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Application of Pushover Analysis to

the Design of Structures ContainingDissipative Elements

Martin S. Williams1 and Denis E. Clment2

1 University of Oxford, UK2 Thomas Jundt Civil Engineers, Geneva, Switzerland

13th World Conference on Earthquake Engineering

Vancouver, August 2004

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Outline

Introduction to knee braced frames

Modelling using Drain-2DX

Five and ten-storey frame designs

Pushover and time-history analyses

Results

Conclusions and future work

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Introduction to knee braced frames (KBFs)

Knee elements can be designed to:

Yield early, maximizing protection to main frame

Yield in web shear rather than flexure

Remain stable under large non-linear excursions

Cross brace

Knee element

Seismic energydissipated through

hysteresis of

short, replaceable

knee elements:

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Modelling a knee element using Drain-2DX

An assemblage of standard truss and beam elements was used torepresent observed shear, flexural and axial behaviour:

Kneeelement

Brace

Connecting

bracket

Brace

Rigid offset

Connection

stiffness

Beam element -

flexural stiffness

Truss element -

axial stiffness

Short cantilever -

shear stiffness

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Hysteresis response of model

Element properties chosen semi-empirically

Comparison with full-scale cyclic test data:

-600

-300

0

300

600

-20 -10 0 10 20

Deflection (mm)

Force

(kN)

Physical testDrain-2DX

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Frame designs

6m6m 6m

6m

6m

6m

4m

4m

4m

4m

4m

4m

4m

4m

4m

4m

Five-storey frame

designed as KBF:Ten-storey frame designed as

ductile MRF, then retrofitted:

5.2m5.2m 5.2m

5.2m

5.2m

Knee braced bay

3m

3m

3m

3m

3m

PLAN:

ELEVATION:

PLAN: ELEVATION:

Designed to EC8, PGA = 0.35g

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Pushover analysis

EC 8:

modal and uniform load patterns

simplify pushover curve to elastic-perfectlyplastic

FEMA 356: other load patterns (e.g. adaptive) permitted,

but not used here

simplify to bi-linear with post-yield stiffnessequal to initial stiffness

ATC 40: capacity spectrum method

Modal pushover(Chopra and Goel, 2002):combine results of pushovers using first few modalload patterns

F

d

F

d

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Time history analyses

30 time-histories generated using SIMQKE

Compatible with EC8 Type 1 spectrum, soil type C

Analysed using DRAIN-2DX (Newmark implicit integration

scheme)

Typical EC8 spectrum-compatible time history

-1

-0.5

0

0.5

1

0 5 10 15 20

Time (s)

Acc

n

(g)

Comparison with EC8 spectrum, TH1-5

0

1

2

3

0 1 2 3

Period (s)

Acceleration

(g)

EC8

TH1

TH2

TH3

TH4

TH5

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Pushover curves

Results shown for 5-storey frame

Post-yield stiffness ~16% of elastic stiffness

As a result, EC8 under-estimates initial stiffness

Modal load pattern

0

500

1000

1500

0 50 100 150Displacement [mm]

Force[kN]

Pushover

EC8

FEMA356

Uniform load

pattern

0

500

1000

1500

0 50 100 150Displacement [mm]

Force[kN]

Pushover

EC8

FEMA356

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Estimated roof displacements

5-storey KBF

10-storey MRF

10-storey KBF

0.8 1 1.2 1.4 1.6 1.8

EC8 Modal

EC8 Uniform

FEMA ModalFEMA Uniform

ATC Modal

ATC Uniform

Multi-modal

Roof disp from pushover

Mean roof disp from TH analyses

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Element yielding

In 5-storey frame, all knee elements yielded and all main elementsremained elastic under design earthquake

In 10-storey retrofitted frame, limited plasticity occurred in main

frame under design earthquake

e.g. 5-storey frame - EC8 pushover analysis undermodal loading:

0.15g

Yielded knee element Plastic hinge

0.35g 0.5g

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Element yielding

5-storey frame

EC8 pushover analysis underuniform loading:

Time history analyses:

first knee element yield at around 0.08g

no hinges in main frame elements below 0.56g

0.15g

Yielded knee element Plastic hinge

0.35g 0.5g

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Inter-storey drifts under design earthquake

5-storey KBF

EC 8

0

1

2

3

4

5

0 0.5 1

Storey drift (%)

Storey

FEMA 356

0

1

2

3

4

5

0 0.5 1

Storey drift (%)

Storey

ATC 40

0

1

2

3

4

5

0 0.5 1

Storey drift (%)

Storey

Mean Mean +/- st. dev.

Uniform Modal

Time history analysis:

Pushover analysis:

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Inter-storey drifts under design earthquake

10-storey MRF (i.e. before retrofit):

EC8

0

1

2

3

4

5

6

7

8

9

10

0 1 2

Drift (%)

Storey

FEMA 356

0

1

2

3

4

5

6

7

8

9

10

0 1 2

Drift (%)

Storey

ATC 40

0

1

2

3

4

5

6

7

8

9

10

0 1 2

Drift (%)

Storey

Modal Pushover

0

1

2

3

4

5

6

7

8

9

10

0 1 2

Drift (%)

Storey

Mean Mean +/- st. dev.

Uniform Modal

Time history analysis:

Pushover analysis:

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Inter-storey drifts under design earthquake

10-storey KBF (i.e. after retrofit with knee elements):

Mean Mean +/- st. dev.

Uniform Modal

Time history analysis:

Pushover analysis:

EC8

0

1

2

3

4

5

6

7

8

9

10

0 1 2

Drift (%)

Storey

FEMA 356

0

1

2

3

4

5

6

7

8

9

10

0 1 2

Drift (%)

Storey

ATC 40

0

1

2

3

4

5

6

7

8

9

10

0 1 2

Drift (%)

Storey

Modal Pushover

0

1

2

3

4

5

6

7

8

9

10

0 1 2

Drift (%)

Storey

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Conclusions

A Drain-2DX knee element model capable of representing shear,

flexural and axial behaviour has been developed and validated.

Pushover analyses of 5 and10-storey knee braced frames

showed that they possess high ductility (~6) and post-yield

stiffness (~16%). In time-history analyses, knee elements began to yield at just

0.08g but remained stable up to 0.56g.

Use of pushover analysis does not necessarily lead to optimal

design. Multi-modal pushover offers some advantages in this

respect.

In comparison with time-history analyses, FEMA 356 pushover

approach gave most consistent results, EC8 approach appears

highly conservative for this type of structure.