Takumi ITO

1
Dynamic Behavior of Safety Domain about Plastic
Collapse of Ductile Steel Framed Structures
Takumi ITO The University of Tokyo Japan
2
Introduction
Purpose
Nonlinear dynamic response analysis on structures
is an effective tool for seismic demand
evaluation, however, which is usually considered
complicated and costly. In this study, A
Simplification of Non-linear Dynamic Design
Procedure on steel framed structures is
proposed. Furthermore, the Hardening Rule is
applied to failure surface model to capture
dynamic behavior of bilinear steel building
structure
Multi-story steel frames
Earthquake input
3
Safety Domain about Plastic Collapse
Normal vector 1, 0T
F2
Normal vector 1, 2T
F2
Normal vector 1, 1T
F1
F1
Restoring force space
Moment-resisting frame
xp
xp
2 xp
xp
xp
Mp
Sp
Plastic hinge
2nd story failure
1st story failure
Combined failure
F1xpF2xp4MpSp
F2xp4MpSp
F1xp2F2xp6Mp?p
4
Inelastic Response Behavior
Locus of modal restoring force during
earthquake (the result of pseudo-dynamic response
test on 2-story 1-bay steel frame)
El Centro NS 1940, 1250 gal
1st story collapse
2nd story collapse
2nd story collapse mechanism occurred
Locus of response
NSP 1st mode pushover
Safety Domain against plastic collapse
Overall collapse
5
Pushover Limit Analysis to Obtain Reference Points
Response of 1st mode component
Modal force of 2nd mode
The most predominant mode pattern only
f r1?1
Response of 2nd mode component
Modal force of 1st mode
SRSS
The most predominant mode pattern plus other mode
Turkstras rule
f r1?1? a12 r2 ?2
Modal force space
6
Yield Surface Model
2nd mode
A lot of unknown failure surface
Reference points on failure surface
Pushover limit analysis
1st mode
7
Simplified Yield Surface Model (Already proposed)
Multi-story steel frames
Simplification 1
Simplification 2
Simplification 3
Yield polyhedron model (Reduced number of plastic
mechanism-based)
Ellipsoidal yield surface model
Parallelogram yield surface model
8
Hysteresis Rule for Proposed Method
Force
Modal force of 2nd mode component
Failure mechanism
Yield polyhedron model
Partial yielding before mechanism formation
Modal force of 1st mode component
Deformation
Yield ellipse model
Assumption Elastic behavior in safety domain
Locus of earthquake response
Member hysteresis based model
Elastic or Failure
Simplified yield surface model
not considering the hardening rule
Not failure mechanism formation or Failure
mechanism formation
9
Application of Hardening Rule to Failure Surface
without hardening
kinematic hardening
isotropic hardening
2nd mode
2nd mode
2nd mode
Yield surface of frame
move
move
1st mode
1st mode
1st mode
not move
move
move
Force
Force
Force
Deformation
Deformation
Deformation
10
Renewal of Force Considering the Hardening Rule
Modal force of j-th mode component
Renewal yield surface after hardening F(r)
F0F1 , or F(r - a) F0
Renewal vector rk
rnew rk-1 K?q
Normal vector ?F / ?r
Increment vector ?r
Cross vector rcrs
?a
Translation vector ?a
At previous step rk-1
Initial center of yield surface a
Modal force of i-th mode component
Initial yield surface F(r)F0
11
Renewal of Force Considering the Hardening Rule
1 modal displacement increment
?qik -gt k1 (1 - h i ?i ?t) ?qik-1 -gt k ?t2
(r i / Mi ÿ0) / (1 h i ?i ?t)
2 modal force increment
If f(rnew) lt F0 then ELASTIC
?r K?q
rnew rk-1 ?r
Else if f(rnew) gt F0 then PLASTIC
3 renewal modal force
rk rcrs ?r
where, ?r I K ? ?F/?r T
?F/?r / (?F/?rT ? ?F/?r)-1 rk-1
K ?q rcrs
4 renewal yield surface
?ar ? ?qp FT CQ f-1 diag(?i Ki)
Cd x F ?qp
12
Pseudo-dynamic response test
Weak structural model constructed on actual
ground at Chiba Experiment Station
Weight of each floor 12,700kg Steel grade JIS
SS400 Base shear coefficient 0.20
Hysteresis damper made of LYP
13
Observation Project and Input EQ
141 E
140 E
1985 10/4 IBARAGI-CHIBA PREFECTURE BORDER EQ.
Normalized response acceleration spectra
1996 9/11 OFF BOSO EQ.
35 N
CHIBA EXPERIMENT STATION
TOKYO
34 N
1987 12/17 OFF EAST CHIBA PREFECTURE EQ.
1986 6/24 OFF BOSO PENINSULA EQ.
PACIFIC OCEAN
33 N
14
Pseudo-dynamic response test
Online computer
Numerical simulation
Concrete slab (rigid)
actuator
Loading test part
Rigid part
Q1
Hysteresis damper
Earthquake resistant stub
Stub
d1
Researchers stay
Measurement
Loading test (1st story)
15
Properties of Frame
Lateral stiffness of each members
Properties of pseudo-dynamic test
16
Yield Surface Model of Weak Structural Model
3rd story local collapse 0.484 r1 1.009 r2
0.715 r3 Qy3
1st story local collapse r1 r2 r3 Qy1
r2
Reference point (Qy 1/(1aij), aij Qy 1/(1aij),
0)
(Qy 1/(1aij), aij, 0, 0)
1st mode 2nd mode
1st mode
r1
1st mode - 2nd mode
Safety domain
Pushover limit analysis
(Qy 2/(0.870.16aij) , -aij Qy 2/(0.870.16aij),
0)
Yield ellipse model 3.851 r12 23.661 r1 r2
5.640 r22 104
2nd story local collapse 0.872 r1 0.162 r2
1.072 r3 Qy2
1st and 2nd modal force space
17
Comparison of Hysteresis Loop of 1st Story
Pseudo-dynamic test
Analysis with yield polyhedron
1)
2)
Analysis with yield ellipse

• Pseudo-dynamic response test
• Full V-mode and plastic yield surface considering
  linematic hardening rule
• Partial V-mode and yield ellipse considering
  kinematic hardening rule

3)
18
Comparison of Loci of Modal Restoring Force
Pseudo-dynamic test
Analysis with yield polyhedron
Initial yield surface
1)
2)
Analysis with yield ellipse
Initial yield ellipse

• Pseudo-dynamic response test
• Full V-mode and plastic yield surface considering
  linematic hardening rule
• Partial V-mode and yield ellipse considering
  kinematic hardening rule

3)
19
Comparison of Response History of Story Drift
1st story
2nd story
3rd story
20
Conclusions

• A simplified method of non-linear dynamic
  response analysis on a ductile steel framed
  structure to a seismic action is proposed
• The method is based on the two kinds of models
  for a safety domain against plastic collapse or a
  global yield surface of a frame
• A yield polyhedron model, and A
  hyper-ellipsoidal model
• Restoring force characteristics are represented
  by the yield surface models, instead of a set of
  member hysteresis based models usually adopted
• The hardening rule is applied to the yield
  surface model of frames, and also the
  formularization is represented.
• The validity of proposed method is checked by
  comparison with a pseudo-dynamic response test
• they are commonly found to provide consistently
  good predictions