Guidelines for seismic design of flexible buswork between substation equipment.pdf
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...
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