optimization of locations of slot connections of ... · 20 optimalsolution type1 type2 joint type...
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OPTIMIZATION OF LOCATIONS OF SLOT CONNECTIONS OF GRIDSHELLSMODELED USING ELASTICA
Makoto Ohsaki (Kyoto University)Kazuya Seki (Azusa Sekkei)
Yuji Miyazu (Hiroshima University)
1
Background2
Gridshell Structure
Planar grid Gridshell
Forced disp.
Bending moment of member and interaction at joints may be large.
Hinge joint
Doubly curved latticed shell by bending planar grid
3
Purpose of study
Define target shape by elastica
- Shape of gridshell with small interaction force at joints
- Add hinge-slot joints to further reduce reactionforce
4
Assembly of elastica
Gridshell with elastica⇒ No interaction force between crossing members
elasticaForced disp.
Distorted shape
Forced disp.
5
Definition of elastica
)()( sEIsM
sss
)()(
s : Arc-length parameterφ(s) : Deflection angleM(s) : Bending momentEI : Bending stiffnessκ(s) : Curvature
Forced disp.
Buckled shape of a rods
φ(s)
No explicit solution(expression using elliptic function)
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Definition of elastica
)()( sEIsM
sss
)()(
Forced disp.
Buckled shape of a rods
φ(s)
Minimize 2( )EI s ds
: penalty parameter related to length of barIgnore axial and shear deformation
Length-constrained minimum energy curve (spline, trajectory)
Step2 : Incrementally computeφ(s), z(s) and M(s) as
0
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
z
x
0
0.5
1
1.5
2
z
0
0.5
1
1.5
2
z
i = 0
7Incremental computation of elastica shape
ii
ii sEIM
1
1 1 1 1
( , ) ( cos , sin )
i i
i i i i i i
x zx s z s
ii zPMM 0
i = 20
i = 40
i = 50
M0
M0
P
P
PM0
M20=2.42
M50=0
M40=0.92
E(s) Target shape
Step1 : φ(0)=x(0)=z(0)=0 , M(0)=M0Assign M0, P, and material constants
is
0
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
z
x
0
0.5
1
1.5
2
z
0
0.5
1
1.5
2
z
i = 0
8
i = 20
i = 40
i = 50
M0
M0
P
P
PM0
M20=2.42
M50=0
M40=0.92
E(s) Target shape
Step3 : Let k=i+1 and go to Step2Stop if Mi = 0
Step4 : Compute length L anddisp. U from
n
n
i i xLUsL ,2)(
Incremental computation of elastica shape
9
Joint types
Y
XZ
Hinge + slot
Y
XZ
Rotation around z axis
Rotation around z axis+
Translation in x direction
Hinge
DOFs
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Example of elastica
0
1
2
0 1 2 3 4 5 6 7 8 9
z [m
]
x-y [m]
Length: L=10 m, Forced disp.: U=0.5m
Material: steel: Young’s modulus = 205 GPa, Poisson’s ratio = 0.3
Width: 0.06 m, Thickness: 0.015 m, P=369 N,M0=720 Nm
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Interaction force at joints
w
vu (slot)
Local coordinates
Cu: u-directional shear forceCv: v-directional shear forceCw: axial force
Hinge
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Gridshell consisting of elastica
Y
XZ
Small interaction force at joints
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Y
XZ
Forced disp.: U=0.50m
1.0m
10.0m
Initial planar grid for optimization
Elastica located diagonally between supports
Target shape = elastica
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Y
XZ
Type 1: uniform cross section
60mm15mm
500mm
Initial planar grid for optimization
Type 2: large section between supports
10020
500
Large section
5010
500
Small section
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Shape of diagonal members
Type 1 Type 2
0
1
2
0 1 2 3 4 5 6 7 8 9
red: elastica, green: Type 1, blue: Type 2.
Type 2 Type 1
Joint type Hinge HingeCu mean 1.162 2.890Cv mean 1.948 4.288Cw mean 0.678 0.833Cu max 5.669 16.17Cv max 5.885 16.40Cw max 2.253 4.802
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Interaction force at joints
Elastica shape for large diagonal member⇒ reduction of interaction force at joint
w
v
u (slot)
Local coordinate
Cu:u-directional shear force
Cv:v-directional shear force
Cw: axial force
(unit: kN)
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Optimization problem
・Design variable x: locations of hinge-slot joints・Objective function: deviation of diagonal curve from target elastica
2
1
219
1))(()(
izij
jijz xceF xminimize
Optimize locations of several hinge+slot joints⇒ Reduction of interaction force with limited construction cost
・Optimization method: Simulated annealing (SA)
ez: z coordinate of target elasticacz: z coordinate of nodes along diagonal members
Simulated annealing (SA)18
termination condition
Initial setting
Find neighborhood solutions
termination
Acceptance decision
m=nm<n
n : Specified number of steps
Assign number hinge-slots and optimize their locations
Case 1: Objective function is reduced → Accept
Case 2: Objective function is increased → Accept with probability defined as
Acceptance criteria of best neighborhood solution
scaleTdiff
・
expnumber random a
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Optimal solution
Type 1 Type 2Joint type All hinge All slot Optimal All hinge All slot OptimalCu mean 2.890 1.765 2.116 1.162 0.674 0.894Cv neam 4.288 3.962 4.031 1.948 1.588 1.732Cw mean 0.833 0.963 0.957 0.678 0.537 0.554
(unit: kN)
Four hinge+slot jointsin 1/4 region
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Optimal solution
Type 1 Type 2Joint type All hinge All slot Optimal All hinge All slot OptimalCu mean 2.890 1.765 2.116 1.162 0.674 0.894Cv neam 4.288 3.962 4.031 1.948 1.588 1.732Cw mean 0.833 0.963 0.957 0.678 0.537 0.554
(unit: kN)
• Drastic reduction of interaction force by optimizinglocations of Hinge+Slots.
• Reduction by several Hinge+Slots is equivalentto results of Hinge+Slots at all joints.
• Type 2 has smaller force than Type 1.
Small-scale model21
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Hinge joint Hinge-slot joint
900mm
applying displacement of 20mm
Small-scale model
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Hinge joints
Hinge –slots joints
Small-scale model
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Conclusions
Joint forces of a gridshell can be reduced by designing the target shape of beam as an elastic.• Elastica can maintain equilibrium shape
without any force from the perpendicularly connected beams.
Joint forces can be reduced by assigning Hinge+Slot joints.
Number of Hinge+Slot joints can be reduced by minimizing the deviation of the shape of curved beam from the target shape of elastica.