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
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
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
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).
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
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
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
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)
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
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
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
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?
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
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.
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
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
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
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
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
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
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
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
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
D
D
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
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
.
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
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
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
Dealing with non-smoothness
1
11 ( )
2 1
m
i i
N
i i
s D Di
E EE E
e
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
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
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
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.
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
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.
Structural Optimization: Size, Shape, and TopologyME 260 / G. K. Ananthasuresh, IISc
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