module 1: lecture 1 classification of optimization problems and … · 2020. 11. 19. ·...

Lecture 15d

Many problems in optimizing a beam

ME 260 at the Indian Inst i tute of Sc ience , Bengaluru

Str uc tur a l O pt imiz a t io n : S iz e , Sha pe , a nd To po lo g y

G . K . Ana ntha s ur e s h

P r o f e s s o r , M e c h a n i c a l E n g i n e e r i n g , I n d i a n I n s t i t u t e o f S c i e n c e , B a n a g a l o r e

s u r e s h @ i i s c . a c . i n

1

Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc

Outline of the lectureSome problems in optimizing the cross-section profile of a beam under transverse loading.

What we will learn:

How to apply the concepts and ideas learned so far to solve problems in calculus of variations with particular examples of transversely loaded beams.

2


How is a beam different from a bar?(in the context of structural optimization)

A bar deforms axially whereas a beam displaces transversely. That is, a beam bends. The governing differential equation for a bar is of second order whereas that for a beam is fourth order. So, we should be prepared for tedious calculations in a beam.

Cross-section area is all that matters for volume and stiffness in a slender bar: its shape does not matter. That is not true for a beam. The volume of a beam depends on the value of cross-section area but the stiffness depends on the second moment of area of cross-section. So, shape of the cross-section matters.

In this set of problems, we consider only rectangular cross-section of beams. A rectangular has two dimensions, breadth, b(x), and depth, t(x). Both of these can be varied independently (provided that such a beam can be manufactured economically). However, most often, we vary only one of them.

If is b(x)varied,

If is t(x)varied,

For most cross-section shapes, we can write:

3

( ) ( ) ;A x b x t=

( ) ( );A x b t x=

3 2 2( ) ( ) /12 ( ) ( /12) ( /12)I x b x t b x t t t A(x)= = =

( )33 2 2 3( ) ( ) /12 ( ) / (12 ) (1 /12 )I x bt x bt x b b A (x)= = =

( )I x A=


Problem 1

4

( ) ( )

( )0

*

0

2 *

( ) : 0 0

: 0

: ,q( ), , ,12

L

A x

L

Min MC q wdx

Subject to

x EIw q E Aw q

Adx V

tData L x E V

=

− = − =

−

=

We assume here that only b(x) is variable.

( )I x A=2

; 112

t

= =

Minimize the mean compliance of a beam for given volume of material.


Problem 2

5

( )

2

( )0

*

0

2 *

1

2

( ) : 0

: 0

: ,q( ), , ,12

L

A x

L

Min SE E Aw dx

Subject to

x E Aw q

Adx V

tData L x E V

=

− =

−

=

Minimize the strain energy of a beam for given volume of material.


Problem 3

6

( )

( )0

2 *

0

2 *

( ) : 0

1: 0

2

: ,q( ), , ,SE12

L

A x

L

Min V Adx

Subject to

x E Aw q

E Aw dx SE

tData L x E

=

− =

−

=

Minimize the volume of a beam subject to an upper bound on the strain energy.


Problem 4

7

( )

( )0

*

0

2 *

( ) : 0

: 0

: ,q( ), , ,MC12

L

A x

L

Min V Adx

Subject to

x E Aw q

q wdx MC

tData L x E

=

− =

−

=

Minimize the volume of material for a given upper bound on the mean compliance.


Problem 5

8

2

( )0

*

0

2 *

2

: 0

: , ( ), , ,12

L

A x

L

MMin SE dx

E A

Subject to

Adx V

tData L M x E V

=

−

=

Minimize the strain energy of a statically determinate beam for given volume of material. For a statically determinate beam, we can obtain the bending moment without knowing the area of cross-section of the beam.


Problem 6

9

2

( ) ( )0

*

0

2 *

1

2

: 0

: , ( ), , ,12

L

A x u x

L

Max Min PE E Aw qw dx

Subject to

Adx V

tData L q x E V

= −

−

=

Min-max formulation for the stiffest beam for given volume of material.


Problem 7

10

( )

( )0

0

*

0

2 *

: 0

: 0

: ,q( ), , ,12

L

A x

L

L

Min MC q wdx

Subject to

E Aw v qv dx

Adx V

tData L x E V

=

− =

−

=

Minimize the mean compliance of a beam for given volume of material with the governing equation in the weak form.


Problem 8

11

( )

( )0

0

*

0

2 *

: 0

: 0

( ) : 0

( ) : 0

: , ( ), , , , ,12

L

A x

L

L

u u

l l

l u

Min MC q wdx

Subject to

E Aw v qv dx

Adx V

x A A

x A A

tData L q x E V A A

=

− =

−

−

−

=

Minimize the mean compliance of a beam for given volume of material and upper and lower bound constraints on the area of cross-section.


Problem 9

12

( )

( )0

3

0

*

0

*2

: 0

: 0

( ) : 0

( ) : 0

1: , ( ), , , , ,12

L

A x

L

L

u u

l l

l u

Min MC q wdx

Subject to

E A w v qv dx

Adx V

x A A

x A A

Data L q x E V A Ab

=

− =

−

−

−

=

We assume here that only t(x) is variable.

( )I x A=

2

1; 3

12b

= =

Minimize the mean compliance of a beam for given volume of material where the depth of the beam is the design variable.


Problem 10

13

( )0

*

0

2 *

: 0

: , ( ), , ,12

L

A x

L

d

d

Min V Adx

Subject to

MMdx

E A

tData L M(x),M x E

=

− =

=

Minimize the volume of a statically determinate beam with a deflection constraint in its span.


Problem 11

14

( )

( )

( )0

*

0

2

( ) : 0

( ) : 0

: 0

:12

L

A x

d d

L

*d

Min V Adx

Subject to

x E Aw q

x E Av q

E Aw v dx

tData L,q(x),q (x),E, = ,Δ

=

− =

− =

− =

Minimize the volume of material of a beam (statically determinate or indeterminate) for a deflection constraint in its span.


Problem 12

15

( )

( )

( )0

*

0

2 *

0

2 *

( ) : 0

( ) : 0

: 0

1: 0

2

: ,12

L

A x

d d

L

L

*d

Min V Adx

Subject to

x E Aw q

x E Av q

E Aw v dx

E Aw dx SE

tData L,q(x),q (x), = ,E,Δ SE

=

− =

− =

− =

− =

Minimize the volume of material of a beam (statically determinate or indeterminate) for a deflection constraint in its span with an upper bound on the strain energy.


Problem 13

16

( )

( )0

0

*

0

2 *

: 0

: 0

( ) : 0

( ) : 0

: , ( ), , ,12

L

A x

L

L

u u

l l

l u

Min MC q wdx

Subject to

E Aw v qv dx

Adx V

x w w

x w w

tData L q x E V ,w ,w

=

− =

−

−

−

=

Minimize the mean compliance of a beam for given volume of material and upper and lower bound constraints on the transverse displacement.


Problem 14

17

( )

( )0

0

*

0

2 *

: 0

: 0

( ) : 0

( ) : 0

: , ( ), , , , ,12

L

A x

L

L

u t

l c

t c

Min MC q wdx

Subject to

E Aw v qv dx

Adx V

x E w

x E w

tData L q x E V

=

− =

−

−

−

=

Minimize the mean compliance of a beam for given volume of material and upper and lower bound constraints on the stress.


Problem 15

18

( )

q( )0

*

0

2 *

( ) : 0

: 0

: , ( ), , ,12

L

x

L

Max MC q wdx

Subject to

x E Aw q

p dx W

tData L A x E W

=

− =

−

=

Determine the world load distribution of a beam. Note that the upper bound on the overall load is specified.


Problem 16

19

( )

2

( )0

*

0

2 *

( ) : 0

: 0

: ,q( ), , ,12

L

A x

L

Min MC q w dx

Subject to

x E Aw q

Adx V

tData L x E V

=

− =

−

=

A general objective function for a beam problem.


Problem 17

20

0

3

*

0

*

1( ) : 0

12

: 0

: ,q( ), ,

L

b(x),t(x)

L

Min MC q wdx

Subject to

x Ebt Aw q

Adx V

Data L x E V

=

− =

−

Minimize the mean compliance of a beam for given volume of material by varying both breadth and depth of the rectangular cross-section of the beam.


The end note

21

ThanksPra

ctic

e p

rob

lem

s in

cal

culu

s o

f v

aria

tio

ns

Constraints on displacements and strains (stresses) can be imposed.

Transversely deforming beam is the second simplest structural optimization problem. Either breadth or depth can be varied; both can be varied too.

Mean compliance and strain energy are measures of stiffness.Volume of material used is a cost-measure.Objective function and functional constraint can be interchanged without affecting the nature of the solution.

Constraints can also be imposed on profile of the area of cross-section.

Equilibrium equation can be posed in strong or weak form without changing the nature of the solution.

module 1: lecture 1 classification of optimization problems and … · 2020. 11. 19. ·...

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