On stress constraints in structural optimization

MECHANICS RESEARCH COMMUNICATIONS

Vol.

0093-6413/81/020061-06502.00/0

ON STRESS CONSTRAINTS

8(2),61-66, 1981. Printed in the USA. Copyright (c) Pergamon Press Ltd

IN S T R U C T U R A L O P T I M I Z A T I O N

Erdal Atrek Faculty of Engineering Engineering,

Istanbul

and Architecture, Technical

Department

University,

of Civil

Istanbul,

Turkey

(Received 8 November 1979; accepted for print 15 December 1980)

Introduction

Stress c o n s t r a i n e d o p t i m u m design of structures has been variously reported to proceed based on i) fully-stressed design (e.g. [1,2]), 2) linear p r o g r a m m i n g with successive linearizations of the nonlinear problem, the constraints on stresses being defined rather in terms of member forces [3], and 3) by w r i t i n g each stress directly as a linear c o m b i n a t i o n of the design variables [4]. The first approach, while practical, has also been shown to lack the o p t i m a l i t y objective (e.g. [i, 2, 5]). On the other hand, the linear p r o g r a m m i n g formulation, u t i l i z i n g the force method for d e r i v a t i o n of the c o n s t r a i n t expressions, requires additional equations based on chosen redundants in order t o enforce compatibility, and e f f i c i e n c y may be impaired if a good choice of redundants is not made. The last formulation provides some practical improvement, h o w e v e r efficient d e r i v a t i o n of all constraint e x p r e s s i o n coefficients, and their m o d i f i c a t i o n would not be possible by direct use of the virtual work principle, as described therein. Therefore, two main issues would tolerate additional c o n s i d e r a t i o n : a) d e r i v a t i o n of the stress expressions in the most efficient manner, and b) m o d i f i c a t i o n of these expressions with the m i n i m u m effort. These issues are treated herein, together with an e x t e n s i o n to c o n s i d e r a t i o n of fabricational constraints.

Theory

While the virtual work p r i n c i p l e

is useful

total reliance may be m i s l e a d i n g

for e s s e n t i a l l y

problems

such as o p t i m u m design.

tigators

to propose

introduced

In fact,

that one virtual

for each stress c o n s t r a i n t 61

for linear

analysis,

non-linear

this has

led inves-

load case needs to be [4].

This

is actually not

62

ERDAL ATREK

necessary, However,

as w i l l

very

s i m p l y be d e m o n s t r a t e d

f i r s t the s t r e s s

are w r i t t e n principle,

in m a t r i x resulting

modification

due

expressions

form with

for the e n t i r e

disregard

in a c h a r a c t e r i s t i c

to r e d e s i g n

ness of this r e l a t i o n s h i p ,

l a t e r herein. structure

for the v i r t u a l w o r k relationship.

is p r o p o s e d ,

Next,

a

b a s e d on a n o n - u n i q u e -

with modifications

due to

different

load c a s e s b e i n g o b v i o u s . An elastic e l e d by stress

structure

finite elements, components

collected

E6]

the b e n d i n g

in the e l e m e n t s

the s e c t i o n converted

, these

stress

under

a

as the o r i g i n a l

consideration.

ones,

loading,

The

representative

a)

nodal

the a x i a l

now,

these

forces which

are

for a c l a s s i c a l stress,

f r o m the n e u t r a l

If,

and m o d -

the s t r u c t u r e

For e x a m p l e ,

at a u n i t d e p t h

the r e s u l t i n g

some

comprising

are t a k e n as

into e q u i v a l e n t

the s t r u c t u r e ,

under

is c o n s i d e r e d .

in a c o l u m n v e c t o r

beam element b)

in e q u i l i b r i u m

axis,

stresses

to

the same

i.e.

= Z a ~ Z S aI

where matrix diagonal

Z

is the d e s c r i b e d

matrix

stresses,

and

of f o r c e s

9I

at

are

are r e a p p l i e d

s t r e s s e s w i l l be e x a c t l y

and

linear

(or m o m e n t s )

is d e f i n e d

(9i)~

as

(i)

transformation,

in d i r e c t i o n

S is a

of r e l e v a n t

i/a~, J where

a 3. is the

J

jth d e s i g n v a r i a b l e

(such as c r o s s - s e c t i o n a l

inertia,

or t h i c k n e s s ) .

of ways,

b u t the a d v a n t a g e s

not a l w a y s First,

changes

(i) m a y be o b t a i n e d by a n u m b e r

to be g a i n e d by this d e s c r i p t i o n w i l l

through

does

enough,

it w i l l be

all r e d e s i g n s

not a f f e c t e x a c t

satisfaction

the s t r e s s

since,

for a g i v e n c o n f i g u r a t i o n , the s t r e s s

equilibrium

although

design.

of E q u a t i o n

Z satisfies

represented

(i), e v e n

This o c c u r s

compatibility,and

by a are in i n t e r n a l

the real c o e f f i c i e n t s

zij w i l l

t h e y n e e d not be r e c o m p u t e d

Revision

to s a t i s f y E q u a t i o n

that holding

configurational

o w i l l be c h a n g i n g .

for e a c h r e d e s i g n ,

the i n i t i a l

sufficient

components

[7]. Thus,

be d i f f e r e n t after

components

observed

not i n v o l v i n g

though

since

m o m e n t of

be o b v i o u s .

interestingly

Z constant

Equation

area,

(i) .

of the f o r c e s S w i l l be

CONSTRAINTS

The problem remaining

IN

is efficient

matrix Z for a given initial structure possible

STRUCTURAL

63

and economical

design.

is now considered.

computation

A statically

I.

clearly the matrix On the other hand,

of

determinate

Since stress redistribution

for such a structure,

equal to the unit matrix

OPTIMIZATION

Z

is not

will be

for a statically

~

indeterminate

structure,

Z is, in general,

fully populated

and,

~

by definition,

substructured: Z =

Removing

chosen

r

I Zrr

Zrd I

LZdr

ZddJ

redundant

(2)

stress components

will yield a basic statically

determinate

from the structure

structure: -i

I =Zdd

+ Zdr

(I - Zrr)

Zrd

(3)

As a result: -i ~dd=~Here

~dr

(~-

~rr )

~rd

(4)

Z T ]T may be obtained by releasing a unit stress ~dr from each redundant stress component in its turn, and redistrib'

[Z~r ~

uting throughout

the entire structure.

{rd

can then be com-

puted from ~dr by consideration of the Maxwell-Betti rocal relationships. Equation

(4).

to generate

Thus,

Finally,

~dd

Law of recip-

would be obtained by use of

only r unit load cases need to be solved for,

the entire matrix

While the above procedure

Z.

gives the correct matrix Z for the ~

initial design,

another matrix Z satisfying

constructed by releasing

only one stress component.

rives from the fact that, as a set of homogeneous set of coefficients all the equations, of each other. and,

in general,

Equation

since Equation

(i) may be This de-

(i) may be considered

equations with the same variables,

satisfying

one of the equations will

except that these equations will be

This even more greatly does not require

simplifies

cedures will yield the same matrix.

satisfy multiples

the procedure

choice of redundants.

for a system of one degree of statical

one

indeterminancy

Obviously,

both pro-

64

ERDAL ATREK

An e x t e n s i o n with

to f a b r i c a t i o n a l

the s e c o n d

Numerical

example,

the p o i n t s

examples

Three-bar

raised

truss

(Figure

i) :

by o p t i m a l i t y

constraint

of n o n - v a n i s h i n g

in the next section.

load cases

in the p r e v i o u s

This

section,

two

also

ib./in~,

approaches

areas 5

shown on F i g u r e

is 0.i

truss has b e e n p r e v i o u s l y

criteria

c o n s t r a i n t s : l o l l ,[o3[!

material

together

are c o n s i d e r e d .

vestigated

stress

is d e s c r i b e d

examples

To i l l u s t r a t e simple

numerical

constraints

(e.g.

at the optimum,

F8,9~)

in-

with

along with

the

the

ksi.,[o21 ! 20 ksi. , for the three i.

The u n i t w e i g h t

and all bars

have

of the

the same m o d u l u s

of

elasticity.

1

10

40 KiP FIG.

leasing column

design

a unit stress of

truss

of all c r o s s - s e c t i o n a l from bar

is o b t a i n e d

as

consideration

of r e l a t i v e

stiffnesses,

[.7071

and

z13 = z 3 1 ( L 3 / L I) .... .2929.

jth

bar,

cant

figures.

the m a t r i x

Z.

obtained 16.04

in e i g h t

ibs., w i t h

algoriLhm iterations

9 =

[6.906

was kept

the m i n i m u m w e i g h t

elsewhere

although

different.

the

Then,

length

during

in [[0!.

the c r o s s - s e c t i o n a l

by

.2929, of the

the rest of all redesigns

The o p t i m u m

in a total w e i g h t

of 15.96

first

(L2/L I)

(4) then gives

2.473J T

re-

off to four s i g n i f i -

constant

resulted 2.775

z21

Lj is the

described

very well w i t h [9!,

equal,

-.2929J T.

z12

rounded

of E q u a t i o n

This m a t r i x

in an o p t i m i z a t i o n

.4142

Here

have b e e n

Application

areas

1 and r e d i s t r i b u t i n g ,

Z

and the n u m b e r s

20 KiP

i

Three-bar For an i n i t i a l

30 KiP

in~. Ibs~ areas

This

design

of compares

reported are s o m e w h a t

CONSTRAINTS IN STRUCTURAL OPTIMIZATION

Cantilever

beam

and cable

however

with

square

viously

[i].

capable

of t r e a t i n g

The

beam

system

cross-section,

approach

described

any type (i)

by w r i t i n g

Equation

structure,

for an initial

areas

cross-sectional of the b e a m

respectively,

a 2 is the m o m e n t of the beam. of the b e a m

(I z)

critical

section

at the b u i l t -

Other

sections

may also be considered,

uniform

critical

resulting

beam profile

on the other

A

45"

in a non-

FIG.

at the opti-

.9731

.0879

.0538

Mbla 2

• 2691

.1030

-.5382

Nc/a 3

.0381

Cantilever

2

beam

Nb

0

and cable

o

indicates verify

that

-.1269

.9239

is the s e c o n d design

both

matrix

sides

0

0

N

in this

area

equation,

and m o m e n t

formulated

A

This

= I

relation

solves

is 3).

z

S

among

a still

further

relationship

between

It also

for f a b r i c a t i o n a l

in an u n c o u p l e d

x

(5) by S -1, w h e r e

a coupled

of inertia.

(One may

to axial

with

an a c c e p t a b l e

to be used

[/a3J

moment.

of b e n d i n g

cross-section

may be obtained.

coefficients been

and M i n d i c a t e s ratio

c m

of E q u a t i o n

that of p r o v i d i n g

cross-sectional

have

force,

the n u m e r i c a l

variables

namely

sensitivity

normal

for a b e a m of u n i f o r m

By p r e - m u l t i p l y i n g

which

(5)

o

I/a2 I

stiffness

issue,

LJa

0

m

all

A-A

cross-

/Nb/a ~

easily

is

as can be seen

mum.

Here N

hand,

10

in end.

if necessary,

pre-

and the

and w h e r e

is taken

herein,

treated

and

of inertia

The

has been

problem,

design

a3

(Axb, Axc)

: A similar

for this

aI

cable,

2)

of c r o s s - s e c t i o n

of a I = a 2 = a 3 , w h e r e are

(Figure

65

manner

provides constraints,

heretofore

[ii].

66

ERDAL ATREK

Conclusions

Stress

expressions

rived using proach makes

a conceptually

rather

analysis

of Equation relating

derivation

straints.

the stress

is also useful

advantageous work

components

for non-linear

lies

design

de-

redistribution

ap-

The

derivation

structure.

As satis-

non-uniqueness

greatly

simplifies

of the initial

and their modification.

extended

generality

are

principle.

of the entire

coefficients

may be easily

Further

structural

(i) may be sufficient,

of stress

relationship

more

than the virtual

use of linear

faction matrix

for use in optimum

to cover

fabricational

in that the given

structural

analysis

The con-

formulation

[7].

References

I. K. F. Reinschmidt, C. A. Cornell, and J. F. Brotchie, "Iterative Design and Structural Optimization", Jo Struct. Div., Proc. ASCE, 92, ST6, 281 (1966). 2. S.Patniak and P. Dayaratnam, "Behavior and Design of Pin Connected Structures", Int. J. Num. Meth. Engng., 2, 579 (1970). 3. B. Farshi and L. A. Schmit, Jr., "Minimum Weight Design of Stress Limited Trusses", J. Struct. Div., Proc. ASCE, ]00, ST1, 97 (19V4). 4. G. Sander and C. Fleury, "A Mixed Method in Structural Optimization", Int. J. Num. Meth. Engng., ]3, 385 (1978). 5. L. Berke and N. S. Khot, "Use of Optimality Criteria Methods for Large Scale Systems", AGARD LS-70, i.i (1974). 6. R. H. Gallagher, Finite Element Analysis: Fundamentals, Prenti~e-Hall, Englewood Cliffs, New Jersey (1975). 7. E. Atrek, "An Iteration Matrix Method for Structural Nonlinear Analysis and Optimum Design", in preparation. 8. V. B. Venkayya, "Design of Optimum Structures", J. Comps. Structs., ], 265 (1971). 9. I. C. Taig and R. I. Kerr, "Optimisation of Aircraft Structures with Multiple Stiffness Requirements", AGARD CP-123, 2nd Symp. Struct. Optim., Milan, Italy, 16.1.(1973). 10. E. Atrek, "Yaplsal Optimizasyonda Optimumluk Ko~ullarl Uzerine", TUB~TAK, VII. Science Congress, Ku~adasl, Turkey (1980). ii. R. A. Gellatly, R. G. Helenbrook, and L. H. Kocher, "Multiple Constraints in Structural Optimization", Int. J. Num. Meth. Engng., ]3 , 297 (1978).