Failure constraints in truss optimization Simultaneous

Lecture 9

Failure constraints in truss optimization

&Simultaneous material selection

and geometry designME260 Indian Inst i tute of Sc ience

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 e n g a l u r u

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

1

mailto:[email protected]

Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc

Outline of the lectureStress and buckling constraints

Material and design indices

Novel material+geometry optimization

What we will learn:

Ashby’s method of material selection

Dealing with strength and stability constraints in truss optimization

Capturing geometric details into a design index

Simultaneous optimization in material and geometry space

Examples that illustrate material+geometry optimization

2


Approaches to structural design

Choose material(s)A

and then

CHow about ?

Simultaneously design the geometry and select material(s).

Usually…

design the geometry

Choose geometryB

and thenselect material(s).


Geometry design vs. material selection

Material axis

Des

ign

ax

is

Simultaneous design andMaterial selection

Starting guess

End point

A

B

C

Optimize geometry for given material

Select material for given geometry


Truss topology optimization assuming a material

Associated with each truss element, define a c/s area variable. This leads to N optimization variables.

Each variable has lower (almost zero) and upper bounds.

Ground structure A possible solution

Kirsch, U. (1989). Optmal Topologies of Structures. Applied Mechanics Reviews 42(8):233-239.

Material selection for the cheapest tensile truss members against failure

= yield or tensile failure strength of the material that is yet to be chosen.

To prevent failure, area of cross-section =

Cost =

y

tt

PA

y

y

t

y

tt

σ

price*ρlPl

PpricelAprice

**

tP tP

Choose a material with lowest .

y

price

*Material index

t

t

y

FoS PA

The material with lowest Material index satisfies the strength criterion and is the cheapest for the component.

With the factor of safety (FoS)


Price*Density (US$/m^3)0.1 1 10 100 1000 10000 100000 1e+006

Te

nsile

Str

en

gth

(P

a)

0.1

1

10

100

1000

Woods

Steels

Aluminium alloys

Glass Ceramic - Slipcast

Titanium/Silicon Carbide Composite

Ashby’s material selection chart using CES (Cambridge Engineering Selector)

*

log log * log

log log * log

y

y

y

priceM

M price

price M

Material selection for the cheapest compression members against (buckling) failure

= Young’s modulus of the material that is yet to be chosen.

To prevent failure, area of cross-section =

Cost =

E

E

pricePL

E

PLLpriceLAprice cccc

ccc

*1212**

2

4

2

2

cP cP

Choose a material with lowest, .

E

price * Material index

2

2

12 ( )c cc

FoSL PA

E

cL

E

PLA cc

c 2

212

With the factor of safety (FoS)

Price*Density (US$/m^3)0.1 1 10 100 1000 10000 100000 1e+006

Yo

un

g's

mo

du

lus^0

.5

0.01

0.1

1

10

Concrete

Steels

Silicone

Cast IronAluminium alloys

Woods

*

log log * log

log log * log

priceM

E

M price E

E price M


Material selection for the entire trussOne option—choose two best materials, one for the tensile members and one for the compression members.This is not attractive from the manufacturability viewpoint.

Second option—choose one material that optimizes the tensile and compression members.But how?


Related work• Lightest, fail-safe truss subjected to equal stress constraints in tension and

compression done by Dorn et al. (1964)

• Achtziger showed that this problem can be solved by taking different stress constraints in tension and compression. (1996). But does not address buckling.

• Stolpe and Svanberg (2004) mathematically proved that at most two materials (one for tension and another for compression) are required when solving this problem.

• Achtziger (1999) has given guidelines for solving truss topology optimization problem using buckling as constraint. But not strength constraint.

The content of this lecture is from:• Ananthasuresh, G. K. and M. F. Ashby, “Concurrent Design and Material Selection for

Trusses,” Proceedings of the Workshop on Optimal Design of Materials and Structures, EcolePolytechnique, Palaiseau, France, Nov. 26-28, 2003.

• Rakshit, S. and Ananthasuresh, G. K., “Simultaneous material selection and geometry designof statically determinate trusses using continuous optimization,” Structural andMultidisciplinary Optimization, 35 (2008), pp. 55-68, DOI 10.1007/s00158-007-0116-4.


A simple truss

1

23

FnF

2

)tan1(;

2

)tan1( nFR

nFR rl

Vertical reaction forces

sin2

)tan1(

sin2

)tan1(

tan2

)tan1(

3

2

1

nFP

nFP

nFP

Internal forces

L

E

PLA

PA cc

c

y

tt 2

212;

Areas of c/s against failure


Basis for a single material selection for a truss

2/12/11

2

1

12)(EE

PL

LPm c

y

t

N

j

j

j

y

N

i

tt

c

c

c

t

ii

Mass =

Design index =t

c

Minimum mass depends on the weighted sum of two material indices.

tN = number of tensile members

cN = number of compression members

Depends only on design of geometry and the loading


1 1

( ) ( )t cN N

t t t c c c

i i

mass AL A L A L

1/ 2 1/ 21 1

( ) 12 | |t c

c

i i c

N Nj

t t j t c

i iy y

Lmass P L P

E E

1 21/ 2

c

t t

f Dm mE S

4f

3f

2f

1f

log f decreases

2log( )m

1log( )m

3M1M

2M

4M

5M6M

7M 8M

9M


Cheapest single material for the entire truss

Price*Density/sqrt(Young's Modulus)1e-004 1e-003 0.01 0.1 1 10

Pri

ce

*De

nsity/T

en

sile

Str

en

gth

1e-004

1e-003

0.01

0.1

1

10

Low alloy steels

Cast Irons

Tool steels

Stainless steels

Aluminium alloys

Aluminium foams

Copper alloys

Beryllium alloysMetals

Cost-limited design

D = 0.01

D = 1

D = 100

CES


D

D


Statement of the problem for simultaneous geometry and material optimization

Geometry variables

1Minimize Strain Energy :

2

Subject to

Static elastic equilibrium equation

T

xMin d

u KU

Satisfying

The failure criteria for tensile and compression members

tt

t

PS

A

Ku = F

2 2

2

12

Loads, boundary conditions, size of the domain

To give

Areas of cross-section, geometry and material

cc

c

EAP

L

.


Algorithm Geometric variables, an initial guess

Function and gradient evaluation

•Calculate the internal forces with unit areas of cross-

sections and unit material properties.

•Calculate the design index.

•Select material using design index.

•Calculate the area of cross-section of individual members

to satisfy failure criteria.

•Calculate the strain energy and its gradient.

Optimization Algorithm

Convergence?

Stop

No

Yes


Dealing with non-smoothness

1

11 ( )

2 1

m

i i

N

i i

s D Di

E EE E

e


Sensitivity Analysis

1

2

T TdSE dE d d

dx x E dx dx dx

K K K A uu u u K

A

2 where,

c tt c

t

dE E dD

dx D dx

d d

dD dx dx

dx

For tensile members

i

i

ti t

t t

dPdA dSP S

dx dx dx

1 2

1.5

For compressive members

We use element equilibrium equation to calculate

1222

i ii i i i

i

c cc c c ci

c

i

i i i

i

L PP dL L dPdA dE

dx E dx dx dxEEP

EAu u P

L

d

dx

P

This gives equations, where

is the number of members in the truss

d

dx

d

dx

m

m

P

C B gu

We use the global equilibrium equation to calculate the rest

2 equations where is the number of nodes in the truss to get n n

dfdxC B

x xd

dx

Ku F

PK K

u uAu

Now assemble

d

dx

d

dx

P

C B g

Q K u h


Results

Example 1

l

5

4

3

21 1

2

3

4

5

6

Optimal layout of structure for force =

1000 N. Selected best material is low alloy

steel. The material cost of the truss is

$1.287 = Rs 59.54. Strain energy calculated

is 18.7325 Joules

Optimal layout for structure with

force = 90 N. The best material is

lightweight concrete.

The cost of the material is $0.0357

= Rs 1.65. Strain energy is 0.034 Joules.

Optimum geometry corresponding to a structure that is ten times as big

as the initial structure. The applied force F = 1000 N. The best material

comes out to be lightweight concrete. The cost of the material is $7.659

= Rs 354.3 and the strain energy is 2.935 Joules.


Example 2

l

12

3

4

1

23

4

Optimal layout for the structure withforce = 1000N. Best single materialis low alloy steel. The material cost

is $ 1.947 = Rs 90.07 and strain energy20.0125 Joules.

Optimal structure for force = 100N Selected material is lightweight concrete. The material cost is $ 0.0397 = Rs 1.84 and strainenergy is 0.0368 Joules.

Optimal structure when only a horizontal load = 1000 N is applied. The internal forces in the dashed members are zero and their cross sections have reached the lower limit. Hence such members may be safely removed from the parent structure. Best material is aerated concrete. The material cost is $0.0378 = Rs 1.75 and strain energy is 0.11708 Joules.

1 2

1

2

3

4

5

1

2

3

4

5

6

1l

2l

1

2

Optimal layout for force = 1000 N. The best material is low alloy steel.The material cost is $2.229 = Rs 103.113and the strain energy is 19.415 Joules.

1

2

3

45

1l

3l

2l

1

2

5

4

3

8 76

1

2

3

4

5 6 7

89

10

11

12

Optimal layout for structure withforce = 100 N. The best materialis lightweight concrete. The materialcost is $ 0.0445 = Rs 2.086 and thestrain energy is 0.0251 Joules.

• CONVERGENCE PROBLEMS:

• STRAIN ENERGY IS A DISCONTINUOUS FUNCTION OF DESIGN VARIABLES

The Strain Energy as a smooth function in the range

10 2mL and

As seen the function is smooth.

10 0.2rad

The Strain Energy in the range

10 2mL and 10 rad

2

As seen the function is nonsmooth. Nonsmoothness occurs when thereis transition from tensile tocompressive members.


The end note

32

Thanks

Tru

ss o

pti

miz

atio

n

Observe how we traversed the discrete material axis using differentiable model

Optimization of geometry and material simultaneously

Geometry optimization for given material vs.Material selection for given geometry

Ashby’s method of material selection

Design index—one number that captures all geometry variables

Strength (stress) and stability (buckling) constraints

Documents

Failure constraints in truss optimization Simultaneous