© 2
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entley S
yste
ms,
Incorp
ora
ted Utilization of Virtual Work Method to Evaluate
Progressive Collapse Analysis of Steel BuildingsBulent N. Alemdar, Ph.D., P.E.
Rakesh Pathak, Ph.D.Infrastructure Systems Conference, Atlanta, GA, June 13-17, 2011
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• The present work is an attempt to identify individual component responses to progressive collapse due to a column removal – Structural Components: beams, columns, connections, panel
zones (2D)
– Nonlinear Behavior: • Material Nonlinearity: connections, panel zones
• Geometric Nonlinearity: P- effects for columns
• It further helps to find which component is next in line in a progressive collapse right after the removal of a column
Goal
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• To achieve the goal
– Well known virtual work method is adopted to computecomponent displacement participation to overall response ofthe structure upon removal of a column
Objective
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• Two step analysis– Step 1:
• The system is solved under gravity loads only
• Column that triggers a collapse is replaced with equivalent externalforces
• The strain energy due to the deformation resulted from the gravityloads is calculated for all members
Proposed Method:
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• Two step analysis– Step 2:
• The strain energy stored in the previous step is preserved
• An axial load is applied incrementally at the point of the columnremoval
• Incremental internal work (i.e. strain energy) is calculated at eachincrement in an nonlinear analysis framework. At each increment, wealso calculate
– Displacement participations for each component, which indicatecontribution of each element to the downward vertical displacement atthe point of column removal.
Proposed Method:
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Step 1: Elastic\Linear System
DL
DL
DL
Pc
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Step 1: Elastic\Linear System
Pc
F
dStrain Energy Stored in Step 1
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Step 2: Inelastic\Nonlinear System
2Pc
F
dStrain Energy Stored in Step 2
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• It should be noted that further contribution of gravity loads from previous step to strain energy stored in this step is not included
• In other words, this step is for strain energy stored at the very instance of the column removed
Step 2: Inelastic\Nonlinear System
F
dStrain Energy Stored in Step 2
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Modeling Details: Joints
Typical beam-column joint
Beam
Column
Rigid Links
Connection Spring
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Modeling Details: Joints-Panel Zone
up
vp
qbp
qcp
qct
qcb
ubRubL
uct
vct
vcb
ucb
vbRvbL
qbRqbL
Pane Zone Modeling: Scissor Joint Model
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• Definition
“work done by either virtual forces acting through real displacement or by real forces acting through virtual displacement”
• Applicable for rigid and deformable systems
• Principle of virtual forces has often been used in the following areas:
– computing displacements
– optimizing frame member sizes based on the computed sensitivity indices of each element
Virtual Work
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• However, so far this approach is only utilized for linear elastic systems
• In this study, it is applied to nonlinear systems
Literature Review
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• It is defined as the displacement contribution due to a component of member flexibility at any given point and at a given direction in the structure.
Displacement Participation: T
n
j
jT
1
sConnection
MinorShear,MajorShear,
TorsionAxial
MinorFlexure,MajorFlexure,
/ColumnBeam
Shear ZonePanelPanelZone
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Bending Moment Diagram Shear Force Diagram
RH R
R
H
M
θ
R
H
My
θy
Column Properties: E, G, I, As
Rotational Spring Property
ΔT
Kr
αKr
EXAMPLE: CANTILEVER COLUMN
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0
10
20
30
40
50
60
70
80
0.00 1.00 2.00 3.00
R (
kip
s)
Displacement (in)
Column Flexure
Column Shear
Spring Flexure
Total
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Numerical Example 1: 2D Model
P = 65.3 kips
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Numerical Example 1: 2D Model
P
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Connection and Panel Zone Models:
y
y
K1
K2
K1=770000 k-inK2= 82800 k-in
y=0.01 rad.
• Panel Zone Behavior
y
y
K1
K2
K1, K2 and y are calculated based on beam and column sizes
• Connection Behavior
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Component Displacement Participations for Vertical Deflection Calculated at the Location of Column Removal
0
10
20
30
40
50
60
70
80
90
100
2P 5P 8P 10P 15P
Percen
tag
e %
Load (kip)
PANEL ZONE
CONNECTION
SHEAR
AXIAL
FRAME
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Numerical Example 1: 2D Model
0
100
200
300
400
500
600
700
800
900
1000
0 20 40 60 80
Lo
ad
(kip
s)
Displacement (in)
FINITE ELEMENT
1
2
3
4
5
6
7
8
9
10
11
12
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Numerical Example 2: 3D Model
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Numerical Example 2: 3D Model
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0
10
20
30
40
50
60
70
80
90
100
1P 2P 5P 8P 10P 15P
Percen
tag
e %
Load (kip)
TORSION
CONNECTION
SHEAR
AXIAL
FRAME
Component Displacement Participations for Vertical Deflection Calculated at the Location of Column Removal
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Numerical Example 2: 3D Model
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 20 40 60 80 100 120 140 160
FINITE ELEMENT VIRTUAL WORK
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• A new method is proposed to identify component contribution to overall response of the building upon column removal.
• This is carried out within a two step nonlinear analysis framework
– Discrete nonlinear behavior is monitored for:• Connections
• Panel Zones
– P-Δ included for columns
Conclusions
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• The proposed method highlights– Individual member response within an overall building response
– Different sources of member responses identified (flexural, shear, axial, torsional, connection and panel zone)
• It is applicable for 2D and 3D structures
Conclusions
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• Column PMM hinges
• Integration of Panel Zone integration within a 3D analysis
• Member Identification for Progressive Collapse– Correlation between analysis results in this study and D/C
ratio of members
• Develop a framework to handle huge amount of data generated during analysis
• The method can be used to identify next component which is next in row to yield under progressive collapse
Future Work