Flexible Scheduling for In-Class Supports: A Blueprint for ...
Optimization of flexible supports for seismic response ...Optimization of flexible supports for...
Transcript of Optimization of flexible supports for seismic response ...Optimization of flexible supports for...
![Page 1: Optimization of flexible supports for seismic response ...Optimization of flexible supports for seismic response reduction of arches and frames Makoto Ohsaki(Kyoto Univ., Japan) SeitaTsuda(Okayama](https://reader033.fdocuments.net/reader033/viewer/2022060100/60b01b778fc20e1f002b63c7/html5/thumbnails/1.jpg)
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.)
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Background
• Complex dynamic property of long-span structure.
• Interaction between upper and supporting structure.
• Interaction of multiple modes.• Dependence on flexibility of support.
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1st mode
2nd mode
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3rd mode
Seismic response
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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
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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
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・Pin-jointed truss・Variable: cross-sectional area・Remove unnecessary members.
Direction of displacement
Ground structure for topology optimization
Previous study(geometrically linear model)
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Previous study(geometrically linear model)
Direction of displacement
Ground Structureapproach
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Deformation like reverse pendulum
Previous study(geometrically nonlinear model)
• Symmetric truss model considering geometrical nonlinearity.
• Optimization of cross-section and nodal location.
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Previous study(geometrically nonlinear model)
Maximize upward displacements for both right and left deformations.
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・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
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( )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)
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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
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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
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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
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Optimal solution
Optimal solution Solution for largerupper-bound area
Simplifiedsolution
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・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
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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.
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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)
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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
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Optimization result
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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
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Vibration modes of optimal solution
1st mode 2nd mode
3rd mode 4th mode
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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
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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
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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
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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
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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
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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
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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
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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
+
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PurposeOptimize flexible base to control mode shape
and reduce response displacement of building frame
Rotate opposite direction against horizontal input
Standard base Flexible base32
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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
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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
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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
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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
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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
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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
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Time‐history response
39
Roo
f dis
plac
emen
tD
rift a
ngle
bet
wee
n1s
t flo
or a
nd ro
of
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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
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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
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Frequency response
42
Roo
f dis
plac
emen
t
Period
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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
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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