Lecture 9 Failure constraints in truss optimization & Simultaneous material selection and geometry design ME260 Indian Institute of Science Structural Optimization: Size, Shape, and Topology G. K. Ananthasuresh Professor, Mechanical Engineering, Indian Institute of Science, Bengaluru [email protected]1
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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
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
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
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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