Optimization techniques play an important role in structural design, the very purpose of
Brics"which
is to find the best solutions from which a designer or a decision maker can derive a
maximum benefit from the available resources."
Optimization in Structural design
By N. G. R. Iyengar
It is fairly accepted fact that one of the most
important human activities is decision making. It
does not matter what field of activity one
belongs to. Whether it is political, military,
economic or technological, decisions have a far
reaching influence on our lives. Optimization
techniques play an important role in structural
design, the very purpose of which is to find the
best ways so that a designer or a decision maker
can derive a maximum benefit from the available
resources.
The basic idea behind intuitive or indirect design
in engineering is the memory of past
experiences, subconscious motives, incomplete
logical processes, random selections or
sometimes mere superstition. This, in general,
will not lead to the best design.
The shortcomings of the indirect design can be
overcome by adopting a direct or optimal design
procedure. The feature of the optimal design is
that it consists of only logical decisions. In
making a logical decision, one sets out the
constraints and then minimizes or maximizes the
objective function (which could be either cost,
weight or merit function).
Structural optimum design methods can also be
according to the design philosophy employed.
Most civil engineering structures are even to-day
designed on the basis of permissible stress
criterion. However, some of the recent methods
use a specified factor of safety against ultimate
failure of the structure. Presently, the approach
is based on the design constraints expressing the
maximum probability of various types of events
such as local or ultimate failure. The objective
function is obtained by calculating each event
and multiply it by the corresponding probability.
The sum of all such products will be the total
objective function. The constraints may also be
probabilistic. These are suitable in situations
when the loads acting on the structure are
probabilistic or the material properties are
random.
During the early fifties there have been
considerable advances in `art` and economy of
the structural design through the use of better
structural materials and refined knowledge of
structural design processes. Thus, the aim was
to put structural design on a scientific basis. The
need for innovation and optimization arose in the
challenging problems faced by the aerospace
industry, which gave a Philip to research
activities in this area.
Requirements for Structural Design
The basic requirements for an efficient structural
design is that the response of the structure
should be acceptable as per various
specifications, i.e., it should at least be a feasible
design. There can be large number of feasible
designs, but it is desirable to choose the best
from these several designs. The best design
could be in terms of minimum cost, minimum
weight or maximum performance or a
combination of these. Many of the methods give
rise to local minimum/maximum. Most of the
methods, in general give rise to local minimum.
This, however, depends on the mathematical
nature of the objective function and the
constraints.
Optimization Problem
The optimization problem is classified on the
basis of nature of equations with respect to
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maximized or minimized. For example, if this
function is cost or weight, then the function is
minimized. On the other hand if it is some other
function, it is maximized.
Structural Optimization Problem
The structural optimization problem can be posed
as:
design variables. If the objective function and the
constraints involving the design variable are
linear then the optimization is termed as linear
optimization problem. If even one of them is nonlinear it is classified as the non-linear
optimization problem. In general the design
variables are real but some times they could be
integers for example, number of layers,
orientation angle, etc. The behavior constraints
could be equality constraints or inequality
constraints depending on the nature of the
problem.
Minimum Weight Design of Structural Elements
(Simultaneous failure mode theory)
One of the earliest techniques employed for the
optimization of structural elements is the
Simultaneous Failure Mode Theory (SFMT). The
approach has been employed to obtain Optimum
Design (minimum strength to weight ratio) of
elements like columns, plates, beams, cylinders,
sheet-stiffener combination etc. The
requirement for optimum design is that all the
failure modes occur simultaneously for the
possible design variables. As the number of
design variables increase or the constraints on
the behaviour variables increase, this approach
does not lead to global optimum solution.
A structural design problem can be represented
as a mathematical model whose constituent
elements are parameters, constraints and
objective or merit function. The design
parameters specify the geometry and topology
of the structure and physical properties of its
members. Some of these can be independent
design parameters and others could be
dependent on the independent design variables.
Some of the design parameters are chosen by
judgment and experience of the designer so as to
reduce the size of the problem. This results in
large savings in computational time, which inturn reduces the cost of the design. From the
design parameters, a set of derived parameters
are obtained which are defined as behaviour
constraints e.g., stresses, deflections, natural
frequencies and buckling loads etc., These
bahaviour parameters are functionally related
through laws of structural mechanics to the
design variables. The objective or the merit
function is formed by the proper choice of the
design parameters. This function is either
Minimize or Maximize
F = F (x1,x2,x3……xn )
Subject to
C1 = C1(x1,x2,x3……xn ) =0
C2 = C2 (x1,x2,x3……xn ) =0
.
.
Cn = Cn (x1,x2,x3……xn ) = 0
and
f1 = f1 (x1,x2,x3,………xn) ³ 0
.
.
fn = fn (x1,x2,x3,……. .xn ) ³ 0
(1)
(2)
x1,x2,……xn are the design variables, C1,C2,….Cn
are equality constraints and f1,f2,…..fn are
the inequality constraints. The nature of the
mathematical programming problem depends on
the functional form of F,C and . If these are linear
functions of design variables, then the
mathematical programming problem is treated as
linear programming problem. On the other hand if
any one of them is a nonlinear function of the
design variable, then it is classified as nonlinear
programming problem. By and large most of the
structural design problems belong to the later. A
hyper surface in the design variable space, such
that all designs represented by points on this
surface are on the verge of failure in a particular
failure mode for a particular load condition, it is
called a behaviour constrained surface. Designs
slightly to one side of the constrained surface will
fail, while designs slightly to the other side will
not fail in the particular mode and load condition
associated with the specific behavior
constrained surface. A hyper surface in the
design variable space such that all designs
represented by points on this surface are on the
verge of being unacceptable for some external
cause, not explicitly related to the behaviour
constraint is called a side constraint surface.
Methods of searching the best design can be
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problem has both integer and real variables. In
order to overcome this difficulty of mixed modes,
optimization studies are carried out for multiple
cell structure treating the number of cells as
parameter rather than a design variable. On the
basis of this study and other studies, one can
categorically state that simultaneously failure
mode theory does not lead to global optimum,
when there is large number of behavior
constraints. When two modes of failure
simultaneously occur, one gets local minima.
However, if there are only two modes of failure
for a solution, minimization by parametric
penalty function proves to be a better technique.
(a) Optimization of Thin Walled Column Elements
under Axial Load
In aerospace structures and other sheet metal
constructions, stiffeners, which are normally
channel and Z-section, are used. The behaviour
of these sections as individual elements differs
from that of the sheet-stiffener combination.
Without precisely understanding the behaviour
of individual elements, we cannot understand
the response of sheet-stringer combination.
Investigations were carried out for-minimum
weight design of columns of channel and Zsections. These two sections were chosen for
the study since the behaviour of these sections
under axial compressive load is very much
different. For such members the number of
behaviour constraints is large since the possible
modes of failure are many. The objective
function which is the weight of the column has
three design variables. Sequential Unconstrained
Minimization technique (SUMT) with interior
penalty function approach is employed for
optimization. As stated earlier this technique has
an advantage that one is always in feasible
domain and at every stage you get a better
solution. The penalty function is minimized by
using Fletcher-Powell method [5].The study
once again brings out that, for the optimum
design all the failure modes do not always occur
simultaneously. In the case of Z-section overall
buckling and local buckling of the flange are the
critical modes at the optimum point. While for the
channel section, the overall buckling and
torsional buckling modes are active at the
optimum point. Here again, the technique leads
classified into simultaneous and sequential
search. The simultaneous classification is
characterized by the fact that all trial designs are
selected before the analysis of any design is
started. On the other hand the sequential search
is characterized by the fact that future trial
designs may be generated by using the results of
the previous trial designs. Over the years a large
number of techniques have been suggested to
solve these equations resulting in an optimal
design. However, these techniques do not
always lead to a global optimum. These at best
lead to local optimum. If the constraint equations
and the objective function are convex functions,
then it is possible to conclude that the local
optimum will be a global optimum. However, in
most of the structural design problem it is
practically impossible to check the convexity of
the function. One of the simplest ways is to start
with different feasible solutions and check the
solutions for global optimality.
Optimization Activities at IIT Kanpur
At this Institute, the work in the area of
structural optimization started way back in 1968
in the departments of Aerospace, Civil and
Mechanical engineering. Depending on the
nature of the problem one picked up for the
study, different methods of analyses have been
employed like closed form solution, finite
element and finite difference etc to obtain the
behaviour constraints. The work was limited to
isotropic materials and small size problems. With
the development of computing facilities, large
size problems were investigated. Since a
structural designer is interested all the time in a
feasible design, Eq. (1) is suitably modified
through the introduction of the penalty term.
This ensures that the design is always in the
feasible domain. The way the penalty term was
introduced depended on the problem. Some of
the work during the period 1967-76 has been
brought out in a book form by Iyengar and
Gupta[1]. Katarya[2] considered the problem of
optimization of multi-cellular wings designed
from isotropic materials under strength and
vibration constraints for simple loadings. The
objective was to minimize the weight of the
wing. Interior Penalty Function approach [3, 4]
has been employed for optimization study. This
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wing skin as an isotropic stiffened plate. Fig. 1
shows the wing structure idealization. Since at
every stage of optimization the analysis has to be
carried out, to save computational time, the
minimum weight design is initially solved
employing linearly approximated re-analysis. To
assess the time saved by employing linearly
approximated re-analysis, the same problem is
solved using the exact analysis in the third and
fourth unconstrained minimization. The study
indicates that the time is reduced by 30 percent;
however, the minimum weight is increased by
1.5 percent. The constrained optimization
problem is solved as a sequence of
unconstrained minimization problems by using
the interior penalty function approach. Cubic
interpolation method of one dimensional search,
which makes use of the gradient, is used for
finding the step length. This study reveals that it
is possible to include multiple constraints for
optimizing the structure. Initial parametric study
carried out before optimization does result in
reducing the number of design variables which in
turn reduces the computational time.
(c) Minimum Cost Design of Grid Floor
The most common form of reinforced concrete
construction of private and public buildings is Tbeam and grid floor. The design of these
structures is generally based on two approaches;
(I) stress design and (ii) strength design. It has
been well established that the strength design is
more logical and also economical. For the design
slabs of various shapes and edge conditions limit
design procedures have also been well
established. These methods result in
considerable economy in the design of reinforced
concrete structures. However, one can further
improve the design if one chooses the
dimensions optimally. The cost of the structure
is often a nonlinear function of the dimensions of
the structure. It is necessary that the structure in
addition to being low cost must meet the safety
and functional requirements. These are also
generally nonlinear. Adidam et.al [7] investigated
the optimal design of T-beam and grid floors
using Nonlinear Mathematical Programming
Technique. The objective function here
represents the cost of one beam and slab
assembly per unit length along the beam span
per unit spacing. This is also expressed as a ratio
to a global optimum [1].
(b) Optimum Design of Wing Structure
Fig.1: Wing Structure Idealization
In real life problems, we encounter multiple
constraints, both behaviour and side constraints.
One such problem investigated and reported here
is the optimum design of wing structure with
design variables including aerodynamic
parameters, such as sweep back angle, aspect
ratio, and thickness to chord ratio, in addition to
the usual geometric parameters [6]. Since
number of variables is large, to minimize the
computer time required for optimization, a
parametric study of the behaviour quantities
e.g., maximum deflection, stresses, buckling
load and natural frequencies is needed to
understand how these are influenced by the
design variables. To start with, there are 13
design variables. This could be reduced to 9
without appreciably changing the behaviour
constraints. Here again, the objective function is
the weight of wing structure. The optimization
problem is to reduce the weight satisfying the
given constraints. As the geometry of the wing is
quite complicated, it is not possible to obtain the
behaviour constraints in the closed form. For the
static and dynamic analysis, the wing is idealized
by finite elements using constant stress
triangular membrane elements and rectangular
shear panels for the skin and web respectively.
The stringers are represented by axial force
members. Elastic buckling constraints are
introduced by treating a typical portion of the
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tank, as shown in Fig.2a. The base of the
structure is subjected to ground acceleration
during an earthquake. For the response analysis,
the structure is idealized as a single-degree-offreedom system as shown in Fig. 2b. The
stiffness of the system is computed by using the
flexibility analysis. The objective function of the
structure is the total volume of the structure. The
lengths of the vertical members of the truss are
treated as design variables with the condition
that the sum of their lengths is a constant.
Hence, instead of three design variables we shall
have two. The total design variables are twelve.
The constraints on the natural frequency of the
structure is so specified that when the tank is
partially filled the first few frequencies of the
liquid oscillations are kept well below the natural
frequency of the structure. This is done to avoid
large amplitude liquid sloshing due to earthquake
excitation. For the stress constraint, the ground
acceleration during an earthquake is assumed to
be stationary random process. For response
calculation, the ground acceleration is locally
considered white noise.
The following observations can be drawn on the
basis of this study:
1.
Stress constraint is the only active
constraint at the optimum point. The optimal
design is sensitive to the way in which this
constraint is defined.
2.
The optimal design is sensitive to the
degree of correlation between member stresses.
(e) Minimum Weight Design of Sheet-Stringer
Panels
of cost per unit area of floor to the cost of one
unit of concrete. An existing square grid of
18.83 meter span was optimized. This results in
a relative cost of 58.76. Further, the optimal
design turns out to be 1.2 meter square grid
instead of existing one meter square. This
indirectly results in saving of form work and
material.
Optimization Under Random Environment
Most physical systems operate under random
environment, e.g., flight vehicles subjected to
gust loading, jet engine noise, boundary layer
turbulence, trains, towers, buildings subjected
to earth quakes. Nigam[8] and Narayanan[9]
have applied the concepts of this design to
structural optimization in random vibration
environment. The problem is formulated by
considering the time dependence of the
response quantities and then reducing it to a
standard nonlinear programming problem (NLP).
The weight of the structure is optimized with
constraints on natural frequencies, buckling
stresses, geometric dimensions, and dynamic
responses such as stresses, acceleration, and
fatigue life of the structure. The constraints are
expressed probabilistically.
Choosing the
probability of failure in such a failure mode is a
matter of engineering judgment based on the
functions of the structural system and on the
possible consequences.
(d) Elevated Water Tank Staging (Earthquake
loads)
2(a)
2(b)
Fig.3: Sheet-Stringer Combination
Fig.2: Support Structure for Water Tank
A typical sheet-stringer panel used in aircraft is
Consider a truss structure supporting a water
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shown in Fig.3. Instead of multiple panels we
shall consider only a single panel between two
stringers. In each frequency band, the lowest
frequency corresponds to the stringer-torsion
mode and the highest frequency to the
stringer-bending mode.
While designing for sheet-stringer panels, the
distance between the frames is specified and
the density of the material is constant. There
are five design variables in this case. In the
stringer-torsion mode, the adjacent panels
vibrate out of phase, whereas in the stringerbending mode they vibrate in phase. The
stringer-torsion mode does not get appreciably
excited by the jet noise as compared with the
stringer-bending mode. To reduce the
response, the pressure spectrum should be so
used that the frequency of the sheet
corresponding to the stringer-bending mode is
well above the frequency of the maximum
sound energy. Fatigue damage is used as the
other constraint. The problem is posed as an
NLP and solved as an unconstrained
minimization problem. The optimization study
results in about 15 percent reduction in
weight.
Optimization Studies in Fibre reinforced
Composites
Fibre Reinforced Composite (FRP) materials are
being employed as primary load carrying
members in aerospace structures in view of
the advantages they offer as compared to
metallic structures. This is because, the fibre
orientation in each lamina can be chosen
depending on the designer's requirements.
This results in changes material properties
which are direction dependent unlike in
metallic materials, which are direction
independent.
Substantial amount of
composites are used in Light Combat Aircraft
(LCA), Advance Light Helicopter (ALH) and the
two seater trainer aircraft (HANSA). This has
resulted in substantial savings in structural
weight in-view of their high strength to weight
ratio and high stiffness to weight ratio. The
advantages can be further improved, provided
these materials are used optimally. The author
and his research students have contributed
significantly to the literature on optimum
design of FRP laminates.
Most of these
studies have been discussed in a book by Iyengar
and Gupta[10]. These studies deal with the
minimization of the weight of composite
laminates subjected to various types of behaviour
constraints. Optimization studies in composites
is little more involved as compared to metallic
materials as the number of variables increase
substantially. Since the design variables will be a
mix of real and integer variables this makes the
analysis complicated.
Genetic Algorithms for Optimization Studies
The problem of mixed mode variables has been
solved by the application of Genetic Algorithms
(GA). These are suitable for complex optimization
problems. Application of genetic algorithms for
optimization studies is gaining wide interest
because of their robustness in locating the global
optimum. Recently this technique has been
applied by Sivakumar [11] for the study of
optimization of FRP laminates with and without
cut-outs undergoing large amplitude oscillations.
The cut-outs are of various sizes and shapes. In a
recent paper, Sivakumar et.al., [12] have clearly
brought out the advantages of GA for the
optimum design of laminated composite plates
with cutouts, over the conventional techniques,
and also the effectiveness of GA in locating the
optimum for problems involving large number of
constraints and variables. They have concluded
from the studies carried out, that the DFP method
is not suitable for the problems considered.
Finding an accurate result requires a large number
of function evaluations than other techniques.
The complex search technique finds the optimum
solution with a small number of function
evaluation than GA when the number of
constraints is not large. When a large number of
constraints are present, it takes a large number of
function evaluation. The disadvantage is that it
cannot handle discrete variables.
GA seems to be the best tool to optimize
composite laminates, since it can handle all types
of variables providing the flexibility needed to
solve such complex problems.
Future Directions in optimization Studies
So far the investigations have been confined to
optimization with single objective function.
Design of a complex system like space vehicles,
aircrafts etc., requires a large number of merit
functions to be satisfied for the best design.
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Furthermore, optimum design of sub-systems
does not lead to optimal design of the entire
system. For example, in the case of aircrafts,
the objective function will be to minimize the
weight. However there could be other
objectives like minimizing the Drag, maximizing
the range etc., Problem of this type has to be
treated as multi objective function optimization.
The work in this direction has already been
initiated.
REFERENCES
[1]Iyengar, N.G.R and Gupta S.K ` Programming methods in
structural design`,Edward Arnold Pub.Ltd.U.K.1980
[2]Katarya, R.` Optimization of multi-cellur wings under
strength and vibrational constraints for simple loading,
M.Tech thesis,IIT Kanpur ,1973
[3]Fiacco,A.V
and
McCormick,G.P`The
sequential
unconstrained minimization technique for nonlinear
Programming: Aprimal-Dual method ,Management sc.
10,360-366,1964
[4]Fiacco,A.V.andMcCormick,G.P
`SUMTwithout
parameters, Operations research,15,820-827,1967
[5]Fletcher,R and Powell, M.J.D., ` A rapidly convergent
descent method for minimization, Computer J. 6,163168,1963
[6] Rao,V.R. Iyengar, N.G.R. and Rao,S. S,Optimization of
wing structures to satisfy strength and frequency
requirement,` Computers and Structures,10,669-674,1977.
[7]Adidam, S.R., Iyengar,N.G.R. and Narayanan,G.V.`
Optimum design of T-beam and grid floors,J. structural
engineering,6,113-124,1978
[8]Nigam,N.C. Structural optimization in random vibration
environment,AIAA.J.10,551-553,1972
[9]Narayanan,S., Structural optimization in random vibration
environment, Ph.D Thesis, IIT Kanpur, 1975.
[10]Iyengar,N.G.R.and
Gupta,S.K.Structural
design
Optimization,Affiliated East west press Ltd. New Delhi, 1997
[11]Sivakumar,K., Optimum design of laminated composite
plates with dynamic constraints,Ph.D thesis IIT Kanpur
[12] Sivakumar,K. and Iyengar, N.G.R and Deb Kalyanmoy,
Optimization of composite laminates with cutouts using
genetic algorithm, variable metric and complex search
methods, Eng. Opt.32,635 –657,2000
About the author:
Dr. N.G.R. Iyengar is a Professor in the Department of Aerospace
Engineering. He did his Ph.D. at IIT Kanpur and has served the institute in many capacities for the
past 37 years. He has played the lead role in establishing the ARDB center of excellence for
composite structures and technology in our institute. His research interests include structural
analysis and optimization, composite structures, composite materials and their failure.
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