Optimization in Structure Design

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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

By N. G. R. Iyengar

Optimization in Structural design

"Optimization techniques play an important role in structural design, the very purpose ofwhich is to find the best solutions from which a designer or a decision maker can derive a

maximum benefit from the available resources."


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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 non-

linear it is classified as the non-linear

optimization problem. In general the design

variables are real but some times they could beintegers 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) ofelements like columns, plates, beams, cylinders,

sheet-st iffener 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 designparameters 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 in-

turn 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

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:

Minimize or Maximize

F = F (x ,x ,x x ) (1)1 2 3 nSubject to

C = C (x ,x ,x x ) =01 1 1 2 3 nC = C (x ,x ,x x ) =02 2 1 2 3 n.

.

C = C (x ,x ,x x ) = 0n n 1 2 3 nand

= (x1,x2,x3,xn) 0 (2)

.

.

= (x1,x2,x3,. .xn ) 0

x ,x ,x are the design variables, C ,C ,.C1 2 n 1 2 nare equality constraints and 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 thedesign 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 behaviorconstrained 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

f1f1

fn fn

f1,f2,..fn


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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 failuremode 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 behaviourof 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 Z-

sections. 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 ofthe 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 startwith 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 toisotropic 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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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

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. Toassess 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 forfinding 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 T-

beam and grid floor. The design of these

structures is generally based on two approaches;

(I) stress design and (ii) strength design. It hasbeen 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 safetyand 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


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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 andmaterial.

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 astandard 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 (Earthquakeloads)

2(a) 2(b)

Fig.2: Support Structure for Water Tank

Consider a truss structure supporting a water

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-of-

freedom system as shown in Fig. 2b. The

stiffness of the system is computed by using the

flexibility analysis. The objective function of thestructure 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 earthquakeexcitation. 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

Fig.3: Sheet-Stringer Combination

A typical sheet-stringer panel used in aircraft is


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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 stringer-

bending 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 objectives like minimizing the Drag, maximizing

does not lead to optimal design of the entire the range etc., Problem of this type has to be

system. For example, in the case of aircrafts, treated as multi objective function optimization.

the objective function will be to minimize the The work in this direction has already been

weight. However there could be other initiated.

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About the author: Dr. N.G.R. Iyengar is a Professor in the Department of AerospaceEngineering. 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.

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. 6,163-

168,1963

[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


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Optimization in Structure Design

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