Utilization of Virtual Work Method to Evaluate Progressive ... · d Utilization of Virtual Work Method to Evaluate Progressive Collapse Analysis of Steel Buildings Bulent N. Alemdar,

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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


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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


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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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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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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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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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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

Documents

Utilization of Virtual Work Method to Evaluate Progressive ... · d Utilization of Virtual Work Method to Evaluate Progressive Collapse Analysis of Steel Buildings Bulent N. Alemdar,