Shape optimization of trusses subject to strength, displacement, and frequency constraints.

Shape optimization of trusses subject to strength, displacement, and frequency constraints.
Calhoun: The NPS Institutional Archive
Theses and Dissertations
Thesis Collection
1981
Shape optimization of trusses subject to strength,
displacement, and frequency constraints.
Felix, Jorge E.
Monterey, California. Naval Postgraduate School
http://hdl.handle.net/10945/20666
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THESIS
SHAPE OPTIMIZATION OF TRUSSES SUBJECT TO
STRENGTH, DISPLACEMENT, AND
FREQUENCY CONSTRAINTS
by
Jorge E. Felix
December 1981
Thesis Ac Ivisor;
Garret N. Vanderplaats
Approved for public release; distribution unlimited
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PERFORMING ORGANIZATION NAME ANO AOORESS
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December 1981
AuTmORi'«J
Jorge
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Shape Optimization of Trusses Subject to
Strength, Displacement, and Frequency
Constraints
7.
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December 1981
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Structural optimization, finite elements, structural configuration,
trusses, frequency constraints.
ABSTRACT
20
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Three-dimensional trusses are designed for minimum weight,
joint
subject to constraints on: member stresses, Euler buckling,
static
Multiple
displacements and system natural frequencies.
load conditions are considered.
_
and
The finite element displacement method of analysis is used
technique.
eigenvalues are calculated using the subspace iteration
All gradient information is calculat ed analytically.
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The design problem is cast as a multi-level numerical optimization problem. The joint coordinates are treated as system
For each proposed configuration, the member sizes are
variables.
updated as a sub-optimization problem.
This sub-problem is
efficiently solved using approximation concepts in the reciprocal
variable space. The multi-level approach is shown to be an
effective technique which conveniently takes advantage of the most
efficient methods available for the member sizing problem.
The optimum
Examples are presented to demonstrate the method.
configuration is shown to be strongly dependent on the constraints
which are imposed on the design.
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Approved for public release; distribution unlimited
Shape Optimization of Trusses Subject to
Strength, Displacement, and
Frequency Constraints
by
Jorge E. Felix
Lieutenant, Ecuadorian Navy
B.S., Naval Postgraduate School, 1981
Submitted in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
from the
NAVAL POSTGRADUATE SCHOOL
December 1981
DUDLEY KNOX UBIt/^Y
NAVAL POSTGRADUATE SCHOOL
ABSTRACT
Three-dimensional trusses are designed for minimum
weight, subject to constraints on:
member stresses, Euler
buckling, joint displacements and system natural frequencies
Multiple static load conditions are considered.
The finite element displacement method of analysis is
used and eigenvalues are calculated using the subspace
iteration technique.
All gradient information is calculated
analytically.
The design problem is cast as a multi-level numerical
optimization problem.
system variables.
The joint coordinates are treated as
For each proposed configuration, the
member sizes are updated as a sub-optimization problem.
This sub-problem is efficiently solved using approximation
concepts in the reciprocal variable space.
The multi-level
approach is shown to be an effective technique which con-
veniently takes advantage of the most efficient methods
available for the member sizing problem.
Examples are presented to demonstrate the method.
The
optimum configuration is shown to be strongly dependent on
the constraints which are imposed on the design.
TABLE OF CONTENTS
I.
INTRODUCTION
12
II.
MATHEMATICAL FORMULATION
16
A.
INTRODUCTION
16
B.
ANALYSIS
16
C.
1.
Static Analysis
16
2.
Dynamic Analysis
18
ANALYTIC GRADIENTS OF THE CONSTRAINTS
19
1.
Gradient of Member Stresses with Respect to
the Reciprocal of Area Variable
20
2.
Gradient of Nodal Displacements with Respect
20
to the Reciprocal of Area Variable
3.
Gradient of Frequencies with Respect to the
Reciprocal of Area Variable
22
4.
Gradient of Stresses with Respect to the
Joint Coordinate Variables
23
Gradient of Displacements with Respect to
the Joint Coordinate Variables
23
Gradient of the Natural Frequencies with
Respect to the Joint Coordinate Variables
24
5.
6.
D.
APPROXIMATION CONCEPTS
25
E.
OBJECTIVE FUNCTION
27
F.
CONSTRAINTS
27
1.
Stress
27
2.
Euler Buckling
28
3
Frequency
28
.
G.
III.
IV.
V.
4.
Limits on Areas
28
5.
Displacement
23
6.
Limits on Coordinate Variables
29
GENERAL FORMULATION
OPTIMIZATION
29
31
A.
INTRODUCTION
31
B.
GENERAL FORMULATION
31
C.
OPTIMUM GEOMETRY DESIGN
32
D.
FIXED GEOMETRY DESIGN
34
NUMERICAL EXAMPLES
A.
INTRODUCTION
B.
CASE
1:
18-BAR PLANAR TRUSS
36
36
36
1.
Case la
37
2.
Case lb
38
3.
Case lc
38
4.
Case Id
38
C.
CASE
2:
47-BAR PLANAR TOWER
39
D.
CASE
3:
25-BAR SPACE TOWER
40
CONCLUSIONS AND RECOMMENDATIONS
42
A.
CONCLUSIONS
42
B
RECOMMENDATIONS
4 3
.
APPENDIX A
44
LIST OF REFERENCES
70
INITIAL DISTRIBUTION LIST
72
LIST OF TABLES
I.
18-Bar Planar Truss, Loads and Constants
56
II.
18-Bar Planar Truss, Design Information. Stress
57
III.
18-Bar Planar Truss, Design Information. Stress,
Euler Buckling
58
18-Bar Planar Truss, Design Information. Stress,
Euler Buckling, Displacement
59
V.
47 -Bar Planar Tower.
Load Conditions
60
VI.
47-Bar Planar Tower.
Load Conditions
CI
VII.
47-Bar Planar Tower.
Area.
IV.
VIII. 47-Bar Planar Tower.
Stress, Euler Buckling-62
Coordinates.
Stress, Euler
Buckling
IX.
X.
XI.
XII.
63
47-Bar Planar Tower.
Coordinates.
Buckling, Displacement
47-Bar Planar Tower.
Displacement
Areas.
Stress, Euler
64
Stress, Euler Buckling,
65
47-Bar Planar Tower. Areas.
Displacement, and Frequency
Stress, Euler Buckling,
Stress, Euler
Coordinates.
47-Bar Planar Tower.
Frequency
and
Displacement,
Buckling,
66
67
XIII.
25-Bar Space Tower.
Load Conditions
68
XIV.
25-Bar Space Tower.
Design Information
69
LIST OF FIGURES
A.l.
A. 2.
18-Bar Planar Truss. Stress, Euler Buckling,
Displacement, and Frequency Constraint
44
18-Bar Planar Truss.
Stress
45
Weight vs. Iteration Number,
A. 3.
18-Bar Planar Truss. Weight vs. Iteration Number.
46
Stress, Euler Buckling
A. 4.
18-Bar Planar Truss. Weight vs. Iteration Number,
47
Stress, Euler Buckling, Displacement
A. 5.
18-Bar Planar Truss. Weight vs. Iteration Number.
Stress, Euler Buckling, Displacement, and
Frequency
48
A. 6.
47-Bar Planar Tower
49
A. 7.
47-Bar Planar Tower.
Stress, Euler Buckling,
Displacement, Frequency
50
A. 8.
47-Bar Planar Tower. Weight vs. Iteration Number.
51
Stress, Euler Buckling
A. 9.
47-Bar Planar Tower. Weight vs. Iteration Number.
52
Stress, Euler Buckling, Displacement
A. 10.
47-Bar Planar Tower. Weight vs. Iteration Number.
Stress, Euler Buckling, Displacement, Frequency
53
A. 11.
25-Bar Space Tower
54
A. 12.
25-Bar Space Tower. Weight vs. Iteration Number.
Stress, Euler Buckling, Displacement, Frequency
55
LIST OF SYMBOLS
Cross-sectional area of member
A^
A:llcl.X
A_.
ill
-LIl
i
i
Minimum and maximum allowable area variables,
respectively
,
.
a,, b.
Constants defined by Eq.
c
Factor defined by Eq.
D
Matrix of direction cosines
E.
Young's Modulus for member
F
Vector of applied loads
G. (X)
Constraint function of coordinate variable
G. (X
m
3
G,
k
(X
m
43
31
i
)
Constraint function of area variable
)
Active constraints of area variable
I
Identity matrix
K
Global stiffness matrix
K.
Element stiffness matrix
zi.
L.
Length of a bar
M
Global mass matrix
M
Global generalized mass matrix
M_
Lumped mass matrix
NAC
Number of active constraints
NIC
Number of initial constraints
S
Direction vector
~G
*L
i
Direction vector on the coordinate design space
S
g
S
m
Direction vector on the area design space
9
u
Nodal displacement vector
u-
ith nodal displacement vector
^/
Minimum and maximum allowable displacement value
^
X
Vector of design variables
X
Vector of coordinate design variables
X
Vector of area design variables
m
1th independent design vector
X„
X,
u,
a,
Minimum and maximum allowable coordinate variable
X,
,
a*,
v
ai, a 2
Direction cosines
Scalar parameter defining distance of move in
the area or coordinate design space.
S
Objective function of direction function problem
5
Displacement constraint value
57
5.
Lower and upper limits on displacement constraints
5
Kronecker delta
„
9.
Push-off factor
g.
Stress in member
a
stress in member
Allowable compressive
c
ci
Allowable tensile stress in member
a.
a.
i
p.
Material density
(J).
Eigenvector corresponding to
to.
ith eigenvalue
$
2
Q
X
i
Allowable Euler buckling stress in member
.
bi
co~,
i
co
i
to.
Minimum and maximum allowable fundamental frequency
Nodal matrix
Expectral matrix
Frequency limit
10
ACKNOWLEDGEMENT
The author wishes to express his deep appreciation to
his faculty advisor, Professor Garret N. Vanderplaats
,
for
his continuous guidance, assistance and encouragement as
well as his great enthusiasm during the course of this
project; to Professor David Salinas who read this project,
and gave of his time and engineering expertise; to the
distinguished members of the faculty and the staff of the
Department of Mechanical Engineering for their support in
very way, my sincere appreciation.
This investigation would not have been possible without
the facilities and support of the W.
R.
Church Computer
Center.
Infinite thanks and grateful appreciation go to my wife,
Yolanda, my son, Paulo, and daughter, Maria Gabriela, whose
understanding and encouragement made the difficult time
bearable.
11
I.
INTRODUCTION
The process of optimization of structures has undergone
important changes since its development in the early 1960s.
Minimum weight of elastic truss structures subject to multiple loadings has been an active area of research.
Attention
has been focused on the problem of least weight, when the
overall layout was known in advance, and when the crosssectional area was the design variable.
Some attention has
been directed toward the optimum configuration of the
structure.
Design improvements in this area often exceed
those in fixed-geometry and so shape optimization is of
major interest.
Pioneer work in shape optimization was conducted in 1964
[Ref.
1]
by Dorn, Gomory and Greenberg.
The optimal con-
nectivity of nodes for truss members, subject to a single
load condition, was found and minimum weight designs were
achieved.
In their work only planar trusses were tested,
and the process was presented as a plastic design problem
using linear programming.
Their work was followed by Dobbs and Felton [Ref.
2]
who in 1969 investigated the effect of multiple load conditions on the optimum configuration of trusses through the
use of non-linear programming methods.
12
Again, only planar
trusses were considered, subject to failure by stress and
elastic buckling.
Later, Pedersen [Ref.
considered the
3]
positions of
the joints as continuous design variables in addition to
the areas of the bars.
Stress, displacement, and buckling
constraints were considered.
Pedersen
1
s
work is significant
because the optimization process is carried out by considering two separate design spaces.
The optimization is
achieved by successive iterations using a gradient method
with move limits.
The optimization process was advanced by Vanderplaats
and Moses
[Ref.
4]
who divided the design space into two sub-
spaces, separating the area variables and the joint position
variables.
Multiple load conditions and constraints on
stresses and Euler buckling were considered.
The optimiza-
tion was carried out alternatively between the two spaces
until convergence was achieved.
Three-dimensional indeter-
minate trusses were designed subject to multiple loading
conditions.
This work was extended by Vanderplaats [Ref.
5]
to include displacement constraints.
In other research,
indeterminate trusses.
Spillers [Ref.
6]
considered statically
The optimization followed an itera-
tive design where the member sizes and node locations were
the design variables.
13
Recently, Imai
[Ref.
7]
treated the sizing and con-
figuration variables simultaneously for either determinate
or indeterminate trusses.
The optimization was achieved
using the Augmented Lagrange multiplier method.
The design problem considered in this study is the opti-
mum configuration of three-dimensional indeterminate
trusses, for multiple prescribed static load conditions.
The objective is to minimize the weight of the structure,
design variables are the node coordinates and the member
sizes.
Constraints include stress, Euler buckling, dis-
placement, and the natural frequency of the system.
Some approximation concepts are introduced in order to
reduce computational effort and to reduce the nonlinear
characteristic of some constraints.
Among these, a first
order Taylor Series expansion is applied to approximate the
constraints.
The optimization proceeds iteratively in two design
spaces:
the member sizing space, where the structure is
optimized for a fix layout, and the coordinate space, where
the geometry is allowed to vary.
In both optimization
processes the minimum weight of the structure was maintained,
subject to the requirements that the constraints remain
satisfied.
The mathematical formulation is presented in Chapter II.
The objective function and constraints are defined in terms
14
of the design variables.
The analytic gradients are also
formulated.
The optimization technique is discussed in Chapter III.
Several examples are presented and explained in Chapter
Conclusions
and recommendations for future work are
presented.
15
IV,
II.
A.
MATHEMATICAL FORMULATION
INTRODUCTION
Several features are desirable when finite element
methods of structural analysis are used in optimization.
First, the number of analyses for the structure should be
kept as low as possible.
Second, the amount of gradient in-
formation required during the design process should be
reduced as much as possible.
B.
ANALYSIS
Static Analysis
1.
The initial layout of the truss, the member sizing
and material properties
(which may be different for each
member), a set of external loads, and support conditions
are initially specified.
The analysis for the stresses and deflections must
be carried out satisfying the conditions of equilibrium of
forces at the nodes and compatibility of deformation.
If
the material of the structure behaves in a linear manner,
Hooke
'
s
law will establish the force-deflection relationship
For a truss, it is also necessary to establish the
following assumptions before selecting the method of computing the internal forces.
The discrete element is treated
as pin-connected, and loads and reactions are supported at
16
the joints.
In this study,
the weight of the members is
not included as loads.
The Displacement (Stiffness) method [Ref.
8]
considers the joint displacement components as the unknowns,
and is written in the most general form using matrix notation,
Ku = F
(Eq.
1)
where K is the global stiffness matrix, F is the vector or
vectors of applied loads, and u is the vector or vectors of
displacements.
Equation
is the set of equilibrium equa-
1
tions, and is formulated such that the compatibility is auto-
matically satisfied.
The generality of the method is important if either
statically determinate or indeterminate trusses are analyzed.
The global stiffness matrix is symmetric and sparce.
These
features are used to write the code for a computer solution,
and the matrix K is stored in compact form for efficient
numerical solution.
Once the displacements at every node are known, the
internal forces or stresses are calculated by applying forceThis is defined as:
deflection relations.
r
E.
*n
ij
=
D
_
T
-VL
\
u„
i
and
j
- u.
^
^
<
V
£j
w„
where
.
.
V
kj
- w.
)
(Eq.
2)
.
are element and load condition numbers respec-
tively, and k and
I
are node numbers associated with the
17
element
and
i.
a.
is the stress,
.
is the length.
L.
direction cosines.
E.
Matrix D =
is the Young's modulus,
f
(a,y,v)
contains the
For brevity, hereafter the second sub-
script is omitted and it is assumed that the stress or
displacement corresponds to the appropriate loading condition,
2
Dynamic Analysis
.
When system natural frequency constraints are imposed
in the design process, the corresponding dynamic analysis of
the structure has to be carried out.
This requires the
solution of an eigen-problem to determine the natural fre-
quencies and normal modes.
For linear elastic structures,
the finite element approach leads to the well-known equation
of motion, considering free vibration conditions,
£Ju +
=
Ky
(Eq.
3)
where M is the global mass matrix, and u is the linear acceleration.
Assuming a solution of the form
ia)t
u = ^e
where
oo
(Eq.
4)
is the angular natural frequency of vibration of
the structure, and
$
is the corresponding eigenvector.
After substitution into Eq.
3,
the generalized eigenvalue
problem becomes,
K<j>
=
2
u)
(Eq.
M$
Written in matrix form for several eigenvalues, Eq.
5)
5
becomes,
KO =
2
tt
m
(Eq.
18
6)
where
$
is the
modal matrix, and 2 1 = diag
2
(oj
)
is the spectral
matrix.
From the static analysis, the global stiffness
matrix is already calculated, then only the global mass
matrix evaluation is needed.
mass options are coded.
Both generalized and lumped
These are defined respectively as:
NE
&
o-A.L.
= .1 p i l l
1=1
^
-
2I
l
3
I-,
3
(Eq.
7)
(Eq.
8)
21
and
NE
1
1=1
A.L.
ill
L.
•
6x6
6
VJhere I is the identity matrix,
method of Bathe and Wilson [Ref.
the sub-space iteration
9]
is used to solve for a
specified number of lowest eigenvalues and the associated
eigenvectors.
large problems.
The method is economically efficient for
The mass matrix may be diagonal or banded.
The method is well suited for re-analysis when small
changes are made in the design.
C.
ANALYTIC GRADIENTS OF THE CONSTRAINTS
The necessity to compute gradients of the relevent
functions in a design optimization process arise from the
fact that efficient mathematical programming algorithms
require information on derivatives.
Furthermore, approxi-
mate methods based on a Taylor Series expansion of functions,
19
requires determination of derivatives.
Based on the static
and dynamic analyses, the gradients of forces, displacements,
and frequencies with respect to the reciprocal of the cross-
sectional areas and the coordinate variables, are formulated.
1
.
Gradient of Member Stresses with Respect to the
Reciprocal of Area Variables
Stress in a member is defined as
F.
1
(Eq.
9)
(Eq.
10)
(Eq.
11)
(Eq.
12)
(Eq.
13)
also
K.ua
=
.
The partial derivative is then
K
3a.
3
ax
3X
A.
l
~l
*i
+
A7 9X-e
u^
The first term of the right hand side is zero, then
—- — ^—
"£
3X„
3X„
A,
(u.
v
~l'
I
This can be written in explicit form as
3£i
s
5s>
T
E.D.
L?
l
2.
Wt
(yi)
Gradient of Nodal Displacements with Respect to the
Reciprocal of Area Variable
Consider the equation
(Eq.
Ku = F
20
14)
The derivative of [K u] with respect to some variable
X,
is
3
[KU]
n
In this case X
=
^"(F)
ax,
= 1/A^
£
(Eq.
15}
and the loads are constant and in-
dependent of the areas (weight of the truss elements are
ignored)
,
then
(?) (u)
t|-
~-(u)
+ ?
=
(Eq.
16)
(Eq.
17)
(Eq.
18)
or
§ 3X
Finally, then
9
3X
where K
-1
(u)
= -k' 1
(K) u
3
?£
is the inverse of K.
It is necessary to compute
the partial derivative of the global stiffness matrix K
defined as:
K =
NE
E A
X
-fc±
i=l
where
D.
i
.
[D
(Eq.
±]
19)
is the matrix of direction cosines defined earlier,
Therefore, the partial derivative with respect to the
reciprocal of the area variables is defined as:
.
NE
E.Af
(Eq.
*l
i=l
i
21
20)
.
where
and
is the kronecker delta defined by
5
=
5
= 1 if i = l
6.
In practice the full matrix,
if i ^ I.
Trvr-(K)
*i
3
is not actually stored.
created directly.
are given in [Ref
3
.
Instead the product, tttt-(K)u is
Details of efficient gradient computations
.
10]
Gradient of Frequencies with Respect to the Reciprocal
of Area Variable
Consider the eigen-problem defined by,
[K - co?M]
=
0.
(Eq.
Taking the derivative with respect to the variable X
21)
[Ref.
11]
gives
4dX
•£
9
{[Ka -
1~
(Eq.
=
$.}
-1
u)|M]
22;
then
- u>?M]}
^|- {[g
T
Pre-multiplying through by
symmetry of the matrix
[K - oojM]
g|-
to.
*I
«
lt h
»!«
i
=
(Eq.
23)
gives
M
^"^
3M
3£
"
<J)
and applying the condition of
<£.
-
Ik
+
$.
{
]
m
2
'*i
~
3u>.
1
3X^
»
(Eq.
*i
24)
The left hand side of the equation is zero because of Eq. 21,
and thus
3
3w?
X
l
3M
K
w
_3x £
T
f
*1
i
Mi~l
*
22
ax
Ji
(Eq.
25)
-
The partial derivative of the generalized mass matrix with
respect to the reciprocal of the area variable is defined as
NE
3.M
~G "
3
p.AfL.
ill
I
?i
2I
°i£
i=l
I
3
3
h
(Ec.
2I
26)
3
The partial derivative of the lumped mass matrix with respect
to the reciprocal of the area variable is formulated following
the same procedure as the generalized mass matrix.
4.
Gradient of Stresses with Respect to the Joint
Coordinate Variables
Since stress in a nember is
K.11.
=
a.
~i
(Eq.
27)
(Eq.
28)
the gradient is calculated as
«£
8X„
In this case,
—?i
3
1
(Jl_k.
u.
)
]
+ _i_
k.
1
1
~l
the stiffness matrix is a function of the
coordinate variables so the first term on the right hand
side of Eq. 28 is not zero.
5
.
Gradient of Displacements with Respect to the Joint
Coordinate Variables
Consider the following equation
Ku = F
(Eq.
29)
(Eq.
30)
The gradient of u with respect to X I is:
(u)
**i
= -K
-1
JL-(K)u
**l
23
.
The partial derivative of the global stiffness matrix K now
is defined as:
NE
n
A.E.
3A.E.
9K.
(K)
l)
(dX
l|
frJ
i
3
i
)
[K
i
(Eq.
where (-1)
*
and
j
]
31)
= 1 or 2 defines the sign of the gradient
at that particular node.
Note that the terms in the summa-
tion are only evaluated for members connected to the joint
defined by the variable X ?
6
.
.
Gradient of the Natural Frequencies with Respect to
the Joint Coordinate Variables
The value of the derivative of the eigenvalue
co
2
found from
2**1 A
u>fM]<j>.
[K -
_
=
n
(Eq
.
32)
(Eq.
33)
is
3M
3K
3X,
9o)f
CO'
SX
•i
T
<f>7
M
h.
I
.
c|).
The partial derivative of the generalized mass matrix with
respect to the coordinate variables are defined as:
NE
.
3X
£
where (-1)- and
""I
i=l
1
j
("I)
l
o
A dX
l
li
.
.
6L.
2I 3
I3
I3
2I 3
(Eq.
34)
l
= 1 or 2 defines the sign of the gradient
at the particular node.
The partial derivative of the lumped
mass matrix follows the same procedure.
24
D.
APPROXIMATION CONCEPTS
At this stage the analysis tools necessary for the design
process under static and dynamic conditions may be regarded
as available.
It has been pointed out that the application
of mathematical programming methods for structural design is
the most widely used because of its great generality and its
simple formulation.
However, it is required that a large
number of structural analyses and sensitivity analyses be
performed.
This has motivated the idea of formulating simple
and explicit approximations for the most relevant response
quantities.
These approximations can only be expected to be of
acceptable quality in some finite region of the design space
surrounding the point about which the approximations were
constructed.
The total number of analyses required to find
an optimum design using approximation concepts is significantly less than the number previously required.
Some sources of simplification can be considered.
First,
the dimension of the design space can be reduced if a proper
subspace can be identified.
Second, linking of design
variables which is imposed because of symmetry or practical
considerations also reduces the dimension of the design
space.
The objective function for the design of trusses is a
relatively simple explicit function of the design variables.
25
On the other hand nonlinear constraint functions are very
complicated.
From a computational standpoint, a small
portion of the constraints play an active role in the opti-
mization process; therefore, deletion of non-critical
constraints avoids effort of evaluation of irrelevant
constraints.
A method to deal with complicated constraint which is
very effective in reducing computational effort is to deduce
simple and explicit expressions for the constraints.
Linearization is directly and efficiently accomplished by
Taylor series expansions.
The order of the expansion
selected is decided based on the degree of nonlinearity and
the approximation required; and a trade-off must be made
between the computational effort required for the highest
order derivatives versus improvement of approximation.
Application of a first order Taylor series expansion on
stress constraints has been found to be sufficient.
This
was not the case for the natural frequency constraints,
which may still be numerically unstable during the optimization.
order.
This suggests further expansion up to a second
However, the computational effort needed to do this
usually exceeds the benefits.
Therefore, a first order
expansion with move limits is used.
First order Taylor series expansion of a function, W,
of the variable, X, about a point, Xo, is written as
W(p)* W(x) = W(Xo) + (X-Xo) VW(Xo)
26
The philosophy underlying the use of linear approxima-
tions is not to transform the problem into a sequence of
linear programs, but to replace the constraints by simple
and explicit approximation functions.
In this study the
Taylor series expansion was used for all constraints with
respect to the reciprocal of the member sizing variables, A..
The objective function, in reciprocal space, is now non-
linear but still explicit and easily evaluated.
E.
OBJECTIVE FUNCTION
The function whose least value is sought in the optimi-
zation procedure is defined as the total weight of the truss
which is given by
NE
W =
V p.A.L.
.£-,111
(Eq.
35)
1=1
Where NE is the total number of members, and p^ is the
material density, both are prescribed constants.
L.
F.
A^ and
are the area and length of the ith member respectively.
CONSTRAINTS
The restrictions to be satisfied in order for the design
to be acceptable are formulated explicitly; behavioral and
side constraints are defined accordingly.
1.
Stress
O
.
ci
<
—
a
.
l
<
—
(Eq.
cr
ti
27
36)
.
.
.
where o^^ and
a
^
are the allowable compressive and tensile
stresses respectively for member
i.
Euler Buckling
2.
The Euler buckling stress is formulated as
cA.E.
= " -q^-
a
bi
<=*•
37
(Eq.
38)
>
where c is a prescribed constant.
Frequency
3.
W
where
^ W K ^ ^K
K
and w K are the lower and upper bounds respectively
oj
on the first natural frequency of the system.
4
Limits on Areas
A
mm
where, A
mm l
.
,
.
<
.
<
A.
i
.
and A
max
A
max
(Eq.
^
39)
are the minimum and maximum allowable
l
cross-sectional areas of the ith member, and are taken to
When symmetry of design is to
be the same for all members.
be preserved,
linking of the variables is required.
This
is defined as
Aj^
where K and
5
i
= A.
(Eq.
40)
(Eq.
41)
are symmetric members.
Displacement
<
ll
<
u*
are the lower and upper bounds respectively
where u«
and u v
K
K
on joint displacements u
.
28
,
6.
Limits on Coordinate Variables
X
X
K " K ~ XK
<
EcI-
42)
where again X and X are the lower and upper bounds
K
K
respectively on the Kth coordinate variable, and if symmetry
must be preserved, linking of the variables is required.
This is obtained by
X
i
= a
i
+ b
l
X
K
<
where a and b^ are constants and X
£
ECI-
43
>
is the coordinate
variable.
G.
GENERAL FORMULATION
The inequality constraints are of the form,
G (X)
.
<
j
= 1
NIC
(Eq.
44)
The constraints are normalized as follows:
Stresses
a
-1*0
a
-1*0
(Eq.
45)
^--1*0
(Eq.
46
°c
Euler Buckling
°b
Note that because
a
and
a,
depend on the member area
this constraint is treated as a nonlinear function.
Taylor series expansion is performed on
of
a,
is continually updated.
29
a
A.
The
and the value
Displacements
-1-0
—
6-
A5
Frequency
1 <
(Eq.
47)
(Eq.
49)
(Eq
50)
+
£-1*0
00
side constraints are
\
" Ai " °
x
-
X
K
x
K
<
X
K " K "
30
°
*
OPTIMIZATION
III.
A.
INTRODUCTION
The main goal of structural engineering is to design
structural systems that efficiently perform specified
purposes.
Selection of a specific algorithm is required
and this algorithm must minimize the number of times the
structure has to be analyzed and the amount of specific
Finally, the algorithm
gradient information required.
should provide reasonable assurance that it will attain an
optimum or near-optimum design.
The next two sections are a brief explanation of the
algorithm used for this work.
B.
GENERAL FORMULATION
The general constrained minimization design problem is
defined as
minimize
W(X)
(Eq.
1)
(Eq.
2)
subject to
G. (X)
where, W(X)
^
= 1,
j
.
.
.
.m
is the objective function.
the set of inequality constraints.
Functions
.
y
31
are
The vector of design
variables X includes member sizing variables X
metric design variables X
G. (X)
and geo-
OPTIMUM GEOMETRY DESIGN
C.
The procedure used here was to treat the geometric design
parameters as independent design variables.
The member sizing
parameters are handled as dependent variables which are
determined as a sub-problem.
Beginning with an initial geometric design vector X
the
,
design proceeds iteratively using the following relationship:
X
g+1 = q
X + a* S*
g
g
g
(Eg.
4
where q is the iteration number and
a
is the search direction
S
y
*
to be determined.
3)'
v
g
is the scalar parameter determining the
distance of travel in the design space.
For each proposed geometric vector, X
,
the structure
was optimized with respect to the member sizing variables,
X
,
by the sub-optimization problem defined in Section D.
Assume that for the initial geometry the structure has
been optimized with respect to the cross-sectional areas,
and that, from this subproblem there are
I
active constraints
of the form:
G. (X)
where, G.(X)
=
k - 1,...£
(Eq.
4)
is defined as active if its value is close to
3
zero.
Now, it is necessary to find the search direction,
S
so that by moving in this direction in the coordinate
design space, the objective function is minimized.
This
direction is found by solving the following subproblem.
32
,
y
.
Minimize
VF(X).S
(Eq .
5)
(Eq.
6)
(Eq.
7)
Subject to:
G
R
(X)
S
k = 1,
<
.
S
.
S * 1
£
At this point, S, provides a search direction in the
combined [X
constraints.
Once,
kj
+
S
space which is projected onto the active
Only the
S
part of
is used.
is known it is substituted back into Eq.
,
and a one-dimensional search on a
X.
the vector,
S
y
3
*
is performed to update
The Optimum-Geometry design problem can be
summarized in the following algorithm.
Given an initial coordinate design vector, X
1.
area design
vector, A
3
o
.
Specify,
D
2
r
max and P max
.
'
(D
,
and
max',
maximum change in the coordinate at each iteration, and
P
,
total number of iterations.)
Solve the fixed geometry problem and calculate the
2
minimum weight
(W*)
for the current geometry.
3.
Determine the set of active constraints.
4.
Determine the search direction vector,
5.
Find the move parameter
S.
*
a,
for a D
change of some
coordinate variable, or such that some coordinate constraints
becomes active.
Solve the fixed geometry problem for coordinates.
6.
X
g
= X
g
and calculate the optimum weight, W(X).
+ a, S
1
g
33
.
*
Find the parameter a-# which minimizes the weight
7.
using 2-point quadratic interpolation.
step
If a. >a.
do to
9.
Solve the fixed geometry design problem for coordi* -a
-a+1
-a
= X^ + ou S M
nates X^
and calculate the optimal
weight,
r
8.
g
g
9.
g
Check convergence; if satisfied, terminate; otherwise,
set q = q+1
D.
2
,
update X and return to step
3
FIXED GEOMETRY DESIGN
As stated earlier, a sub-optimization problem has to be
solved for a proposed geometry design vector,
X.
The
structure is now optimized based on the cross-sectional
This is defined in general as:
area subspace.
Minimize,
W(X)
(Eq.
8)
(Eq.
9)
Subject to,
G. (X)
<
j
= 1,
I
The design proceeds iteratively.
of design variables X
,
Given an initial vector
find the X vector at the (q+1)
ith
iteration defined as:
X<3
m
*
where, a m
+1 = q +
X* S q
X
m
m
(Eq.
m
-
,
-
is a scaler multiplier, and, S m
,
10)
is a vector move
direction in the design space.
Now the problem becomes one of finding the direction,
S,
and the move parameter, a*.
34
Zoutendijk [Ref. 12] shows
that the direction may be found by solving the following
problem.
Maximize
8
(Eq.
11)
(Eq.
12)
(Eq.
13)
(Eq.
14)
Subject to
VF(X)
G. (X
m
S
S
.
S
.
S +
.
)
<
+
-
3
9
6
<
j
= 1,...NAC.
1
where the scalars
6.
are named as "push-off" factors.
If some of the constraints are violated, this algorithm
is modified in order to find a feasible design
[Ref.
13].
For the fixed geometry sub-problem, the reciprocals of
the member sizes are used as design variables and approxima-
tion techniques are employed [Ref.
5]
.
The optimization
program CONMIN [Ref. 14] is used to solve this sub-problem.
35
IV.
A.
NUMERICAL EXAMPLES
INTRODUCTION
Design of planar trusses and space towers are presented
here and the corresponding numerical results are summarized
to demonstrate the purpose of this study.
The examples begin with an 18-bar truss.
In this case,
it is shown how the optimum geometry is dependent on the
constraints imposed.
A single load condition was considered
and the cross-sectional areas are linked.
Next, a
4
7-bar planar tower is designed to
of load conditions given in Table VI.
support a set
The design is subject
to constraints on the member stresses, Euler buckling, dis-
placement and first natural frequency.
Linking was imposed
for symmetry in both cross-sectional area and coordinate
variables.
Finally, a 25-bar space tower was designed.
Constraints
on stresses, Euler buckling, displacement, and frequencies
were imposed.
The truss was required to support two
different load conditions.
Symmetry for the member areas
and coordinates was established.
B.
CASE
1:
18-BAR PLANAR TRUSS
A cantilever truss, as shown in Fig. A.l, has been used
previously as an example for the design of trusses of
36
a
;
specified geometry [Ref
.
7]
This structure was designed
.
for optimum geometry subject to a single set of load condi-
tions given in Table
The allowable stresses are speci-
I.
fied as
-20,000
<
a
20,000 psi
<
i
Young's modulus is taken as 10
p
= 0.1 lb./cu.
in.
psi and the material density
The allowable stress at which Euler
buckling occurs is
a.
b
=-SM
L2
The independent coordinate variables were taken as X3,
Y3, X5, Y5,
X7,
U7,
X9
,
in the following groups:
Y9
The member areas were linked
.
A1=A4=A8=A12=A16
A3=A7=A11=A15; A5=A9=A13=A17
.
;
A2=A6=A10=A14=A18
There are a total of eight
independent coordinate variables and four independent area
variables.
1.
Case la
The structure was designed subject to stress constraints
only.
The resulting geometry is shown in Fig. la, and the
design information is given in Table II.
iteration history is plotted in Fig. A.
2.
Weight versus
The number of
analyses for this design is 59, and the execution time 2.53
seconds on an IBM 30 33 computer.
37
Case lb
2.
Stress and Euler buckling constraints were imposed
for this case.
Design information is given in Table III.
Figure lb shows the final geometry layout and the weight
versus iteration history is shown in Fig. A.
3.
The number
of analyses for this design is 78 and the execution time
was 3.94 seconds.
Case lc
3.
This design is based on stress, Euler buckling and
displacement constraints.
number
The latter was applied at node
in the Y-direction.
1
Table IV.
The results are shown in
Figure lc presents the final layout and the
weight versus iteration is plotted in Fig. A.
4.
The number
of analyses is 91 and the execution time was 3.94 seconds.
4.
Case Id
This final case includes all the constraints mentioned
before plus a constraint that the first natural frequency
be greater than or equal to
3
Hz.
A non-structural mass
of fixed value W = 1,000 lbs was placed at node 1.
Steady
convergence is achieved and results are summarized in Table V.
The final geometry and weight iteration history are shown in
Figures Id and A.
5,
respectively.
The number of analyses is
96 and the execution time was 4.7 3 seconds.
When the mass is removed from the structure, the
design fails to converge even when move limits are imposed.
First, it is known by definition that the frequency is
38
proportional to the stiffness and is inversely proportional
to the mass.
Second, both the K and M matrices are functions
of area and coordinate variables.
Therefore, as K increases,
M also increases, and the frequency is kept closed to the
initial range.
CASE
C.
47-BAR PLANAR TOWER
2:
The initial layout of the tower is shown in Fig. A.
6.
Stress constraints were imposed as well as constraints on
Euler buckling, displacement, and first natural frequency:
-15,000
<
a
<
i
20,000 psi
i=l,47
and
IO.IttEA.
°
0i '
bi
=
"
1^
psi
l
X
* 12 hz.
respectively.
The members are assumed tubular with D/t=10.
Young's modulus of 3.x 10
&
=
.
3
Ibs/cu. in.
imposed.
7
psi, and material density
Minimum allowable area of 10
— fi
Symmetry about the y-axis was desired during
optimization so linking of variables is necessary.
15,
16,
in. was
17,
22 are kept fixed,
to move in the x-axis direction
and joints
(Y=0)
.
1
and
2
Joints
allowed
This gave a total
of 27 area design variables and 17 coordinate variables.
Nonstructural masses were attached at nodes 19 and 20, of
W=500 lbs each.
through XII.
The results are tabulated in Tables VII
The final geometry for the case where all
39
—
:
constraints were imposed is shown in Fig. A.
iteration history is shown in Figs.
A. 8
7
and the
through A. 10.
When
all constraints are imposed, the design required 220 analysis
and the execution time was 160.7 seconds.
CASE
D.
3:
25-BAR SPACE TOWER
The 25-bar space tower shown in Fig. A. 11 was designed
to support two independent load conditions given in Table
The allowable stresses were specified as
XIII.
-40,000
<
a
<;
i
40,000 psi
i-1,25
Young's modulus was selected as 10
density
p
= .1 lb./cu.
psi and the material
The members are assumed to be
in.
turbular with a nominal diameter to thickness ratio of
D/t = 100., so that the stress at which Euler buckling occurs
is
10.1
°
bi
tt
EA.
~
psi
-
iZf
Symmetry with respect to both x-z plane and the y-z plane
was imposed, so linking of variables were made as follows:
the member areas were grouped in the following sequence
Al, A2=A3=A4=A5, A6=A7=A8=A9
A10=A11, A12=A13, A14=A15=A16=A17
,
A18=A19=A20=A21, and A22=A23=A24=A25.
X4
1
,
Y4
and
,
2
Z4,
X8
,
For the coordinates
Y8 were considered as variables.
were held fixed and joints
to lie in the x-z plane.
were attached at nodes
1
7
The joints
through 10 were required
Nonstructural masses of W=500 lbs.
and
2,
40
respectively.
The first
,
natural frequency was limited to a value
A
^
16 Hz.
Results of this example are shown in Table XIV and the
iteration history is shown in Fig.
41
A. 12.
V.
A.
CONCLUSIONS AND RECOMMENDATIONS
CONCLUSIONS
The final layout dependency on constraints has been
presented for the elastic design of trusses for optimal
geometry.
The truss may be planar or three-dimensional and
may be indeterminate.
sidered.
Multiple load conditions were con-
The design procedure was separated as:
analysis,
design for fixed geometry, design for optimum geometry.
The displacement method for static analysis and the
subspace iteration method for dynamic analysis were applied.
The sequential optimization based on two design sub-
spaces present substantial advantages in the reduced
number of analyses and allow the designer to keep control
of the optimization process.
Several examples were considered.
In every case,
cant weight reduction was efficiently achieved.
signifi-
Also, the
geometries obtained appear quite acceptable from an
aesthetic
as well as structural point of view.
The graphs of Weight vs. Iteration number show that an
acceptable design can be achieved in few iterations.
42
B.
RECOMMENDATIONS
The following recommendations may be of theoretical
and practical value.
1.
Weight due to the structure itself and other design
dependent external forces can easily be taken into account
and should be considered in future studies.
2.
Application should be made to reasonable sized
structures such as offshore towers and long span roof
trusses.
3.
The principles and the procedure described herein
can also be used for optimal design of frames as well as
other structures.
43
APPENDIX A
@
Figure
1.
r
©
18-Dar Planar Truss. Initial Geometry.
4,780.
(lbs)
Hf« 4,524.
(lbs)
Wi= 6,430.
(lbs)
Wf- 5,724.
(lbs)
W1= 7,413.
(lbs)
Wf» 6,465.
(lbs)
HI »
Figure la. Stress,
Figure lb. Stress, Euler 3uckling.
Figure lc. Stress, Euler 3uckling, Displacement.
Wi= 13,686. (lbs)
Wf= 11,528. (lbs)
Figure Id. Stress, Euler Buckling, Displacement, and
Frequency.
Figure A.1.
18-Ear Planar Truss. Stress, Euler Buckling,
Displacenent, and Frequency Constraint.
44
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TABLE I
18-Bar Elanar Truss. Loads and Constants
LCAE CONDITION
Jcint
(lbs
.)
Ex
Ey
1
0.
2
0.
-20,000
-20,000
4
0.
-20,000
6
0.
-20,000
a
0.
-20,000
TAELE OF CONSTANTS
Young's Modulus
E=
Allowable Stress
5" =
Density
P
Buckling Coefficient
=
1EC8
psi.
(.69E08 KN/m
0.2E05
psi.
(.138E06 KN/m
)
lbs/cu.in
(.276E04 Kg/m
)
(.276E09 KN/m
)
0.
0.
c=
56
1
.4E08
psi.
)
TABLE II
18-Bar Flanar Truss, resign Inf or nation.
AREA*
Member
A1=A4=A8=A12=A16
A2=A6=A10=A14=A18
A3=A7=A1 1=A 15
A4=A9=A13=A17
(sg.in.)
Initial
Final
10.0
1
15.0
15.07
5.0
a. 54
7.07
5.33
COORDINATES
1.05
(in.)
Initial
Joint
*
Stress
Final
X
y
X
3
000
0.0
99
5
750
0.0
745.88
7
500
0.0
494.61
34.538
9
250
0.0
249.54
23.609
1.
y
17
19.686
15. 12
Optimum Weight for Initial Geometry
=
4,780.5
Optimum Weight for Final Geometry
=
4,524.7 (lbs.)
(Its.)
Areas ars the optimum values fcr the initial and final geometry.
57
TAELE III
Stress, Euler
18-Bar Planar Truss. Design Information.
Buckling.
AREA*
Kember
A1=A4=A8=A12=A16
A2=A6=A10=A14=A18
A3=A7=A11=A15
A4=A9=A13=A17
(sg.in.)
Initial
Final
10.00
11.34
21.65
19.28
12.5
1C.97
7.071
CCOBDIMTZS
(in.)
Initial
Joint
z
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5.306
Final
X
y
y
3
1000
0.0
994.57
162.31
5
747.36
102.92
750
0.0
7
500
0.0
482.90
32.962
9
250
0.0
221.71
17. 105
Optimum Weight fcr Initial Geometry
=
6,430.7 (Its.)
Optimum Weight for Final Geometry
=
5,713.C (lbs.)
Areas are tie optimum values for the initial and final geometry.
TABLE IV
Stress, Euler
18-Bar Planar Truss. Design Information.
Buckling, Displacement.
AREA*
Member
(sg.in.)
Initial
Final
A1=A4=A8=A12=A16
A2=A6=A10=A14=A18
A3=A7=A11=A15
14.18
16.27
21.66
20.06
12.5
11.18
A4=A9=A13=A17
10.32
COORDINATES
Joint
7.863
(in.)
Initial
x
Final
x
y
5
750
0.0
962.19
703.95
7
500
0.0
452.7
9
250
0.0
208.48
3
1000
0.0
Optimum Weight for Initial Geometry
Optimum Weight for Final Geometry
=
=
7,413.1
6,466.9
y
109.93
55.309
27.307
-
2.884
(lbs.)
(lbs.)
* Areas are the optimum values fcr the initial and final gcemetry.
59
TABLE
V
18-Bar Planar Truss. Design Information.
Stress, Euler
Euckling, Displacement, Frequency.
AREA*
Hember
A1=A4=A8=A12=A16
A2=A6=A10=A14=A18
A3=A7=A11=A15
A4=A9=A13=A17
(sg.in.)
Initial
Final
31.66
2S.66
40.58
35.29
12.5
13.03
21. 10
16.09
CCCEDIHATES
Initial
Jcint
*
(in.)
Final
X
X
1
000
0.0
972.52
88.0
750
0.0
717.08
27.8 4
500'
0.0
250
0.0
468.75
200.98
y
-
3.91
-19.29
Cptimun Weight fcr Initial Geometry
=
13,686.0 (Its.)
Optimum Height for Final Geometry
=
11, 528.
C
(lbs.)
Areas are the optimum values for the initial and final geometry.
60
TABLE VI
47-Bar Planar Tower. Load Conditions
LOAD CONDITION
1
Fy
JOINT
Fx
17
6, COO.
22
(lbs.)
-14,000.
0.
0.
LOAD CONDITION
2
(lbs.)
Fy
Joint
Fx
17
0.
0.
22
6,000.
-14,000.
LOAD CONDITION
3
Fx
Joint
(Its.)
Fy
17
6, COO.
-14,000.
22
6,000.
-14,000.
61
TAELE VII
47-3ar Planar Tcwer.
Heater
Area.
Stress, Euler Buckling.
Initial
irea *
Final
(sg.in.)
Area *
3
3.764
2.727
It
3.315
2.468
5
0.787
0.727
7
0.864
0.2
8
0.856
0.938
10
1.754
1.076
12
1.691
14
2.087
1.188
0.695
15
1
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1.058
18
2.087
1.412
20
0.648
0.263
22
0.843
0.811
24
1.700
1.060
26
1.700
1.052
27
1.354
0.820
28
0.847
0.302
30
3.609
2.766
31
1.435
0.657
33
0.638
0.207
35
2.842
2.697
36
0.676
0.266
38
1.596
1.408
40
3.686
3.429
13
41
1.526
0.991
43
45
0.677
4.486
0.170
3.650
46
1.532
1.005
* Optimum areas for initial and
62
(sg.in.)
final geometry.
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X
47-3ar Planar Tcuer. Areas. Stress, Euler Buckling,
Displacement.
Hemter
Initial
Area * (sg.in.)
Final
Area *
15.28
4.478
4
16.55
3.671
5
7
3.066
2.691
0.561
8
2.435
1.156
10
3.706
1. 170
12
5.263
2.318
14
4.048
0.669
15
5.077
1
3
(sq.in.)
1.037
.385
18
4.780
1. 169
20
1.650
0.562
22
2.231
1.295
24
4.665
1.526
26
4.816
1.511
27
3.693
1.036
28
2.923
0.566
30
17.91
5.231
31
7.091
0.949
33
2.566
0.558
35
17.59
6.175
36
6.640
1.217
38
3.385
1.076
40
28.11
7.730
41
6.792
1.570
43
2.701
0.555
45
37. 11
9.878
46
7.363
2.153
* Optimum ares
for initial and final geometry.
65
)
TABLE XI
47-Bar Planar Tower. Areas. Stress, Euler Buckling,
Displacement, and Frequency.
Henter
Initial
Area * (sg.in.)
3
16.59
4.825
4
15.37
4.605
5
1.321
0.S38
7
1.871
0.348
8
1.356
1.081
10
5.152
1.089
12
8.593
3.025
14
5.988
1.088
15
8.468
1.691
18
8.593
1.656
20
1.416
0.635
22
1.836
1.724
24
8.470
2.480
26
8.470
2.186
27
5.999
1.342
28
1.949
0.452
30
14.54
4.833
31
7.096
0.855
33
1.709
0.307
35
36
9.710
1.414
4.579
0.368
38
1.385
1.439
40
10.43
4.3 96
41
1.484
1.126
43
1.465
0.270
45
11.17
4.365
46
1.482
1.664
Final
Area *
(sg. in.
* Optimum areas for initial and final geometry.
66
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A.
TAELE XIV
25-Ear Space lower. Design Information
COORDINATES
(in.
Initial
Strs-6c)c.
X
37.5
21 .487
1
37.5
48 .271
602 .83
47.271
z
100.
1C0.27
1
.26
128.77
X
100.
22 .146
47. 83
107.56
1
100.
96 .353
140 .53
110.077
Joint
AREAS
•
STRESS- BOCKl
INITIAL
FINAL
EENBER
6
0.9E-3
0.782
0.754
0.013
0.414
0.842
10
0. 1E-3
0.033
12
0. 130
14
0.558
0.982
0.801
1
2
18
22
Height
)
(lbs.)
I
)isplaceaent
33. 512
14
Frequency
64.9 13
(sq.in.)
DISPLACEMENT
INITIAL
FINAL
0.668
0. .029
PREQUENCI
INITIAL
FINAL
0.54 b
0.07U
0, .450
2.533
,219
2.635
1.209
0.533
0. .015
0.547
0.054
0.101
0.549
0.
118
C.496
0.206
0.121
0.668
0. ,084
0.533
0.350
0.739
1.592
0, .751
1.449
0.720
0, .668
3.36 3
1.
2.
0.554
Stress-Buckl.
Pinal
Initial
128.55
229.52
032
3. 100
1,
2.6S3
.
Displacement
Initial
Fianal
169.9
565.2
0.73a
130
Frequency
Initial
Final
587.73
261.52
Iteration
12
44
9
Tiie
5.38
37.14
21.09
*
(sec.)
Optimua areas for initial and final geoaetry.
69
LIST OF REFERENCES
1.
Dorn, W. S., Gomory, R. E., and Greenberg, H. J.,
Automatic Design of Optimal Structures Journal de
Mechanique, Vol. 3, March 1964, pp 25-52.
,
2.
Dobbs, M. W.
and Felton, L. P., Optimization of Truss
Geometry Journal of the Structural Division, ASCE,
Vol. 95, No. ST10, Proc. Paper 6797, October 1969,
pp 2105-2118.
,
,
3.
Pedersen, P., Optimum Joint Positions for Space Trusses
Journal of Structural Division, ASCE, Vol. 99, No. ST12,
December 1973, pp. 2459-2475.
4.
Vanderplaats, G. N.
and Moses, F.
Automated Design
of Trusses for Optimum Geometry Journal of Structural
Division, ASCE, Vol. 98, No. ST3, March 1972, pp. 671-
,
,
,
,
690.
5.
Vanderplaats, G. N.
Design of Structures for Optimum
Geometry Technical Memorandum TM-X-62,462, National
Aeronautics and Space Administration, August 1975.
,
,
6.
Spillers, W. R.
Iterative Design for Optimal Geometry
Journal of Structural Division, ASCE, Vol. 101, No. ST7,
July 1975, pp. 1435-1442.
7.
Imas K.
and Schmit, L. A., Jr., Configuration Optimization of Trusses Journal of the Structural Division,
ASCE, Vol. 107, No. ST5, Proc. Paper 16251, May 1981,
pp 745-756.
,
,
,
,
,
8.
and Topp,
Martin, H. C.
Turner, M. J., Clough, R. W.
Stiffness and Deflection Analysis of Complex
L. J.
Structures J. Aeronaut. Sci. Vol. 23, No. 9, pp 805824, September 1956.
,
,
:
,
,
9.
Large Eigenvalue
Bathe, K, J., and Wilson, E. L.
of the Engineering
Journal
Problems in Dynamics Analysis
Mechanics Division, ASCE, Vol. 98, June 1973, pp 213-226.
,
,
10.
Aurora, J. S. and Haug, E. J., Methods of Design Sensitivity Analysis in Structural Optimization AIAA Journal,
Vol. 17, September 1979, pp 970-974.
,
11.
Nelson, R. B., Simplified Calculation of Eigenvectors
Derivatives AIAA Journal, Vol. 14, No. 9, September
1976, pp 1201-1205.
,
70
12.
Zoutendijk, K. G.
Methods of Feasible Directions
Elsevier Publishing Co., Amsterdam, Netherlands,
,
,
1960.
13.
Vanderplaats, G. N.
and Moses, F., Structural Optimization by Method of Feasible Directions, Computers
and Structures Vol. 3, Pergamon Press, 1973, pp 739,
,
755.
14.
Vanderplaats, G. N., CONMIN - A FORTRAN Program for
Constrained Function Minimization - User's Manual
NASA TMX-62,282, August 1973.
,
15.
Martin, H. C.
Truss Analysis by Stiffness Considerations, Trans. Am. Soc. Civil Engrs., Vol. 123, pp 1182,
1194,
1958.
71
.
INITIAL DISTRIBUTION LIST
No.
Copies
1.
Defense Technical Information Center
Cameron Station
Alexandria, Virginia 22314
2
2.
Library, Code 0142
Naval Postgraduate School
Monterey, California 93940
2
3.
Professor G. N. Vanderplaats Code 67Vn
Department of Mechanical Engineering
Naval Postgraduate School
Monterey, California 93940
3
4.
Professor David Salinas, Code 69Zc
Department of Mechanical Engineering
Naval Postgraduate School
Monterey, .California 93940
1
5
Naval Attache
Embassy of Ecuador
2535 15th Street NW
Washington, DC 20009
1
6.
LT Jorge E. Felix
Box 5147
Guayaquil, Ecuador, SOUTH AMERICA
2
7.
LT David J. Mulholland, USN
COMNAVSHIPYD PEARL HARBOR
Pearl Harbor, HI 96782
1
8.
Dr.
Richard H. Gallegher
Dean of College of Engineering
University of Arizona
Tucson, Arizona 85702
1
9.
Department Chairman, Code 69
Department of Mechanical Engineering
Naval Postgraduate School
Monterey, California 93940
1
,
72
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