
u(x,y)
v(x y)
=
b n (H-y)
c
i
(
function can be selected.
series
The
of terms of the general form:
— -x)
(j--x)
(3-
c
(3.2)
These expressions were carefully chosen and represent a compromise due to the
culties of satisfying all
boundary conditions with the physical phenomena of
lem.
19
trial
this
diffi-
prob-
From
the above expressions,
boundary conditions
x=L2
at
and
immediately obvious that the geometric
is
it
y=H
quirements of the Rayleigh-Ritz method.
have been
This satisfies the
satisfied.
In addition, there are a
re-
number of important
characteristics that illustrate the advantage of this selection:
and
are negative thus simulating a compressive environment in the vicinity of
The importance of this is obvious. If normal strains were not
negative, the resulting requirement would be for the coefficient to be less than zero
ix
1.
£
y
the point of contact.
to simulate
It is obvious that this would result in deformations opwas desired by the choice of the trial function.
compression.
posite to that which
This selection for deformation fields has the important characteristic of decreasing,
deformation as we move away from the point of contact. Also note that deformation is maximum at the point of contact.
2.
The exponents a,b.c. and d can be varied to simulate subsurface stress fields. For
example, if the analytical solution indicates a large y gradient for a, at x = 0, the:
objective would be to increase the rate at which c, decreases as the distance from
the point of contact increases. This would be easily simulated by raising the value:
of a. If this change had a detrimental effect on the behavior of a the exponents
of the vertical deformation could be varied to restore the solution.
3.
,
v
It is
important to note that
this selection
significant limitation of the trial functions
u.
Physically,
However,
This
is
it
is
expected the
in this selection
of
trial
is
behave as an odd function as discussed above.
w(.rj,')
function, a positive value of
contrary to the physical behavior of the problem and
small, this error
is
Another
less severe limitation
is
this
jc
=
,
deformation
c are at least
It is
range
is
compromise
is
in the elastic
this
error.
a restriction on the order of the exponents in
Since a, and a y are combinations of
Since the normal strains are
equal to
first
ex
derivatives, this requires that
2.
worthwhile to note that most of the considerations discussed above
ceed the requirements stipulated by the Rayleigh-Ritz method.
to utilize trial functions that closely
fort to
some
both normal strains must be zero at the boundaries to ensure that stresses are
zero at these locations.
b and
will lead to
the origin.
0.
order to maintain zero stress at the boundaries.
y
w(.vj-) exists at
Another consequence of
believed to be limited.
the existence of non-zero shear strain at
i
not without compromise. The most
with regard to the horizontal deformation
However, considering that the magnitude of
and
is
The
far ex-
objective has been
match the physical nature of the problem
in
an
ef-
maximize accuracy.
The
specific trial functions used in this study are listed below.
deformation was assumed to be;
20
The horizontal
rri
Ya (H-yf +n \ \ - x)
=
u( Xxy)
2
(3.3;
n
n=\
where
//
and L represent the height and length of the bearing foundation,
Summation was done
for n equal
and
1
4.
The
vertical
respectively.
deformation was assumed to be;
m
Yb
=
v{xy)
+n)
(H-yf( -j - xf
n
(3.4)
.
1=1
As discussed
to be
previously, manipulation of the exponents enables the
matched with the analytical
ponents chosen
in the
above functions achieve
With the deformations chosen, the
in the calculation
As
solution.
Y-2a
=
n
function results
in the results, the ex-
this goal sufficiently.
and
stresses
of the foundation strain enersv.
tx
be illustrated
will
trial
(H-y)
{
strains
can be calculated for use
These values are shown below:
* +r,)
(-^-x)
(3.
n=\
n
Zy
=
Y -2b (H-y)(j--xf
j
a
where
stress
E and
and
v
are
+n)
n
(1
+v)(l-2v)
^(l
+ v,fl-2v)
1(\-v)e x +ve v ]
C(1
-
V)
Young's modulus and Poisson's
^
+V ^ ]
ratio, respectively.
(3.6)
(3.7a)
(3.7b)
For the shear
strain,
m
|t.y-(4 + .K(II-/«(frf
oy
(3.S)
i—i
JL = Yj
-(l +
n)b n (H-y)
21
2
(j--xf
(3.9)
+ f.
-f.
cy
ox
y y
T
where
G
is
(3.10)
xy=Gv Xy
(3.11)
the shear modulus.
Using the above quantities, the strain energy
U
can be calculated.
of strain energy applied to a two dimension deformation
From
the definition
field,
ii
—
\\
(
axi x
+ oy zy +
z xy y xy )dydx
(3.12)
o
Because of symmetry about the origin, strain energy can be calculated for half of the
domain and doubled.
With
strain energy calculated, the total potential for the
system
can be found from.
n = U-FS
(3.13)
where,
F=
=
(3
external force per unit length applied to the roller
vertical distance traveled
The quantity FS represents
B.
work done by
the roller on the bearing foundation.
Total Potential Derivation
The
finite
roller.
ELEMENT APPROACH
FINITE
1.
the
by the
finite
element method can be employed to solve the contact problem.
A
element mesh can be constructed to approximate the behavior of an elastic foun-
dation subjected to line contact loading from an rigid
between the foundation and
roller
roller.
The
enables the calculation of the foundation's strain en-
ergy and the subsequent calculation of the total potential energy.
optimization techniques discussed in Chapter
mined.
Thus
The
resultant interaction
II,
By again
utilizing the
the equilibrium position can be deter-
the contact stresses can be calculated throughout the body.
objective
by application of the
is
to derive a
finite
means of calculating
the total potential of the system
element technique. Total potential energy
22
is
defined
as,
n = U-
F5
(3.14)
where,
U=
internal strain energy
F=
=
J
The
external force per unit length applied to the roller
vertical distance traveled
strain energy of the
Q
roller.
system can be found from.
U=
where
by the
—
+
\o xz x
represents the problem domain.
oy e y
+
r xy y xy )JQ
This can be expressed in matrix form
T
-T{c} {a}ciQ
3.15)
as,
3.16)
Q
where.
£)
-
{a}
=
l
On
i
cx
c
y
[o x a
"<'xy}
t
}
the element level.
T
±{e} {a}dQ
U=
(3.17)
Q!
where
The
i
represents analysis of the
stress
h
i'
matrix can be expressed
element.
as,
{*}
=
[£]{£}
where [D] represents the material property matnx.
23
(3.18)
For
a condition of plane strain, the stiffness matrix
—
1
can be expressed
V
V
E
[0]
V
1
+v)(l
1
—
(3.19a)
V
2v]
1
For
as,
-
2v
a plane stress condition.
=
[D]
1
V
V
1
3.1%)
(1-v'
The development of
this
technique
1
-
V
will use linear triangular
elements.
The method,
however, can be applied to any type element. For linear triangular element, the defor-
mations take the followin2 form:
u
=
Hiii\
i"i
H-^ih
+ H-u
3"3
(3.20)
v
=
//.v,
111*22
,», -r H,\
+
H,\
^ M
3'3
(3.21)
+
+
where the shape functions H, are defined
as:
=
C(-v ^'3
~ x 3>'2) +
CV'2
~}'3) x
+ x ~
H =
2
[(-ViXi
- xy3 +
CV'3
-}'\) x
+ x ~ x3 ]y]
H =
[(*i>2
~ x iy\) +
Oi -y-d*
//,
i
)
where,
x ,y, = coordinates for node
i
(
A = element
The
strain matrix
area
[Ref. 14
].
can be expressed
as,
24
(
(
+
3
*:)>']
\
(*2
-
*ilv]
~
(3.
-
(3-23)
IA
37
>->}
(3.24)
4-
o
o
f
ex
w.=
g
c
cv
dx
Substituting equations (3.20) and (3.21) into equation (3.25) yields,
r
o//,
6H2
dx
ex
c//3
-i
"l
V
l
c
cH
=
{
a//2
c// 3
u2
cy
cy
v2
°y
c//,
cy
cH
x
c// :
o//2
BH
dH
cy
ex
cy
ex
expressed
as,
ex
In abbreviated notation, equation (3.26)
is
For the linear triangular element, [5] reduces
>2->'3
[Z?]
=
17
r3
-
- x2
>'3
,v
3
- x2
}'2
-}'2
3
''3
3
V3
to,
-yi
x - x2
y,
\
.V!
-
}'3
~}'\
-y
2
x2 -
,v
3
x2 ~ x
\
}'\
-V,
i.28)
~}'2
Returning to the element strain energy calculation, equation (3.17),
U=
±{c}
T
{o}d£l
&
and substituting
(3.18) into (3.17),
±-{,}\D^z}dV.
i
25
(3.29)
Substituting equation (3.27) yields,
U=
\{lB-]{d])\D^B^{d)dV
q:
r = 4-{^ r[5] r[D][5]{c/}f
l-
where
i
=
=
T
-T{cf}
d\
[Bf[D]iBl4i{J}
(3.30)
-
Defining the element stillness matrix
unit depth.
[A1
=
[A. ]
r
[*] [D][5].-!/
13.31)
then the strain energy per element equals,
U = ±-{d}tK\{d\
The element
With
strain energy
stiiTness
now
(3.32)
matrix can be expanded into the global stillness matrix.
determined, total potential can be determined from equation
(3.14).
2.
Optimization and Static Condensation
As with
method
mum
will
the previously developed model, the
augmented Lagrange
multiplier
be utilized to determine the equilibrium condition via the theorem of mini-
potential.
The
objective function
ever, the design variables are the
is
again the total potential.
nodal deformations, u and
v„
t
In this case,
how-
for non-fixed nodes.
Constraint equations are developed in a similar manner as discussed in Chapter
II
to:
ensure that one body does not violate space occupied by the other.
Since the nodal deformations are represented as the optimization design variables, the
number of design
For a simple mesh, a
However,
it
is
direct application of this
known
number of design
variables will equal twice the
procedure
number of non-fixed nodes.
will likely yield
accurate results.
that the accuracy of the optimization routine declines as the
variables increases.
Hence, for complicated
solution accuracy will be adversely affected by the large
finite
element meshes,
number of design
variables.
Therefore a procedure must be adopted to eliminate the need for assigning design vari-
26
ables to nodes where information
cedure
is
known
is
not necessary for evaluating a solution.
This pro-
as static condensation.
Static condensation has
been
duce computer computational time.
utilized
The
ganization of the global stiffness matrix.
by References
2
and
3
in
an
idea behind static condensation
A
finite
[KJ{u}
=
effort to reis
the reor-
element problem can be expressed
{F}
as,
(3.33)
where.
[K,]
It is
=
the stiffness matrix
{«}
=
the deformation vector
{F}
=
the force vector
desired to reorganize this system of equations into the following:
The vector
m,
^11
^12
n
Kt,
Ao->
of
(3.34)
contains the essential nodes while vector u2 contains non-essential nodes.
Essential nodes are those where
boundary conditions are applied and nodes that are
signed optimization design variables.
By matrix manipulation.
= -lK22 TXK2l -]{u
{u2 )
as-
x
(3.35)
}.
Therefore, the displacement vector can be expressed as,
[/]
u2
where
I
}"\-LK22 T
l
(3.36)
- IK22 T\K2 {\
LK2l 2M]
represents the identity matrix.
Substituting this equation into the global counterpart of equation (3.32),
,;
V
=y
1
f
(
,71
M i)
IK22 Y\K2 {]
\K
~]
fo
S
[^22]
[^21]
-
By defining the reduced
stiffness
matrix [A ], the above reduces
27
to.
(3.37)
U = ±-{u
As discussed
at the
beginning of
{
}
T
[K}{ Ul )
(3.38)
this section, the
signed as the horizontal and vertical deformations at
all
ADS
design variables are as-
non-fixed nodes.
With the
in-
tegration of static condensation, design variable assignments are further restricted ten
non-fixed, non-condensed nodes.
The procedure
is
now
in
place for calculation of strain:
energy and total potential energy. In summary, a flow chart of the solution procedure'
utilized in this chapter
is
shown
in
figure
9.
28
]
(
V
Initialize Program
I
y
Define Optimization Design Variables
Calculate Element Stiffness Matrix
[K]
Assemble Global Stiffness Matrix (KG]
Reduce Degree of Freedom by Static Condensation
Assign Design Variables to
Calculate Strain Energy
U={d}
t
(d)
U
(KG]{d)
Alter Design
Variables
Calculate Total Potential
Define Constraints
Call Optimizer to Minimize
Total Potential
Post-Process
C
Figure
9.
Finite element
method applied
End
to contact
29
)
problem
[
KG
RESULTS AND DISCUSSION
IV.
PROCEDURE VALIDATIONS
A.
Method
Rayleigh-Ritz
1.
In
problem
Chapter
III
an approximation technique was developed
via the Rayleigh-Ritz
mated the deformation
Results
field
method.
As
were selected
discussed, two
in
terms of
trial
unknown
to solve the contact!
functions that approxicoefficients.
The
hori-
zontal deformation was assumed to be;
m
n=]
The
vertical
deformation was assumed to be;
m
vi
Y bn(H-y) (-T-x)
2
x vv) =
0+n)
j
n=\
Using the above deformation
theory of elasticity relationships and the definition
fields,
of strain energy were employed to obtain an expression for the total potential energy of
the system
to
shown
in
Figure
1.
Numerical minimization techniques were then employed
determine the equilibrium condition and the contact
To
of
illustrate the application
this
stresses.
method, an isotropic material with the
fol-
lowing properties was selected:
E = 200
=
v
G
As
stated in Chapter
II,
this
because an analytical solution
GPa
0.3
=
76.9
GPa
problem was selected
is
for
development of
available as a result of the
Contact
stresses as well as the size of the
strongly affected by the size of the contact zone.
30
technique
work done be Hertz.
desired to use this analytical solution to choose a roller size, load,
that can be used to accurately simulates the contact
this
It is
and problem domain
phenomenon.
surrounding region of influence are
Naturally, as the size of the contact
zone increases, the load
The
is
distributed over a larger area and contact stresses decrease.
extent of the affected subsurface zone also decreases.
the contact zone size
is
and the diameter of the
roller [ Ref. 13
Hence
].
and the
defining the
Figure
3.
examining
it
o,
domain beyond which
is
resulting stresses.
Therefore the decay o^ ay
.
shows the ana-
3
is
the limiting factor in
From
strain energy contributions are negligible.
phenomenon can be
estimated that the contact
a region equal to
Figure
zones (a) awa\ from the contact point. As shown.
stresses as a function of half-contact
a y decreases more gradually than
and
This figure shows the decrease of the normal
of the contact problem.
lytical solution
for a given material, the roller size
defined the width of the contact zone as 2a.
2
the analytical solution,
defined by the externally applied force, the material properties
the external load define the contact zone size
Figure
From
approximately
accurately modeled by
five half-zones (5a).
Since a numerical integration technique was used to perform the double inte-
gration required by Equation 3.12. the dimensions were selected for numerical convenience.
Referring to Figure
respectively.
Due
to the
8.
height
H and
length L were selected as
H
and
2
meters,
problem's symmetry, half the foundation was analyzed.
enabled the double integration to be conducted between the limits of
foundation height
1
has been
set to
1
m,
it
is
and
1.
This
Since the
desired to have this distance equal to 5
contact zones (5a) as described above.
Using the analytical solution, a load and
roller
ated a contact zone such that the foundation height
The only
additional restriction
was that the
roller radius
combination used
equal to a distance of 5a.
resulting contact stresses remained within
MPa.
90
Radius: 75
MN
m
values were obtained using the analytical solutions found in Reference
latter
dimension
is
of base dimensions
unrealistic.
However, as stated above,
in the interest
The load
study are;
in this
Load:
The
H was
Yield stress was assumed to be 300
the elastic range of the material.
and
diameter were selected that cre-
it
of numerical convenience.
is
13.
The
a result of the selection
This model simply repres-
ents a scaled-up examination of the base material in a small region adjacent to the contact zone.
Comparisons of the approximated contact
along the axis of symmetry are shown
in
stresses with the analytical solution
Figures 10 and
31
11.
Both
figures are normalized
graphs of the stress distribution.
the point of contact.
Figure 10
is
a
As shown,
comparison
for
ay
As shown
this stress
As
stated in
Chapter
the exponents to
ax
It is
.
III,
match
the
maximum
apparent that
1
shows the analytical and approximate
1
this
method over approximates
function
this trial
is
it
in this
A
stress in
In the case of Figure 11,
it
brief expla-
solution technique follows.
must be understood that a and a y are both funcx
vertical deformations.
deformation to alter the
this stress.-
the ability to change
analytical solutions with approximate solutions.
and
at
x
one of the benefits of
While selecting exponents,
occurs
stress
equal to a and o y at the point of contact:
is
Figure
nation of the choice of exponents used
tions of horizontal
3,
This figure shows a stress distribution that closely
.
approximates the analytical solution.
stress distribution for
Figure
in
Therefore changing the exponent of one
one direction
behavior of the other.
will influence the
appears that a modification
is
required.
An
increase in the
exponent of the y-portion of the horizontal deformation function seems appropriate
increase the rate at which a x decays.
fect
However,
of decreasing the rate of decay of a y
exponents of the
maintain a zero
vertical deformation.
stress
is
the
at
and
Chapter
y=H
III. in
ef-
order to
the exponents of
compromise must be reached.
11.
all
It
is
on ensuring a y
is
Accordingly, a decision was
Figure
in
a
2
in
priority should be placed
as close as possible to the analytical solution.
shown
have the undesirable
possible to counter by decreasing the
x=L
Thus
2.
dominant term,
cept the over estimation of a x as
is
However, as stated
boundary condition
terms must be greater than or equal to
believed that since a y
It
.
this action will
to
made
to ac-
In this case, the estimation of a x
conservative.
is
The contours of the Rayleigh-Ritz
It
solution are
shown
in
Figures 12 through
15.
has been determined that this method can accurately predict contact stresses resulting!
from
line
contact between a roller and
flat
plane. Table
1
compares the approximated!
contact stresses and the analytical results at the point of contact.
Table
RAYLEIGH-RITZ RESULTS AT CONTACT POINT
1.
Current Model
Analytical Solution
a,
(MPa)
285.6
289.7
o
(MPa)
301.9
289.
v
32
Pi
'
LEGEI D
* - APPRO CIMRTE
en
u
m
• - EXACT
aa: a
(-
m
a
u
S
z
a"
'I
a
a.
0.0
0.5
1.0
1.5
3.0
NORMALIZED DEPTH BELOW CONTACT SURFACE
Figure
10.
Comparison of a y
in
a roller contact problem
33
2.5
3.0
*•
rw
o
LEGD
in
u
U1
U1
B
« -
APPRO (inflTE
• -
EXACT
fri
in
a
u
M
_1
•
_
§
z
o"
ri
a
—
3.0
1.0
.5
l.S
•—
3.0
NORMALIZED DEPTH BELOW CONTACT SURFACE
Figure
1
1.
Comparison of a
x
in a roller contact
34
problem
2.5
1
1
2
.0
Figure
2.
Stress contour of
12.
Finite
«r
y
from Rayleigh-Ritz method
Element Method Results
In Chapter III, an approximation
problem using the
finite
element method.
method was developed
A
to solve the contact
method of calculating
ergy and the total potential energy was investigated.
a system's strain en-
In addition, the use of static
condensation to improve optimization efficiency was described.
As
discussed, the nu-
merical minimization techniques were again utilized to determine the equilibrium position.
A
means of employing these techniques
introduced.
In this section, a simple contact
algorithms
used
condensation.
problem
will
a.
to
calculate
With confidence
strain
to evaluate the contact
problem
energy
and
will
be investigated to validate the
those
in these algorithms, the
phenomenon was
used
to
implement
static
more complex roller-foundation
be examined.
Two Thin
Plates in Contact
The procedure developed was
the solution of which
was known.
first
validated
In this example,
35
on
a simple contact
two thin bodies
in
problem
plane stress were
Figure
studied.
Stress contour of
13.
As shown
in
Figure
<x,
16,
from Rayleigh-Ritz method
one body, restrained on one edge and subjected to
a
horizontal load, comes in contact with a second body rigidly supported on three sides.
The
Finite
element model developed to solve
angular elements as shown in Figure
problem was
to solve this
finite
17.
The
this
problem
is
composed of
objective of the fortran
static
tri-
program developed
to calculate the total potential energy of the
element technique and the method of
14 linear
system using the
condensation. With this accomplished,
the equilibrium position can then be found via the
augmented Lagrange
multiplier
method.
The
Equation
3.14.
expressed
as:
objective
function
for
this
problem
is
the
total
potential
energy,
Referring to Figure 16 and 17 the constraints imposed on the system are
u&
<
0.005
36
+ uu
Figure
Strain contour of
14.
t
y
from Rayleigh-Ritz method
tig
<
0.005
+
u l2
where u represents the horizontal deformation of node
As shown
in
Figure
seventeen nodes were used to model the system.
17,
Each node has been assigned horizontal and
ingly, the degree
in
Chapter
III,
of freedom for
static
non-essential nodes.
this
/.
system
is
vertical
deformation variables.
twice the
number of
nodes.
Accord-
As discussed
condensation requires the identification of essential nodes and
Essential nodes are those nodes where
assigned and boundary conditions are applied.
ADS
design variables are
Referring to Figure
point of load application and must be assigned a design variable.
17,
Nodes
node
8, 9,
3
is
11,
the
and
12 are assigned design variables in order to define the constraint equations described
above.
After eliminating
all
fixed
only candidates for condensation.
nodes from consideration, nodes
As
discussed in Chapter
37
III,
2, 5,
and 6 are the
the global stillness
ma-
Figure
trix is
Strain Contour
15.
et
from Rayleigh-Ritz method
rearranged according to Equation 3.34.
the non-essential nodal information. u2
{w2 }
where
u,
and
v\
Strain energy
3.14.
The
r
is
,
={w2
For
this
problem, the vector containing
arranged as follows;
u5
v2
v5
u6
v6 }
represent the horizontal and vertical deformation of node
and
total potential energy
scribed in Chapter
The
results obtained
de-
compared with
the
II.
from Y.W. Kvvon and
place to solve the
method
multiplier
solution obtained from this simple problem were
The solutions were
respectively.
were calculated according to Equations 3.38 and
was minimized using the augmented Lagrange
latter
/,
in
agreement.
It
J.E.
Akin
[
Ref. 15
was concluded that a
]
in
Table
satisfactory procedure
more complex roller-foundation problem.
38
and are shown
2.
was
in
V/
'/_
W/y/////////////zzzv7//,
Figure
Table
Tno
16.
2.
FINITE
Load (N)
thin plates in contact
ELEMENT RESULTS: TWO PLATES
l.OxlO
No
structed to
Contact
.503x10-'
"12
4
.336X10"
.319x10^
"9
.702xlO- 2
.734x10^
1
finite
No
Contact
.503xlO- 2
Roller- Foundation Contact
A
Reference 15
"9
un
b.
CONTACT
Current Model
Deformation
1.0x10 s
1.0x10°
IN
.201xlO~
2
.235xlO"
2
Problem
element grid composed of 512 linear triangular elements was con-
model the roller-foundation assembly shown
in
Figure
symmetry of the problem, one-half of the foundation was modeled.
constructed in the vicinity of the point of contact.
The mesh
is
1.
A
Because of the
refined
shown
in
mesh was
Figure 18
where the origin represents the point of contact. The domain dimensions are similar
39
to
Figure
Finite element
17.
mesh
for
two plates
in
contact
those chosen in the Rayleigh-Ritz method discussed in Part
radius
was chosen
as 75 meters
A
of this chapter. The
and the half-domain dimensions are
roller
1.40 x 1.40 meters,
j
As
discussed in Part A, these dimensions represent an analysis of the region immediately
adjacent to the contact zone and are a result of the local nature of the contact problem.
Constraint
Chapter
•
II
Boundary' conditions were imposed
in the
the
discussion
of
following manner:
Horizontal and vertical deformations were prohibited on the remote mesh boundaries
•
Part B.
equations were constructed according to
(i.e.,
m( 1.40, y)
= v( 1.40, y) =
and
u(jr,1.40)
=
v<x,1.40)
=
0).
Horizontal deformation was prohibited on the axis of symmetry
40
(i.e.,
u(0,y)
=
0).
*•
n
I—
U
(C
Z
o
CJ
u.
o
""
o
iz
E-o"
E
o
a:
u.
u
IS1—
to
E
*
tae
u
>•
a"
o
1
1
3.0
1
0.4
0.2
0.8
1
0.8
1
1.0
1
1.2
HORTIZONTflL D I STANCE FROM POINT OF CONTACT
i
Figure
18.
Finite element
mesh
for roller-foundation
41
problem
i
1
Since the system has 289 nodes, the resulting 578 degrees of freedom necessitated the utilization of static condensation.
Referring to the contact surface, the
deformation variables that correspond to nodes on
straint equations.
all
subject to
The nodes
this surface are required for con-
that comprise the other three borders of the
boundary conditions. Consequently, the
nodes that are candidates
1
for static condensation.
interior
domain
arc:
nodes of the mesh are the
In this model,
all
interior
nodes were
condensed. The original system was reduced from 578 to 128 degrees of freedom. After;
application of the boundary conditions, there were 46 possible deformations, a
number of
ciently small
design variable was used to represent the distance of travel by the
needed
to calculate the
As
One
design variables for the optimization algorithm.
work done by the
before. Equation 3.38
roller
was used
suffi-
additional
This value
roller.
is'
on the bearing foundation.
to calculate the system's strain energy.
Following calculation and the subsequent minimization of the total potential energy, a
post-processing procedure was followed to determine the contact stresses.
of the optimization routine represents the nonzero components of the
The output
{«,} vector.
In
order to calculate stresses throughout the body, the remaining deformations contained
within the condensed vector
using Equation 3.35.
must be determined.
{;/,}
is
calculated directly
With deformations known throughout the domain,
determined by applying Equation 3.27 to each element.
Equation 3.18 enables determination of
To
This vector
illustrate the capability
stress for
strains
can be
The subsequent application of
each element.
of this method, an isotropic material with the
following properties was chosen:
E = 240
v
=
GPa
0.3
G =
92.3
GPa
Load = 90.0
MPa
Figure 19 shows the deformation resulting from the loading.
For
clarity, the defor-
mations have been magnified 100 times their original values. Comparisons of the
distributions with the analytical solution along the axis of
ures 20
and
21.
These figures are similar to Figures 10 and
versions of contact stresses.
mation of the
As shown
stress distribution in the
symmetry are shown
1
in these figures, this
foundation of a loaded
42
1
stress
in Fig-
and represent normalized
method
is
a
good approxi-
roller bearing.
If the
mesh
was more refined near the contact zone and the domain extended
further, the
between the numerical and analytical solutions would be
figures 22 and 23 rep-
resent normal stress contours of this problem.
Table
3
shows
a
comparison of the
better,
show normal
Figures 24 and 25
results of this
agreement
strains.
model and the analytical solution
at
a selected element in the region of contact.
Table
COMPARISON OF STRESSES NEAR THE POINT OF CONTACT
3.
FEM
Analytical Solution
Solution
a,
(MPa)
at x
= 0.0137. y = 0.0273
281.3
272.0
ex,
(MPa)
at x
= 0.0137,
y = 0.0273
315.7
313.8
APPLICATION
B.
The preceding
lated using the
show how
this
section illustrated that the contact problem can be accurately simu-
methods developed
in
Chapter
approach can be applied
III.
It is
the objective of this section to
problem
to a contact
composite plate sub-
in a
jected to low-velocity impact.
A
multi-ply laminate model has been constructed to investigate the response of
composite materials to low velocity impact.
subject to impact
damage commonly
fail
It
due
has been found that composite bodies
currently being considered for use as turbine blades.
It
an understanding of the behavior of sandwich materials
In order to accomplish this task, a
picted in Figure 26 has been modeled.
cm.
Sandwich composites are
to delamination.
would be
to impact
clamped composite beam
The beam length
is
25 cm.
beneficial to acquire
damage.
similar to the
Beam
thickness
is
Because of symmetry, half the beam was modeled with 256 bilinear elements.
shown
in
Figure 27, the
mesh
is
The major assumptions of
strain
and that the dynamic
this
effect
model are that the beam
condition of plane
of the impact can be neglected.
Reference 10 ap-
cordingly, the stress distributions obtained
is
maximum normal and
from
this
time of the response of the body at the instant of
finite
element program developed to solve
alter the material stiffness
As
in a
the peak load will be examined to study the
The
2.5
refined near the point of contact, the origin of the mesh.
proximated the loading resulting from low velocity impact as a sinusoid.
in
one de-
matrix [D] shown
43
in
In this study,
shear stresses.
Ac-
study will represent a 'snap-shot'
maximum
this
loading.
problem
is
sufficiently flexible to
Equation 3.18 during construction of the
1—
cc
o
U
o
Z
o
£o
a:
U.
LJ
83-
o
_]
"
d-
>
o
"
—
,
..
d
.
i
i
0.0
i
0.4
0.2
-i—
0.8
-i
0.B
i
1.0
i
\
1.2
HORTIZONTflL 01 STANCE FROM POINT OF CONTACT
Figure
19.
Deformed
finite
element solution: deformation magnified 100 times
44
ri
LEGE* Y0
m
U
x - APPRO K I MATE
X
• -
aa: o-
x
(—
en
a
U
EXACT
N.
x
>
x
(si
>_
L
IE
O
z
X
X
X
*•
o"
ri
o"
o
0.0
0.S
1.5
1.0
2.S
2.0
3.0
NORMALIZED DEPTH BELOW CONTACT SURFACE
Figure 20.
Analytical solution
vs.
approximate a y from
45
finite
element results
rt
o
LEGEIO
en
U
a3= O
« -
APPRO
• -
EXACT
(I
HATE
1—
in
o
u
CE
O
z
o"
n\
<<•
X
X
X
o
X
- •
0.0
O.S
1.0
X
•— r-
3.0
l.S
2.S
•
3.0
NORMALIZED DEPTH BELOW CONTACT SLRFACE
Figure 21.
Analytical solution vs. approximate
46
<x x
from
finite
element results
0.4
0.2
0.8
0.S
1.0
HORIZONTAL DISTANCE FROM POINT OF CONTACT
Figure 22.
Stress contour a
from
finite
element results (increment 0.01 GPa)
47
0.0
0.2
0.4
0.B
0.0
1.0
1.2
HORIZONTAL DISTANCE FROM POINT OF CONTACT
Figure 23.
Stress contour
<x,
from
finite
element results (increment 0.02 GPa)
48
Figure 24.
Strain contour
sy
from
finite
element results (increment 0.000035)
49
Figure 25.
Strain contour
£,
from
finite
element results (increment 0.00001)
50
Figure 26.
Clamped composite beam
51
fl
B
C
FINITE ELEMENT MESH
5.0
2.5
5
7.S
10.
HORIZONTAL DISTANCE FROM POINT OP CONTACT
4
o
H
0=
Figure 27.
finite
Finite element
element global
mesh
for
stiffness matrix.
clamped beam model
Therefore by defining the layup for the laminate,
the lamina stiffness matrices can be varied from element to element to accurately model
the behavior of the body.
This enables a variety of laminate layups and lamina thick-
nesses to be studied.
The sandwich materials used
in this study are
stiffness
matrix
is
given by Equation 3.19a.
orthotropic exterior laminae, the material stiffness matrix
[Ref. 16
isotropic interior
Since a condition of plane strain was as-
material and orthotropic exterior laminae.
sumed, the isotropic material
composed of an
].
52
is
For the
given by Equation
4.1
1
—
V23V32
v
i
E2 E3 T
[Z>]
v
=
\2+
+
2
V
32
V I3
£,£ 3r
v 32 v 13
1
" v 13 v
3
1
(4.1)
E E3 T
£,£ r
3
{
r
12
where.
r=
v
1
::
v :,
-
V3VJ,-
the
r*
.
=
-
2l\,v 3: v 13
£,£2 £3
£ = Young's Modulus
v
v 31 v 13
in /" direction
Poisson's ratio for lateral contraction in/'" direction resulting from loading
in
direction
The derivation of
done using
the total potential energy calculation in Chapter III. Part B
Since bilinear elements were used
linear triangular elements.
calculation of the element stiffness matrix
was more computationally
in this
wise, the calculation of total potential energy
Chapter
was
model,
Indi-
intensive.
vidual entries of the element stiffness matrix were obtained from Reference
17.
was
Other-
identical to the procedure outlined in
III.
There were two groups of boundary conditions applied to the problem.
clamped edge, deformation was prohibited.
As
prohibited along the axis of symmetry.
In addition, horizontal deformation
the
was
before, the application of static condensation
requires the identification of essential nodes the information of which
the {w,} vector.
Along
is
contained within
In addition to the essential nodes associated with the
above boundary-
conditions, the nodes along the contact surface are needed for the constraint equations.
All other nodes were
To
condensed
out.
illustrate this application
of problem solving, sandwich material with the
lowing properties were studied:
Exterior
Laminae
£u =
170
Ex =
11.8
Gu =
5.2
v i2
=
GPa
GPa
GPa
0.33
53
fol-
Isotropic
Core
E =
2.24
GPa
G =
0.84
GPa
v
=
0.35
The beam was loaded by contact with
was 250 N.
Some unexpected
mined by the optimizer.
a 10
cm
The transmitted
radius ball.
force
trends were observed in the equilibrium position deter-|
By examining the deformations along the
gradually decreasing trend in deformation moving
away from
interrupted in lamina of significantly decreased stiffness.
It is
axis of
symmetry, a
was
the point of contact
believed that this difficulty
resulted from an inability of the optimization routine to approximate deformations
through regions containing very different orders of strain energy.
ficulties,
some
beginning of
In spite of these dif-
information was obtained from this program. As discussed
critical
this study,
one of the greatest
difficulties
termination of the size of the contact zone.
of the contact problem
and the distance between the
(r)
can be determined
node
distance
r'
to the
i
if
m
a
node
is
is
r'
is
greater than
r,
is
By com-
and the node
-
(r
illustrated in Figure 28.
),
it
The
given by the equation:
r'
If
ball center
This condition
in contact.
the de-
is
Fortunately, the size of the contact zone can
be readily determined by examining the output from the constraint equations.
paring the ball radius
in the
the node
is
=v
not
(r
in
-
+ vf +
<5
x?
contact with the body.
With the extent of the contact zone known, the solution
to this
tained by applying the contact boundary condition to a direct
finite
Since the validity of the optimization program was in question, this
problem was obelement program.
method was applied
using a 0-90-0 layup similar to one used in a study conducted by Sun and Rechak [Ref.
The solution obtained from the current approach was very
10].
close to the other re-
sults.
With confidence
model
in the
to various layups.
procedure,
The
objective
be used for meaningful research.
it
was desired
was
to illustrate
concern
direction, the
is
examine the behavior of
how
this solution
this
technique can
The study conducted by Sun and Rechak analyzed
methods of reducing the likelihood of composite
ticular
to
failure
due to delamination.
the magnitude of shear stress distribution
two predominant causes of delamination
54
and
failure.
Of
par-
tensile stress in the y-
Using materials with the
B
z
8
o
SHEAR STRESS
/—
a.
n-
o
-
0.0002
'
tr.
H"!
in
1
o
fcB#5TO
/
3.0
a. 5
^T"
0.0002
5.0
7.S
10.0
ia.s
HORIZONTAL DISTANCE FROM POINT OF CONTACT
o
fa:
U
Figure 30.
Stress contour:
x xy for
clamped beam model
thickness of the exterior layer increases, the
However,
this stress
is
always compressive
the magnitudes of the shear
appear that
if
by high shear
41.
value of this stress increases.
with the core.
at the interface
and transverse normal
delamination was to occur at
By comparing
stresses at the interface,
this interface,
it
is
more
likely to
stresses at cross section
A
are
shown
in
be caused
These figures show that as the thickness of the exterior layer increases,
indicate that
lamina.
would
Figures 39, 40,
linear stress distribution intensifies in the layer closest to the contact surface.
would
it
stresses.
Comparisons of the bending
and
maximum
beam
theory
is
This nonlinear behavior
a
non-
This trend
unsuitable for estimating bending stress through this
is
local to the contact zone.
Figure 42 shows the
counterpart for Figure 41 at cross section C. The stress distribution in the contact layer
is
approximately
bending
linear.
stress graphs.
Another
As
significant observation
can be made by examining the
the thickness of the exterior layer opposite to the contact
layer increases, the stress distribution within this layer transforms
57
from purely
tensile
8
STRESS-X
S.S
5.0
7.5
10.0
HORIZONTAL DISTANCE FROM POINT OF CONTACT
Figure 31.
Stress contour:
<t
x
for
clamped beam model
behavior to compressivc-tensile behavior.
by
initiated
tensile stresses tends to
is
significant because a
bending crack
propagate to the core interface and cause delami-
The presence of compressive
nation.
This
stress within this layer will tend to
slow the growth
of this crack toward the core.
Thus
far
symmetric layups have been studied.
metric layups respond to contact loading, two
To analyze how beams
beams with
were studied: 0(3)-ISO(6)-0(7) and 0(8)-ISO(6)-0(2).
lamina closest to the contact surface.
utions for these two layups.
As
The
Figures 43 and 44
before, the
maximum
with asym-
the following designations
designated layer
first
show
is
the
the shear stress distrib-
shear stress
is
relatively unaffected
by the different layups. As was the case for the symmetric beams, a thicker exterior layer
close to the contact zone results in a
core.
The
As seen
same
as
in
result
is
more gradual
transition of shear stress into the
a lower shear stress at the interface for the 0(8)-ISO(6)-0(2) case.
Figures 45 and 46, the trends for the transverse normal stress, a y were the
,
those
found
in
the
svmmetnc beams.
58
Deflection
was
less
for
the
~ xi
I
r^y
6
vi
t
t
rrr
/ -v
^
r
r'
Figure 28.
Determination of nodal contact
properties outlined above, a sandwich composite
degrees was
beams
Since the
studied.
first
finite
studied will be described with the
element model
number of
is
finite
outer fibers aligned to
composed of
element layers
For example, a 0(3)-ISO( 10)-0(3) beam
following the layer description.
10 isotropic core layers within
beam with
3
16 layers, the
in
parenthesis
composed of
is
layers of material with the fibers oriented at
degrees
on the top and bottom of the beam.
Three symmetric layups of varying core thickness were
have
the
following
designations:
mations magnified 100 times.
and
stress
ites, is
As
is
32.
0(3)-ISO( 10)-0(3),
Stress contours for this
beam
The beams
are
beam
shown
in
with deforFigures 30,
Figure 30 shows the shear stress contour for the loaded condition.
of particular concern since delamination, a
commonly
initiated
by high shear
beam
increases
failure
mode
stresses or tensile transverse
the figure shows, a very high stress gradient
distance along the
common
away from
is
and
0(4)-ISO(8)-0(4),
Figure 29 shows the deformed 0(3)-ISO(lO)-0(3)
0(5)-ISO(6)-0(5).
31,
initially studied.
This
for
compos-
normal
stresses.
present near the contact zone.
As
the
the contact zone, the magnitude of the
gradient decreases until the cross sectional shear stress distribution becomes parabolic.
The
transverse normal stress
Figures 33, 34, and 35
three symmetric
is
also concentrated
show
around the contact zone.
the cross sectional shear stress distributions for the
beams described above.
Three separate cross sections are shown on
55
s
—
a'S
Figure 29.
'
7 S
ia-5
10 -°
POINT OF CONTACT
HWIZDKIfL DISTBNDE FROM
magnified 100 times
Loaded beam: deformation
"™*^^
Ftgure 27. The
of which are indicated in
each oraph the locat.ons
she, that
interfaces. These figures
lamma
the
identify
gr
Ph
onea'ch
.
<h«ir stress
"o?
is
a,, stgntfi
an
c-
However,
to the core thickness.
relatively insensitive
stress transtuon
tncreases, the shear
to the contact surface
ea
a reduction ,n the s
core. The resuit is
the tnterfa'c, w,th the
that
lamma. it ts also noteworthy
w.th th.c.er extenor
lavups
for
interface
atthe
sect.on A.
sect.on B vice cross
stress occurs at cross
shear
maxtmum
lh e
same
normal stress. ., for the
for the transverse
graphs
are
38
and
F.sures 36 37
nes
the th
s
show an increase in .. as
layups. These graphs
stronger
a
produces
these layers
The increased th.ckness of
,
o lavers increases
1
:
-est
mlg li «
I;:
J—
str esses increase.
Is,
.
of*
At
cross sect.on C,
some
stress
stated. tens„e transverse
is
stress
tens.le transverse
,s
euden,
of delam.nat.on.
a potentia, source
56
A.
As
P«
STRESS-Y: ZOOM OF CONTfCT ZONE
i
I
i
3.0
1.0
3.0
I
4.0
5.0
HORIZONTAL DISTANCE FROM POINT OF CONTACT
2
Stress contour: o y for clamped
Figure 32.
0(8)-ISO(6)-0(2) case.
As seen
in the
resulting smaller contact
symmetric cases, an increase
contact zone resulted
stress,
The
in
beam model
an increase
seen at cross section C.
in the
(In region of contact)
zone lead
in the thickness
magnitude of the
However, the
stress at the
to higher contact stresses.
of the layer closest to the
maximum
tensile transverse
laminate interface was always
compressive.
With regard
to
bending stresses for these layups, Figures 47 and 48 clearly show the
nonlinear behavior as the contact layer thickness increases.
In addition, the thickness
of the exterior layer opposite to the contact surface shows similar results as the symmetric cases.
As
the thickness of this layer decreases, the
bending crack that propagates into the interface.
59
beam
is
more
susceptible to a
0.2
0.4
0.S
0-8
NORMALIZED DEPTH BELOW CONTACT SURFACE
Figure 33.
Stress distribution: zxy for 0(3)-ISO(10)-0(3) laminate
60
^3*
LEGEND
GROSS SECTION
(ROSS SECTION
laDss-SECikm
0.4
0.2
0.8
O.S
NORMAL I ZED DEPTH BELOW CONTACT SURFACE
Figure 34.
Stress distribution:
t
for 0(4)-ISO(8)-0(4) laminate
61
o
i
•
'-i
m
•-
i
S-
»
*-~~^M
a
o
/
a.
co
—
I
>-?"
J
X
1
cs
m
J
a
j
T"
LEGEND
o«-
1
a
• -
(
ROSS SECTION A
C
ross Section B
r
Rrrvi
rfttthn r
i
o
0.0
0.2
0.4
0.8
0.0
NORMALIZED DEPTH BELOW CONTACT SURFACE
Figure 35.
Stress distribution: r
for 0(5)-ISO(6)-0(5) laminate
62
t
.0
CROSS SECTION
CROSS SECTION
[ROSS SECTION
a.o
0.2
0.4
O.B
0.8
NORMALIZED DEPTH BELOW CONTACT SURFACE
Figure 36.
Stress distribution: a y for 0(3)-ISO(10)-0(3) laminate
63
1.0
o
__J
1
a
a
T
//
/
<E
a.
r:
en
/
o
T
i
a
njFkin
'
o-
c
ROSS SECTION
* -
(
ROSS SECTION a
• -
[
ROSS SECTION c
a
0.0
0.2
0.4
O.S
0.8
NORMALIZED OEPTH BELOW CONTACT SURFACE
Figure 37.
Stress distribution:
<r
y
for 0(4)-ISO(8)-0(4) laminate
64
1.0
en
a.
a - [ROSS
X -
SECTION
(Ross Section
(ROSS SECTION
0.0
0.2
0.4
0.8
0.8
NORMAL I ZED DEPTH BELQU CONTACT SURFACE
Figure 38.
Stress distribution:
<t
for 0(5)-ISO(6)-0(5) laminate
65
1.0
J
/
V
/
/
LEGENO
Q-
0.0
0.4
o.a
[
ROSS SECTION
o.s
in
o.a
NORMALIZED DEPTH BELOW CONTRCT SURFACE
Figure 39.
Stress distribution:
<j x
for 0(3)-ISO(10)-0(3) laminate
66
i.o
t
o
J
to
x
'
ca
in
/
o
LEGEND
1
a-
'
[
I
ROSS SECTilQN n
o
0.0
0.2
0.4
0.8
0.8
NORMALIZED DEPTH BELOW CONTACT SURFACE
Figure 40.
Stress distribution: a x for 0(4)-ISO(8)-0(4) laminate
67
1.0
/
o
m
a
»
e
•
a
Co
X
C3
'
O
LEGENO
1
Q-
(
RQSS SECTION A
o
N
0.0
0.2
0.4
O.t
0.8
NORnflLIZED DEPTH BELflH CONTACT SURFACE
Figure 41.
Stress distribution:
<r,
for 0(5)-ISO(6)-0(5) laminate
68
1.0
/
o
E
Q_
X
CJ
'
o
i
LEGI
• -
c
ROSS SECTION c
o
w.
1
0.0
0.2
0.4
0.8
o.a
1.0
NORMALIZED 0EPTH BELOW CONTACT SURFACE
Figure 42.
Stress distribution: a x for 0(5)-ISO(6)-0(5) laminate at cross section
69
C
o
1
\
V^/
o
a
ri"
a
r
LEEENO
ROSS SECTION n
ROSS SECTION B
o
i-
r
RRSS SFTTrnN
in
o
0.0
0.4
0.2
3.8
O.S
NORMALIZED DEPTH BELOW CONTACT SURFACE
Figure 43.
Stress distribution:
t,,
for 0(3)-ISO(6)-0(7) laminate
70
l.a
*
A
^^-*c
I
—
i
w
i
U
U-
—
— — d£z*
—•
*
v^
^S^
a
EC
Q.
E
w
a
J
><
C3
en
a
T
LEGENO
o*o
• -
C
ROSS SECTION R
c
ROSS SECTION B
r
Rnqq SFTTTriN n
1
0.0
0.4
0.2
0.8
0.8
NORMALIZED DEPTH BELOW CONTRCT SURFACE
Figure 44.
Stress distribution:
x xy for
0(8)-ISO(6)-0(2) laminate
71
1.0
o
-**
•^
^
i
a
i
o
f
a.
9
hit
>-
in
o
T
i
a
rerun
oi-
[
ROSS SECTION a
*!"
1
ROSS SECTION B
•
(
ROSS SECTION C
1-
i
a
0.0
0.2
0.4
0.8
0.8
NORMALIZED DEPTH BELOW CONTACT SURFACE
Figure 45.
Stress distribution: a y for 0(3)-ISO(6)-0(7) laminate
72
1.0
1
w
o
n
•—
1
•
1
*f
o
r
o
T
O
T
i
o«O
• -
n;rwn
c
ROSS SECTION R
C
ROSS SECTION B
c
ROSS SECTION C
1
o
ri
0.0
0.2
0.4
3.8
0.8
NORMAL I ZED DEPTH BELOW CONTRCT SURFACE
Figure 46.
Stress distribution:
<r
y
for 0(8)-ISO(6)-0(2) laminate
73
L.O
»
e
LEGEND
GROSS SECTION
o.o
0.2
0.4
0.8
0,8
NORMAL I ZED DEPTH BELOW CONTRCT SURFACE
Figure 47.
Stress distribution:
<r x
for 0(3)-ISO(6)-0(7) laminate
74
1.0
/
o
in"
X
C9
^
in"
'
<n
o
LEGEND
t
a-
[
ROSS SECTION n
o
H.
1
0.0
0.2
0.4
O.B
o.a
NORMALIZED DEPTH BELOW CONTACT SURFACE
Figure 48.
Stress distribution: a x for 0(8)-ISO(6)-0(2) laminate
75
i.o
V.
CONCLUSIONS AND RECOMMENDATIONS
This study has developed two methods for approximating contact stresses using the
augmented Lagrange
multiplier method.
methods accurately approximate the
and plane
surface.
As
illustrated in Part
stresses that result
This study has also illustrated
how
A
of Chapter IV, these
from contact between
this
a cylinder
approach can be applied
to
understand the behavior of an actual contact problem by examining the response of a
composite plate to low velocity impact.
RAYLEIGH-RITZ APPROACH
A.
In the process of developing these methods, a
number of comments can be made
regarding the application of the Rayleigh-Ritz method to solving contact stress problems.
1.
The
is an extremely challenging process.
If it is dedetermine the deformation in a contact problem, the proper stress field
must be first satisfied. Because of this, the selection of possible trial functions is
limited.
For example, when selecting a trial function for the vertical deformation
of the foundation of Figure 1. a suitable selection is given by the following
selection of the trial function
sired to
equation:
v{xy)
=AX
)
cos
( -fff )
This equation exhibits the favorable characteristics of maximum deformation at the
contact surface and diminishing deformation as the distance from the contact surface increases. If contact stresses are to be modeled, this trial function is inappropriate. Calculation of t y is as follows:
~ dy-
=
-/Wl7T sin (ltf
)
This function exhibits zero strain at the point of contact increasing to
strain at the lower boundary.
2.
3.
maximum
Since the selection of trial functions is difficult, the task is further impeded by
complicated geometries. Furthermore, selection of a trial function necessitates that
some knowledge of the deformation field exists. Without a sensible selection of
trial functions, an accurate approximation is unlikely.
This method assumes the trial function in the form of an infinite series. Solution
accuracy theoretically should improve with an increased number of terms. However, precautions must be taken to ensure the solution is numerically stable as the
number of terms increases. Since the strain energy calculations require integration,
there are choices of trial functions that will increase without bound as the number
of terms is increased. This problem can be controlled by normalizing dimensions
or limiting the choice of trial functions.
76
4.
An
increase in accuracy was observed as the number of constraints was increased
and the distance between consecutive constraints was decreased. It is believed that
the improved accuracy results from a better definition of the contact surface.
FINITE
B.
The
ELEMENT APPROACH
results in
ment method
Chapter
III
illustrated that this
to contact stress analysis
is
effective.
approach of applying the
A number
finite ele-
of comments can be
made
regarding this approach to problem solving.
1.
this method accurately approximated the isotropic
However, some difficulties were encountered during the
bearing problem.
modeling of the multi-ply composite. In this model, smooth trends of decreasing
deformations were often interrupted by spurious deformations or groups of deformations.
These interruptions occurred within layers of significantly reduced
stiffness.
It is believed that the optimization routine had difficulty approximating
the deformations through these layers because of their very small contribution to
strain energy. As stated in the results, the contact boundary conditions were obtained from the optimization program and applied to a direct finite element program to solve the problem. The above difficulty is recognized as a limitation of this
As
illustrated in the results,
roller
approach.
2.
3.
This approach is much more flexible for complicated geometries than
Rayleigh-Ritz approach. In addition, detailed knowledge of the deformation
is not needed as required by the Rayleigh-Ritz approach.
the
field
The application of
static condensation is crucial to the successful implementation
method. Every effort should be made to reduce the number of design variables to improve optimization efficiency.
of
this
COMMENTS ON OPTIMIZATION
C.
A number
of observations were
Design Synthesis System and the
made
specific
regarding the general use of the
Automated
usage of the augmented Lagrange multiplier
method.
1.
The global optimum was more
was normalized.
likely to
be determined
when
the objective function
2.
For both the Rayleigh-Ritz approach and the finite element approach, the initial
choice of design variables had a significant affect on the possibility of obtaining the
global optimum. For the Rayleigh-Ritz method, initial selections of design variables can result in largely dissimilar values of strain energy and external work, the
two components of the objective function. It was determined that convergence was
more likely when optimization commenced with these two terms on the same order
of magnitude. With regard to the finite element approach, sensible choices of the
initial design variable vector was necessary for convergence to the global optimum.
This was accomplished by intuitive selection of design variables to model the likely
deformation.
3.
Solution accuracy can be improved by scaling constraint equations. It has been
stated that in some circumstances, some constraints change more rapidly than
77
others and can influence the solution excessively while others have
[Ref. 6: p. 136].
4.
little
influence
With regard to the usage of the augmented Lagrange multiplier method, it was
frequently necessary to tune' the optimization algorithm to a specific problem.
This was done by varying the initial penalty term p and the initial Lagrange multiplier term /
As stated by Vanderplaats, commencing optimization with a small
value of p should theoretically suffice for most problems [Ref. 6: pp. 137-138].
However, it was frequently necessary to select an initial value for p due to convergence to unrealistic solutions. Similarly an initial choice of the Lagrange multiplier term can effect the solution.
Commencement with a small value is again
recommended. [Ref. 6: p. 141]. This need to tune' the problem is a significant
drawback to using this optimization method. The ideal way to overcome this limitation is to first tune the optimization routine using a known solution. With this
accomplished, this approach can be used for meaningful data collection.
.
,
SANDWICH COMPOSITE MATERIAL STUDY
D.
The behavior of sandwich composite materials
to
low velocity impact loading was
successfully investigated by the application of the finite element approach.
of observations can be
1.
is
observed.
Tensile transverse normal stresses exist at some cross sections away from the contact zone. However, this stress is always compressive at the interface. Compressive
transverse stresses increase in beams with smaller cores due to reduced deflection
and contact zone
3.
the results.
The maximum shear stress is relatively insensitive to layer thicknesses. However,
as the thickness of the contact layer increases, a reduction of the interface shear
stress
2.
made from examining
A number
size.
As
the thickness of the layer closest to the contact zone increases, a nonlinear distribution of bending stress within this layer intensifies. This phenomenon is local-
ized to the region of contact.
4.
As
the thickness of the layer opposite to the contact zone increases, bending crack
propagation toward the core is less likely due to increased compressive bending
stresses within the layer.
E.
RECOMMENDATIONS FOR FURTHER STUDY
The methods developed
can grow.
Chapter
II
A
in this
reasonable direction
Part B.
frictionless surface
is
study offer a basis from which additional research
the relaxation of
For example, relaxation of the
some of
the assumptions
rigid roller
assumption would provide challenging research.
plex geometry could be created.
made
in
assumption and the
Models with com-
For example, a model of a pin loaded bolt connection
could be created with rigid or non-rigid pins.
Implementation of these changes would
provide a versatile and highly applicable model for contact stress analysis.
78
.
.
REFERENCES
1.
and Kubomura. K.. "Formulation of Contact Problems by Assumed
Hybrid Elements,' pp. 49-59: Wunderlich, W., et.al., Fds., Xonlinear Finite
Element Analysis in Structural Mechanics: Proceedings of the Europe-L'.S. WorkPian, T. H. H..
-
Stress
shop,
Ruhr-L
niversitat at
Bochum, Germany, Julv
28-31, 1980. Berlin, Springer,
1981.
2.
Cheng. W. Q., Zhu. F. W., and Luo, J. W., "Computational Finite Element Analand Optimal Design for Multibody Contact System," Computer Methods in
Applied Mechanics and E/igineering, v. 71, pp. 31-39. November 1986.
ysis
3.
4.
Nour-Omid, B.. and Wriggers, P.. "A Two-Level Iteration Method for Solution of
Contact Problems." Computer Methods in Applied Mechanics and Engineering, v.
54. pp. 131-144, February 19S6.
Guerra.
F.
\F, and Browning. R. V., "Comparison of Two Slideline Methods L'sing
Structures, v. 17, pp. 819-834, June 1983.
ADINA." Computers and
5.
Bischoff, D., "Indirect Optimization Algorithms for Nonlinear Contact Problems,"
pp. 533-545: Conference on the Mathematics of Finite Elements and Applications
(5th: 1984: Brunei University), London, Academic Press, 1985.
6.
Yanderplaats. G. N., Xumerical Optimization Techniques for Engineering Design
with Applications,
/
S.
McGraw-Hill Book Company,
D. A., and Lowe, M. J., Mathematical Programming
Lagrangians, Addison-Wesley Publishing Company, Inc., 1975.
Pierre.
10.
\'ia
Augmented
Rothert, H.. Idelberger. H., Jacobi. W., and Niemann, L., "On Geometrically
Nonlinear Contact Problems with Friction," Computer Methods in Applied Mechanics and Engineering,
9.
1984.
v.
51, pp. 149-154,
September 1985.
and Sun, C. T., "Impact Induced Fracture in a Laminated Composite,"
Journal of Composite Materials, v. 19, pp. 51-66, January 1985.
Joshi, S. P.,
Sun, C. T., and Rechak, S., "Effects of Adhesive Layers on Impact Damage in
Testing and Design
Composite Laminates," pp. 97-123, Composite Materials:
ASTM STP 972, J. D. Whitcomb, Ed., American Society for
( Eighth Conference)
Testing and Materials, Philadelphia, 1988.
,
1 1
Choi, H. Y., Wang, H. S., and Chang, F. K., "Effect of Laminate Configuration and
Impactor's Mass on the Initial Impact Damage of Graphite Epoxy Composite
Plates due to Line-Loading Impact," Journal of Composite Materials, v. 26. No. 6,
pp. 804-827, 1992.
A. C, and Fenster, S. K., Advanced Strength
Elsevier Science Publishing Co., Inc., 1987.
12. L'gural,
and Applied
Elasticity,
2d
ed.,
13.
Shislev, J. E., and Mishke, C. R., Mechanical Engineering Design, 5th ed., pp.
71-74. McGraw-Hill Publishing Company, 1989.
14.
Burnett, D. S., Finite Element Analysis from Concept
Addison-Wesley Publishing Company, 1987.
79
to
Applications,
p.
568,
15.
Kwon, Y. W., and Akin, J. E., "Materially Non-Linear Analysis of Body Contact
Using a Finite Element Method," Engineering Compulations, v. 3, pp. 317-322,
December
1986.
16.
Jones, R. M., Stechanics of Composite Materials, pp. 37-41, Hemisphere Publishing
Corporation, 1975.
17.
Babiloglu, E., A Numerical Study of Dynamic Crack Propagation in Composites,
Master's Thesis, Naval Postgraduate School, Monterev, California, September
1992.
80
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