Optimization of flexible supports for seismic response reduction of arches and frames

Makoto Ohsaki (Kyoto Univ., Japan)Seita Tsuda (Okayama Pref. Univ.)Yuji Miyazu (Hiroshima Univ.)

Background

• Complex dynamic property of long-span structure.

• Interaction between upper and supporting structure.

• Interaction of multiple modes.• Dependence on flexibility of support.

1st mode

2nd mode

3rd mode

Seismic response

1. Optimization of supporting structure of arch subjected to seismic excitation.

2. Reduction of acceleration and deformation of upper structure.

3. Utilization of flexibility of support.

Purpose

Three-step optimization of supporting structure

Step 1: Maximization of vertical/horizontal displacement ratio against static horizontal load.

Step 2: Minimization of structural volume.Step 3: Dynamic response reduction of upper structure.

Seismic inputSeismic input

Tangential direction

Normal direction

Supporting structure

Roof structure

・Pin-jointed truss・Variable: cross-sectional area・Remove unnecessary members.

Direction of displacement

Ground structure for topology optimization

Previous study(geometrically linear model)

Previous study(geometrically linear model)

Direction of displacement

Ground Structureapproach

Deformation like reverse pendulum

Previous study(geometrically nonlinear model)

• Symmetric truss model considering geometrical nonlinearity.

• Optimization of cross-section and nodal location.

Previous study(geometrically nonlinear model)

Maximize upward displacements for both right and left deformations.

・Pin-jointed truss・Young’s modulus: 2.05×105 N/mm2

・Mass at node A: 1800kg・Mass at nodes 3～10: 600kg

・Variable: cross-sectional area・Standard ground structure approach

Direction of displacement

Groundstructure

Geometrically linear model

A

( )Maximize ( )

( )

subject to

hv

hh

Lgh gh

Lgv gv

Uhh hh

L Ui i i

dR

d

d d

d d

d d

A A A

AA

A

A

A

A

Maximize upward/horizontal disp. ratio due to horizontal forced disp.

← Disp. Ratio

← Stiffness for self-weight

← Stiffness for self-weight

← Stiffness for horizontalload

Optimization problem (Step 1)

Penalization of intermediate cross-sectional area

Cross-sectional area for volume

Cro

ss-s

ectio

nal a

rea

for

stiff

ness

Underestimate stiffness→ Error in structural response

Penalization of intermediate cross-sectional area

Cross-sectional area for stiffness

Cro

ss-s

ectio

nal a

rea

for

volu

me

Overestimate volume→ No error in structural response

Maximize ( )

subject to opt

Lgh gh

Lgv gv

Uhh hh

L Ui i i

V

R CR

d d

d d

d d

A A A

A A

A

A

A

A

Minimize volume under constraint on vertical/horizontal disp. ratiodue to horizontal forced disp.

Optimization problem (Step 2)

← Structural volume

← Disp. Ratio

← Stiffness for self-weight

← Stiffness for self-weight

← Stiffness for horizontalload

Optimal solution

Optimal solution Solution for largerupper-bound area

Simplifiedsolution

・Span L=19.5 m・Rise H=2.613 m・Open angle θ=60°・Member section H-300×150×6.5×9・Member length 2.04 m・Young’s modulus 2.05×105 N/mm2

・Mass: nodes A, T 800 kgnodes 1～8 800 kg

Attach arch to opt 1, and carry out further optimization

Arch model

19 2

A A11

Minimize

subject to

ni

i

Lgh gh

Lgv gv

L Ui i i

F A u A

d d

d d

A A A

A

A

Objective function:Square norm of acceleration in normal direction.Modal analysis: CQC methodRayleigh damping with h=0.02 for 1st and 2nd modes.

←Stiffness against self-weight

： Vertical disp. at node A against self-weightAvD： Horizontal disp. at node A against self-weightA

hD

Optimization problem (Step 3)←Norm of acc. In normal dir.

3

2 2

8 4

1 4 1 4

s r r s s r r ssr

s r

h h h h h h h h

h h

22 2 2 2 21 4 1 4s r s rh h h h

N N

1 1

, ,N N

N i ii s s s s s sr r r r r r

s r

S T h S T h

：normal displacement component at node iN is

：participation factor

sr ：modal correlation coefficient

sT ：natural period sh ：damping factor

sS ：acceleration response spectrum s ：natural circular frequency

r s

Max. acceleration of node i： Ni

s

CQC method(complete quadratic combination)

Response spectrum

0

1

2

3

0 0.5 1 1.5 2

Acc

eler

atio

n [m

/s2 ]

Period [s]

-1.5

-1

-0.5

0

0.5

1

1.5

0 10 20 30 40 50 60

Acc

eler

atio

n (m

/s2 )

Time (s)

0.96 9.0 for 0.161.5( , ) 2.4 for 0.16 0.864

1 102.074 / for 0.864

s s

a s s ss

s s

T TS T h T

hT T

Optimization result

Vibration properties of optimal solution

Mode Period Ts [s]

Damping factor hs

Effective mass ratio 

in X‐dir Ms X [%]

Effective mass ratio 

in Y‐dir Ms Y [%]

1 0.4488  0.0200 49.194  0.000 

2 0.3593  0.0200 1.235  0.000 

3 0.2878  0.0210 0.000  50.771 

4 0.1637  0.0284 0.000  2.395 

Vibration modes of optimal solution

1st mode 2nd mode

3rd mode 4th mode

0

0.5

1

1.5

2

2.5

3

-10 -5 0 5 10

m/s2

m 0

0.5

1

1.5

2

2.5

3

-10 -5 0 5 10

m/s2

m

(a) Tangential acceleration (b) Normal acceleration

(c) Tangential displacement (d) Normal displacement

0

0.005

0.01

0.015

-10 -5 0 5 10

m

m 0

0.005

0.01

0.015

-10 -5 0 5 10

m

m

△: Stiff-model, ×:Flexible-model

Mean-maximum responses of acceleration and displacement

Attachment of viscous damper

0

0.5

1

1.5

2

2.5

3

0 5000 10000 15000 20000 25000

m/s2

N∙s/m

Tangential direction

Normal directionRelation between damping coefficient and acceleration response

Location of the dampers

0

0.005

0.01

0.015

-10 -5 0 5 10

m

m 0

0.005

0.01

0.015

-10 -5 0 5 10

m

m

0

0.5

1

1.5

2

2.5

3

-10 -5 0 5 10

m/s2

m 0

0.5

1

1.5

2

2.5

3

-10 -5 0 5 10

m/s2

m

(a) Tangential acceleration (b) Normal acceleration

(c) Tangential displacement (d) Normal displacement

Attachment of viscous damper

○: Flexible-model with dampers×: Flexible-model without dampers△:Stiff-model

Extension to single‐layer grid

X

YZ

u

v

w19.5 m19.5 m

edge archedge arch

Minimize interaction force between roof and supporting structure

Geometry of supporting structure

(1)

(2)

(3)

(4)(5)

(6)(7)(8)

(9)

(10)(11)

(12)(13)

1

2 34

5

6

7 XX

Optimal objective function value:about 78 % of stiff-model.

Diagonal locationof top node ofsupport

time (s)

Iner

tia fo

rce

of th

e ro

of st

ruct

ure

in th

e X

-dire

ctio

n (N

)

-3.0E+05

-2.0E+05

-1.0E+05

0.0E+00

1.0E+05

2.0E+05

3.0E+05

0 10 20 30 40 50 60

Attachment of viscous damper

Inte

ratio

nfo

rce

at s

uppo

rt

Red: with damperBlue: without damper

Conclusions

• Flexibility of supports can be effectively utilized for reduction of seismic responses of structures.

• Three‐step procedure:– 1st step: static optimizationmaximization of vertical displacement

– 2nd step: static optimizationminimization of structural volume:

– 3rd step: dynamic optimizationseismic response reduction

Optimization of flexible base for reduction of seismic response of buildings

31

Compliant bar-joint structureIsolate structure using a flexible support

Rocking mechanismDissipate seismic energy using rocking of frame and plastic dissipation at column base

Reduce seismic response combiningrocking mechanism and compliant mechanism

+

PurposeOptimize flexible base to control mode shape

and reduce response displacement of building frame

Rotate opposite direction against horizontal input

Standard base Flexible base32

Details of flexible base

33

Material: steelA (members 1‐4,3‐5) 200.0B (members 2‐4,2‐5) 50.0C (members 1‐2,2‐3) 1.0

Truss members (cm2)

Manufacture member Cusing a spring

Optimization problem

34

max · ·

Design variables: nodal coordinate cross-sectional area

Objective function： roof displacement maxminimize

Response spectrum approach (SRSS rule)

Constraint ： lowest natural period

Optimization result

35

・Damping factor 2%・Young’s modulus：200kN/mm2

・Story mass: 8000kg

Beam section (SN400B)2～R H－400×200×8×13

Column section ( BCR295)1～4 □－350×350×16

・Base beam (points 4,5) 45000㎏

Design variables:location of pin support (1, 3) X(m)

3.0 X 3.0Cross-sectional area of horizontal

member (C) A(cm2)0.1 A 10

Optimization result

36

Objective function (m) X (m) A (cm2)

1.078×10‐2 ‐0.719 1.191

X=－0.72 A=1.19 X=0.0 A=100.0

Optimal model Stiff model

Maximum responses

37

0

1

2

3

4

5

0 1 2 3 4Displacement (cm )

stiffopt

0

1

2

3

4

5

0 1 2 3 4 5Acceleration (m/s2)

stiffopt

Acceleration

R R

Floor Floor

Displacement

38

1

2

3

4

0 0.001 0.002 0.003

stiff

1

2

3

4

0 100 200 300

stiff

1

2

3

4

0 20 40 60

stiff

Interstory drift angle (rad) Story shear (kN) Column axial force (kN)

層 層 層

Rocking response is enhanced

Maximum responses

Time‐history response

39

Roo

f dis

plac

emen

tD

rift a

ngle

bet

wee

n1s

t flo

or a

nd ro

of

Trajectory of drift angle and rotation of base

40

Positive rotation angle

右に変位したときを正

Drif

t ang

le b

etw

een

1st f

loor

and

roof

Rotation of base

Positive drift angle

Eigenvalue analysis

41

Optimal model

Stiff model

Period Participation factor

Effectivemass ratio

T(s) X‐dir. (%)1st 0.556 157.30 20.282nd 0.160 90.66 6.743rd 0.107 186.00 28.36

Period Participation factor

Effectivemass ratio

T(s) X‐dir. (%)1st 0.712 16.53 0.222nd 0.379 287.77 67.883rd 0.155 13.75 0.15

Eigenmode

2nd1st 3rd

Eigenmode

2nd1st 3rd

Frequency response

42

Roo

f dis

plac

emen

t

Period

Base with viscous damper

43

‐1.5

‐1

‐0.5

0

0.5

1

1.5

0 5 10 15 20

Displacem

ent (cm

)

Time (s)

‐4

‐3

‐2

‐1

0

1

2

3

4

0 5 10 15 20

Acceleratio

n (m

/s2 )

Time (s)

0

1

2

3

4

5

6

0 0.5 1 1.5 2最大変位 (cm)

0

1

2

3

4

5

6

0 1 2 3 4 5最大加速度 (m/s2)

R

R

階

階

Red: with damper, Blue: without damper

Conclusions

• Flexibility of supports can be effectively utilized for reduction of seismic responses of structures.

• Two‐stage procedure:– 1st stage: static optimizationmaximization of vertical displacement:minimization of structural volume:

• 2nd stage: dynamic optimizationseismic response reductionvariable: cross‐sectional area