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)

6

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

10

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

11

Interaction force at joints

w

vu (slot)

Local coordinates

Cu： u-directional shear forceCv： v-directional shear forceCw： axial force

Hinge

12

Gridshell consisting of elastica

Ｙ

ＸＺ

Small interaction force at joints

13

Ｙ

ＸＺ

Forced disp.: U=0.50m

1.0m

10.0m

Initial planar grid for optimization

Elastica located diagonally between supports

Target shape = elastica

14

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

15

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

16

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)

17

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

19

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

20

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

22

Hinge joint Hinge-slot joint

900mm

applying displacement of 20mm

Small-scale model

23

Hinge joints

Hinge –slots joints

Small-scale model

24

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.