Near optimum solutions and sensitivity analyses based structural
optimization applied to truss structures under arbitrary constraints
STRATIS KANARACHOS, DIMITRIOS KOULOCHERIS, HARRIS VRAZOPOULOS
Mechanical Engineering Department
National Technical University of Athens
Polytechnioupolis Zografou, 15780 Athens
GREECE
Abstract: - This paper discusses a new structural optimization method and its application to the weight
minimization of truss structures under arbitrary constraints. The method introduces a “min-max” principle for
determining a near optimum solution for weight minimization under arbitrary constraints, using results obtained
from the solution of weight optimization problems under a single behavioral constraint. The optimal solution
can then be easily obtained using a sensitivity analysis based optimization, or even mathematical programming.
Test cases, broadly used in the literature, are selected to illustrate the performance of the proposed
methodology.
Key-Words: - Structural optimization; Truss structures; Finite element method; Optimality criteria
1 Introduction
The history of structural optimization goes back to
the 1960s and is characterized by three
methodological impacts: (i) the optimality criteria
method, (ii) the non-linear mathematical
programming techniques and (iii) the evolution
strategy/genetic algorithms. The literature is too
vast to be all acknowledged; however the basic
ideas of Venkayya and Prager [1]-[2], Schmit [4]
and of Rechneberg, Holland et all [5]-[7], should
not be left unmentioned.
With respect to the optimality criteria method,
Prager [2] developed optimality criteria for such
structures as beams and trusses subject to a single
behavioral constraint. Venkayya [1] derived a strain
energy criterion, also based on the principle of
minimum potential energy. Many authors in the
1970s derived directly the optimality criteria,
mainly for trusses subjected to static loading, using
a Lagrangian approach where the optimality criteria
become similar to the Kuhn-Tucker necessary
conditions. The major difficulties arise in the
presence of multiple constraints, as it is not an easy
task to find whether a constraint is active and, if so,
what the contribution of the constraint is to the
overall requirement. In this context, several
methods have been proposed, e.g. [3]. Later
research [8-9] tries to establish robust iterative
algorithms for stress- and displacement-constrained
trusses on the basis of FSD, using numerical
experiments [9]. However, other researchers use
the evolution strategy/genetic algorithms, e.g. [1011], in order to solve similar structural optimization
problems for multiple behavioral constraints.
The basic argument for not using mathematical
programming
methods,
or
for
applying
evolution/genetic or other algorithms, is that it is
not possible to construct an initial solution in the
(convex) neighborhood of the global optimum.
Thus, algorithms have to be developed to overcome
local optima and to converge to the global one.
In contradistinction to the above, the proposed
method introduces a “min-max” principlefor
determining near optimum solutions for weight
minimization under arbitrary constraints. This is
achieved using results obtained from the solution of
weight optimization problems under a single
behavioral constraint. The optimal solution can
then be easily obtained applying a sensitivity
analysis based optimization, or even mathematical
programming techniques. Two test (benchmark)
cases from the literature are used to illustrate the
performance of the proposed methodology.
2 Problem formulation
Let us consider a truss that consists of M nodes and
N bars with cross-sectional areas: Ai, i=1,2,…,N.
Normally, the design variables of real truss
structures are, due to cost, assembly and production
aspects, never equal to N. However for the purpose
of this paper (mainly comparison with other
authors) it will be assumed in the following, that all
cross-sectional areas Ai compose the vector x of the
optimization parameters:
x  [ A1 A2 ... AN ]T
(1)
The problem to be solved refers to the computation
of the vector x achieving the minimum weight W of
the truss
W
N
 A L
(2)
i i
i 1
for given stress- and deflection-constraints
(displacement-constrained trusses):
 i   0i
1 i  N
u i  u 0i
1 i  M
(3)
In equation (3), s0i and u0i denote the allowable
stress and displacement upper limits, respectively.
The behavioral constraints can be more
generally formulated:
 i  max( 0i , S ki )
1 i  N
ui  u 0 i
1 i  M
(4)
 2  2
In eq. (4) the buckling critical loads Ski and the first
eigenfrequency ω2 are added (Ω2 denotes the
eigenfrequency lower limit of the structure).
The static and dynamic behaviour of the
structure is given by the FEM equations:
K u  f j
 2  φT Mφ  φT Κφ  0
(5)
In equation (5), u denotes the displacement vector,
K and M denote the stiffness and mass matrix,
respectively, fj the j=1,2,…,J load cases, while ω2
the first eigenfrequency and φ the first eigenmode
of the truss structure.
The sensitivity analysis of (5) leads to the
following equations:
(K  K )  (u  u)  f j

(   )  φ [(Μ  Μ)  (Κ  Κ )]φ  0
2
2
  2  φ Μφ   2  φ  Μφ  φ  Κφ  0
3.1 Computation of the near optimum
solution
For the computation of a near optimum solution,
~
x , let us assume that the following solutions are
known:
x
(6)
x , j
(7)
In equation (8), xσ,j denotes the weight minimal
solution for the stress-constrained, xu,j for the
displacement-constrained and xσB,j for the
stresses/buckling-constrained structure, for each jth load case. Further in eq. (8), xω denotes the
weight minimal solution for the eigenfrequencyconstrained structure.
Then, the near optimum solutions ~
x are
Neglecting higher order terms, (6) leads to:
u  K 1  K  u
optimization algorithm depends heavily on the
properties of the starting (or initial) solution x0:
 If x0 is near the global minimum xopt, then any
iterative algorithm or a parameter optimization
method will be able to converge to xopt.
 On the contrary, if x0 is not near the global
optimum xopt, then iterative algorithms or
parameter optimization methods will face
increased convergence difficulties.
Therefore, emphasis is put in this paper in
computing a starting solution ~
x near the global
minimum xopt (“near optimum solution”).
For the computation of a starting solution ~
x in
the (convex) neighborhood of the global minimum
xopt, a new “min-max” principle is introduced. The
above principle utilizes results obtained from the
solution of weight optimization problems under a
single behavioral constraint. In other words, results
obtained from the solution of
 stress-constrained,
 displacement-constrained and
 eigenfrequency-constrained
weight optimization problems. For truss structures,
the above results can be easily obtained using
optimality criteria methods (e.g. [1]-[2]).
Starting now from ~
x , the optimal solution xopt
can be obtained using an appropriate mathematical
optimization method, considering that the search
area is, with the highest possible probability,
convex. If a minimization of the number of FEM
analyses is sought, we propose in this paper an
optimization algorithm on the basis of the
sensitivity analysis (7). This sensitivity analysis
based algorithm does not need any additional FEM
analyses. It defines a parameter optimization
problem using only (7) and (4).
3 The Optimization algorithm
It is well known, that the performance of any
x u, j
x B, j
(8)
computed according to the following procedure:
1. First, if the structure is subjected to J load
cases, the near optimum solutions are computed
using the following “min-max” principle (9):
~
xi  max j ( xi , j )
~
xiu  max j ( xiu , j )
~
x B  max ( x sB, j )
i
j

 ~
x

 ~
xu
(9)
~
xi   i  xi  x new

 ~
xB
i
In eq. (9), ~
x , ~x u and ~
xB denote the near
optimum solutions for the stress-, displacementand stress/buckling-constrained structures.
2. Second, in a similar manner, the near optimum
solutions for combined constraints are computed
using the following “min-max” principle (10):
~
xiu  max( xi , xiu )
~
x u  max( x , x u , x )

 ~
x u

 ~
x Bu
~
xiBu  max( xiB , xiu , xi )

 ~
x Bu
i
i
i
i
are applied simultaneously. The xi -values are
defined by the user and correspond to the search
area of the algorithm, while αk are the restricted
(  1   k  1 ) optimization parameters
The obtained solution αk defines a new design
vector
(10)
In eq. (10) ~
xu , ~
x u and ~
xBu denote the near
optimum solutions for the combined stressdisplacement, stress-displacement-eigenfrequency
and stress/buckling-displacement-eigenfrequency
constraints.
(13)
leading to σ new , u new and  new , obtained through
analysis (5). Then a new iteration step (11)-(12) is
performed, using ~
x  x new , etc.
The proposed solution methodology proves to
be robust and effective. For the solution of (11)(12) the Nelder-Mead algorithm (FMINSEARCH
of MATLAB) was used.
4 Numerical Implementation
In this section, the efficiency of the proposed
methodology is investigated by applying it to two
test cases, the results of which are compared with
results from the literature. The first example refers
to three and the second to ten design variables
(three- and ten-bar trusses).
4.1 Three-bar Truss
3.2 Optimization algorithm
If now a near optimum solution ~
x is known, then
the optimal solution x=xopt can be obtained using
the sensitivity analysis equations (7) and defining
the following optimization problem:
W (~
xi   i  xi )  Minimum
(11)
under the constraints:
A three-bar truss with Young’s modulus E=30,000
ksi, density ρ=0.1 lb/in3 and allowable strength
σ0=20 ksi is shown in Fig. 1. The truss is subjected
to two load cases:
 Px=-50 kip, Py=-100 kip (j=1)
 Px=+50 kip, Py=0 (j=2)
The displacement constraints for the non-restrained
node are given as u0=0.20 in and v0=0.05 in. In
addition, at the non-restrained node a mass m=50 lb
is attached and an additional constraint of ω2≥5 is
imposed.
 i ( ~xi   i  xi )
  i (~
xi ) 
Ν

k
  ik  max( 0i , S ki )
k 1
ui ( ~
x i   i  x i )  u i ( ~
xi ) 
N

k
 uik  u 0i
(12)
k 1
Ν
 2 ( ~xi   i  xi )   2 ( ~xi )    k   ik2   2
k 1
The above sensitivity analysis based
optimization problem refers to the minimization of
the weight W (11) under the constraints (12), which
Fig. 1: Three-bar truss.
4.1.1 Construction of ~x
In the following, we proceed to the computation of
near optimum solutions ~
x . We begin with the

u
~
~
x and x solutions for the two load cases j=1-2
according to (9) (Tables 1 and 2).
A1
A2
A3
W
Case
0.100 2.501 3.535 76.41
j=1, x
1.768 0.100 1.767 50.99
j=2, x
1.768 2.501 3.535 100.00 j=1-2, ~
x
Table 1. ~
x for stress constraints.
A1
A2
A3
W
Case
0.114 3.575 1.817 63.06 j=1, x u
1.174 0.100 1.183 34.33 j=2, x u
1.174 3.575 1.817 78.05 j=1-2, ~x u
Table 2. ~x u for displacement constraints.
The above results are now used to compute the near
optimum solution ~
xu for the combined stress and
displacements constraints (eq. (10)) (Table 3).
A1
1.768
1.174
1.768
1.103
A2
2.501
3.575
3.575
3.778
A3
3.535
1.817
3.535
3.301
W
Case
100.00 j=1-2, ~x s
78.05
j=1-2, ~x u
110,74 j=1-2, ~
xu
100.07 j=1-2, xu
σ1
σ2
σ3
u1
v1
ω2 j
-2.73 15.91 18.64 -0.071 -0.053
1
15.46 3.19 -12.27 0.092 -0.011 4.84 2
Table 3. ~
x us and exact solution xus for stress and
displacement constraints and properties of ~
xu .
Further, using (10), the near optimum solution
~
x u for a more complicated case concerning stress,
displacement and eigenfrequency constraints
(Table 4) is computed.
A1
3.908
1.178
3.908
3.063
A2
0.100
3.654
3.654
3.205
A3
3.908
3.535
3.908
4.951
W
Case
76.92
j=1-2, x
103.18 j=1-2, ~
xu
147.07 j=1-2, ~
x u
145.39 j=1-2, x u
j
σ1
σ2
σ3
u1
v1
ω
-1.25 15.59 16.84 -0.060 -0.052
1
9.05 0.01 -9.04 0.060
0 7.00 2
Table 4. ~
x u and exact solution x u for stress,
displacement and eigenfrequency constraints and
properties of ~
x u .
2
Finally, one may compute the near optimum
solution ~
xuB for a displacement-constrained truss
structure, considering stress/buckling constraints
(Table 5).
A1
1.090
0.272
1.090
0.793
A2
A3
W
Case
1.894 6.526 126.65 j=1-2, xB
3.654 2.448 75.00
j=1-2, x u
3.654 6.526 144.24 j=1-2, ~
xuB
3.012 6.915 139.15 j=1-2, xuB
Table 5. ~
xuB and exact solution xuB for stress,
displacement and buckling constraints
As it may be seen from Table 5, the stress/buckling
constraint dominates and shifts the solution to
another x-region (the stress limits are not any more
constant!). This important constraint case, is not
considered in the available literature (see chapter
4.1.2).
4.1.2 Comparison with the Literature
The performance of the proposed optimization
method (11)-(12) is compared with results available
from the literature, presented in [9], dealing only
with stress and displacement constraints (no
buckling constraints). The results of the
optimization algorithm (11)-(12) are shown in
Table 6 ((*)Starting vector=[2 2 2], (**) Starting
vector= ~
xu ). They are comparable to the other
methods and are obtained with a lower number of
reanalyses.
Design
A1
A2
A3
W
Reana
method
lyses
MFUD 1.088 3.841 3.265 99.97
24
FUD
1.574 3.336 4.706 122.18
10
SUMT 1.088 3.848 3.267 100.07
62
FD
1.092 3.855 3.250 99.95
47
OC
1.053 3.913 3.345 101.33
80
[9]
1.087 3.844 3.267 100.02
16
Present 1.103 3.778 3.301 100.07
5(*)
paper
1.040 3.766 3.440 101.03 3(**)
Table 6. Three-bar truss: comparison between
present paper and other methods(s-u-constraints)
Notation: MFUD (Modified Fully Utilized Design, FUD (Fully
Utilized Design, SUMT (Sequence of Unconstrained
Minimizations Technique, FD (Feasible Directions), OC
(Optimality Criteria)
4.2 Ten-bar truss
This is the well-known cantilever truss illustrated in
Fig. 2. Each member’s area is treated as an
independent design variable. Young’s modulus is
E=107 psi, density ρ=0.1 lb/in3 and allowable
strength σ0=25 ksi.
The truss is subjected to one load case, P1=100
kip, as depicted in Fig. 3. The displacement
constraints are given as u0=2.0 in and v0=2.0 in.
Also: A4≥0.1 in, A6≥0.1 in and A9≥0.1 in.
~
σi
umax
σi
umax
xiu
x i
1 23.48 -10.3
9.94 -25.0
2 14.10 -7.2
3.83 -25.0
3
2.08 23.5
2.08 25.0
4
0.10 -10.6
0.10 13.6
5 24.59
6.3 -2.01 6.01 25.0 -7.19
6
0.10 38.8
0.10
0.2
7 14.55 14.5
2.74 -25.0
8
9.05 22.7
8.60 25.0
9
0.10 15.1
0.10 -18.2
10 19.88
7.2
5.70 25.0
W 4.539
1,667
su
~
Table 8. Properties of the x and the x solutions.
i
Fig. 2: Ten-bar truss.
The same procedure is repeated for the case of
stress/buckling and displacement constraints (Table
9). As it can be seen, the xB solution is near to the
global optimum xBu . Also in this case, the
buckling constraints are dominating.
Fig. 3: Load case for the ten-bar truss.
~
xB
xBu
xu
x Bu
A1 64.01 23.48 64.01 63.61
A2 39.89 14.10 39.89 40.37
A3
2.70 0.96 2.70 4.68
A5
7.60 24.59 24.59 23.12
A7 48.00 14.55 48.00 12.83
A8 22.63 9.05 22.63 15.65
A10 17.66 19.88 19.88 11.22
W 8,618 4,499 9,343 8,308
Table 9. ~
x Bu and exact solution xBu for stress,
displacement and buckling constraints
4.2.1 Construction of ~x
4.2.2 Comparison with the Literature
In the following, the computed near optimum
solution ~
xu according to (10) and its properties
are shown in Tables 7 and 8. As it can be seen, the
~
xu solution is very near to the global optimum xu .
The performance of the proposed optimization
method (11)-(12) is compared with results available
from the literature, dealing only with only stress
and displacement constraints (buckling constraints
are not available).
The results of the optimization algorithm (11)xu ).
(12) are shown in Table 6 (Starting vector= ~
They are comparable to the other methods and are
obtained with a lower number of reanalyses.
~
xu
xu
x
xu
A1
9.94 23.48 23.48 25.32
A2 3.83 14.10 14.10 14.57
A3
2.08 0.96 2.08 1.97
A5
6.01 24.59 24.59 23.12
A7
2.74 14.55 14.55 12.83
A8
8.60 9.05 9.05 12.35
A10 5.70 19.88 19.88 20.51
W 1,667 4,499 4,539 4,678
Table 7. ~
x su and exact solution xu for stress and
displacement constraints.
A1
A2
A3
A5
A7
A8
Schmit
Harless
24.29
13.65
1.97
23.35
12.54
12.67
23.59
14.45
1.97
24.96
12.82
12.45
[4]
[9]
23.53
14.37
1.97
25.29
12.83
12.39
23.79
14.18
1.97
25.55
12.90
12.40
Present
Paper
24.25
14.13
1.99
23.91
12.81
12.59
A10
21.97
20.43 20.33 20.04
20.65
W
4,692
4,678 4,677 4,678
4,677
Itera
22
12
9
40
4
tions
Table 6. Ten-bar truss: comparison between present
paper and other methods for stress and
displacement constraints.
5 Discussion-Conclusions
Since Schmit, who applied first in 1960s
mathematical optimization methods for the solution
of structural optimization problems, the structural
optimization research has been conducted toward
two distinct methodologies:
i) optimality type algorithms (including FSD,
MFUD, etc.) and
ii) evolution and genetic search algorithms.
The basic reason for the development of these two
methodologies apart from the mathematical
optimization methods, is to overcome local
minima, since it has been (generally) not easy to
define or estimate a starting or initial solution
“near” (within the convex area of) the global
minimum.
As it has been shown in this paper, the proposed
“min-max” principle succeeds in computing near
optimum solutions for arbitrary constraints. This is
achieved using results from weight minimal truss
structures subjected to a single behavioral
constraint. These results can be easily obtained
applying optimality criteria methods. In addition a
mathematical optimization algorithm has been
proposed, based on the sensitivity analyses of the
current designs, capable of converging to the global
minimum within a few iteration steps.
The results of this investigation are very
encouraging, so that future research is planned for
other types of structures.
Finally, it should be mentioned that the test
cases used in this paper do not contain an excessive
number of design variables, e.g. 100 and more
(which would not have been a problem for the
proposed solution methodology). The reason is that
the number of design parameters in real truss
structures remains relative small due to
manufacturing, assembly, logistic and cost reasons,
and is never equal to the number of the members of
the truss structure. Therefore, most important for an
algorithm is its ability to handle simultaneously
arbitrary constraints than an excessive number of
design variables.
References:
[1] V.B. Venkayya. Design of optimum
structures. Computers and Structures 1
(1971) 265-309.
[2] W. Prager, Conditions for structural
optimality. Computers and Structures 2
(1972) 833-840-309.
[3] R.J. Allwood, Y.S. Chung. Minimum weight
design of trusses by an optimality criteria
method. Int. J. Numer. Methods Engrg. 20
(1984) 697-713.
[4] L.A. Schmit. Structural design by systematic
synthesis in Proceedings of the 2nd
Conference of Electronic Computation,
ASCE, New York (1960) 105-132.
[5] T. Baeck, H. P. Schwefel, An overview of
evolutionary algorithms for parameter
optimization, Evolutionary Computation,
1(1), pp. 1-23, (1993).
[6] H. P.Schwefel, Evolution & Optimum
Seeking, John Wiley & Sons, Inc., (1995).
[7] Z. Michalewicz, Genetic Algorithms + Data
Structures = Evolution Programs, SpringerVerlag, New York, (1996).
[8] S.N. Patnaik, J.D. Guptil, L. Berke. Merits
and limitations of optimality criteria method
for structural optimization. Int. J. Numer.
Methods Engrg. 38 (1995) 3087-3120.
[9] P.A. Makris, C.G. Provatidis. Weight
minimization of displacement-constrained
truss structures using a strain energy criterion.
Comput. Methods Appl. Mech.Engrg. 191
(2002) 2159-2177.
[10] K. Deb, S. Gulati. Design of truss structures
for minimum weight using genetic
algorithms. Finite Elements in Analysis and
Design 37 (2001) 447-465.
[11] N.
Lagaros,
M.
Papadrakakis,
G.
Kokossalakis. Structural optimization using
evolutionary algorithms. Computers and
Structures 80 (2002) 571-589.
[12] S.N. Patnaik, A.S. Gendy, L. Berke, D.A.
Hopkins. Modified fully utilized design
(MFUD) method for stress and displacements
constraints. Int. J. Numer. Methods Engrg. 41
(1998) 1171-1194.
.