Comparison of Simplified Procedures for

Nonlinear Seismic Analysis of Structures

Dan Zamfirescu

To evaluate the suitability for practical application of different procedures based on:

• Accuracy

• Simplicity

• Transparency

• Theoretical background

Objective

Investigated Procedures

• ATC 40

• TriServices’ Manual 1996 (Freeman)

• FEMA 273 and 356

• BSL 2000

• N2 (EC8 – draft 2001)

• Yield Point Spectrum

• Chopra-Goel (Modal Pushover Analysis)

• Priestley

• Fardis - Panagiotakos (Elastic)

Investigated Procedures

Common Features

Common steps (exceptions Priestley and Fardis):

• Pushover analysis of the MDOF to the target displacement

• Bilinear approximation of the pushover curve (exception BSL)

• MDOF to SDOF transformation

• The use of the response spectrum for the assesment of target displacement

Common restriction

• Planar model (partial exceptions FEMA, N2)

Differences

• Response spectrum

– Inelastic (Chopra, FEMA, N2, Yield Spectra)

– Equivalent (overdamped) elastic using “substitute structure” (ATC 40, TriServices, BSL, Priestley)

• Elastic stiffness of structural components

• Distribution of lateral forces for pushover

• Displacement shape along the height

• MDOF to SDOF transformation

• Idealization of pushover curve (bilinear)

• Iterative procedure for the target displacement and for the bilinear idealization needed

• Graphic presentation

FEMA 273 & 356

• Pushover: two lateral load distributions

• Special rule for bilinear idealization (iteration needed for FEMA 356)

• Inconsistent MDOF to SDOF transformation

• Target displacement from nonlinear spectra:– T > TC – equal displacement rule with possible

correction

– T < TC – elastic displacement amplified

• No graphic representation

Chopra-Goel (Modal)

• Modal Pushover Analysis - the effect of higher modes is taken into account by combining several individual peak modal responses obtained from a pushover procedure.

• The pushover procedure is iterative due to the proposed bilinear idealization of the pushover curve.

• Advocates the use of computed, or simplified (Newmark-Hall) inelastic spectra.

ATC 40 & TriServices

• Known as “capacity spectrum method”

• Pushover: lateral load distribution according to the first elastic mode (basic variant)

• Target displacement (performance point) from equivalent elastic spectrum by a graphical iterative procedure

• Equivalent damping determined from

– dissipated energy based on idealized hysteretic loops (ATC 40)

– Newmark-Hall reduction factors for inelastic spectra (TriServices’ Manual)

• Iteration is needed

N2 - BSL 2000Comparison

Load Pattern

• According to assumed displacement shape

• 2 different shapes

MDOF SDOF• According to assumed

displacement shape

N2 BSL 2000

Load Pattern

• Consistent with the previous BSL

MDOF SDOF• According to first mode

shape

N2 - BSL 2000Comparison

Response Spectra

• Inelastic

For T < Tc

–

For T > Tc

– Sd = Sde – equal displacement rule

N2 BSL 2000

Response Spectra• Equivalent elastic

Simplified approach –m-displacement ductility of eq. SDOF

))1(1(*T

TR

R

SS cde

d m

m

im

imeqim

eq

eqim

W

Whh

h )1

1(4

1

m1)

2)

BSL 2000 – equivalent damping

Reduction Factor

0

0.2

0.4

0.6

0.8

1

1.2

0.00 0.05 0.10 0.15

Top Displacement (m)

Red

ucti

on

Facto

r

1) Fh 2) Fh SDOF

Reduction Factor

0

0.2

0.4

0.6

0.8

1

1.2

0.00 0.05 0.10 0.15 0.20 0.25

Top Displacement (m)

Red

ucti

on

Facto

r

1) Fh 2) Fh SDOF

Frame Structure Wall Structure

Inelastic vs. Equivalent Elastic Spectrum

Sd

Sa

T*

Sae

m

Sd = Sde

Say

Sad

Dd*

Dy*

m1 (elastic)

S a

S d

= 0 ,5 %

= 3 0 %

sp e c t r e d e r a sp u n sb i p a r t i t e

42 2

sp e c t r u l c a p a c i t a t i i

F u

F

F n

y

K1

K e

R K i

INELASTIC EQUIVALENT ELASTIC

Simple Version of N2 in A-D Format

Damped spectra

Capacity diagram

N2 - BSL 2000Comparison

Response Spectra

• Initial elastic period (bilinear idealization)

• Solution – bilinear idealization function of the target displacement (FEMA 356). Result –iterative procedure.

N2 BSL 2000

Response Spectra

1) - Complicated for large structures. Advantage – no bilinear idealization.

2) - The yielding displace-ment has to be specified

Iteration is needed if perfor-mance has to be assesed.

Is heq easier to determine than inelastic spectra?

Priestley

MAIN STEPS:

• The plastic mechanism of the structure is assumed, based on simple calculations

• The global displacement and ductility capacity of the structure is determined, based on simplified formulas for yield and ultimate element rotations, assumed (PREDETERMINED) displacement shape, and drift limit values

• Substitute structure characteristics are based on ductility capacity

• The displacement demand is established using the substitute structure (equivalent elastic) method

• Compare demand and capacity (End of original procedure -1997)

• Iteration is needed for determination of actual demand! (For direct displacement-based design (new buildings), no iteration is required)

Fardis-Panagiotakos

ELASTIC ANALYSIS

similar to FEMA LSP and NZ code

• Determine secant stiffness (at yielding) of components using empirical relations for chord rotations

• Estimate the peak inelastic chord rotations from linear analysis using the equal displacement rule

Restrictions:

• The method is appropriate for structures having moment diagram in the inelastic range similar to the elastic one

• Restricted for structures having T > Tc

EXAMPLES

• Test example– 5 storey RC frame designed according EC8 (PGA =0.2g)

– 5 storey RC wall structure (only wall & wall + frame is considered to resist lateral forces)

• Seismic demand in terms of:– Top displacement

– Storey drifts

– Rotations and ductilities of components

• Was determined by:– All investigated procedures

– Nonlinear time-history analysis using 11 accelerograms calibrated to EC8 acceleration spectra

• Two ground motions intensities considered:– Spectral acceleration equal to EC8 elastic spectrum

– Spectral acceleration equal to twice EC8 elastic spectrum

Test Example

28 KN/m

21 KN/m

74.5 KN 99.7 KN

102.2 KN 100.6 KN

2.85m

2.85m

2.85m

2.85m

2.85m

5.4m 5.4m

25x55 cm

3f22 4f22

2f22

2f222f22

2f16

12f16

12f18

12f18

12f20

40x40 cm

12f16

Test Example

28 KN/m

21 KN/m

74.5 KN 99.7 KN

102.2 KN 100.6 KN

25x55 cm

2f20+1f16

2f16 8f16

40x40 cm

2f20

6f16

2f8/15cm

530x30 cm

2f20

6f16

390 KN

370 KN

2.85m

2.85m

2.85m

2.85m

2.85m2f20

6f16

2f8/15cm

530x30 cm

2f20

6f16

390 KN

370 KN

5.4m 5.4m

European Accelerograms Recorded on Stiff Soil

Frame Structure

Te= 0.38 s

Te= 0.95 s

Wall Structure

MDOF a SDOF

N2

mod1

N2

uni1

FEMA

mod2

FEMA

uni2

ATC

403

Triser-

vices4

MPA

Mode I5

MPA

Mode II5

BSL6

Fardis7

Priest

ley8

T(s) 0.95

0.97 0.88 0.86 0.87 0.90 0.91 0.19 - 1.59 1.55

cy 0.25 0.25 0.22 0.25 0.20 0.22 0.19 1.26 - - -

Sh 0.00 0.00 0.08 0.08 0.06 0.00 0.00 0.00 - - -

D F 1.26 1.00 1.30 1.20 1.26 1.26 1.26 0.37 1.26 1.00 1.35

Limit State 2

T - - - - 0.90 0.90 0.91 0.32 - - -

cy - - - - 0.21 0.23 0.23 1.26 - - -

Sh - - - - 0.10 0.00 0.00 0.00 - - -

D F - - - - 1.26 1.26 1.26 0.37 - - -

Frame Structure

MDOF a SDOF

Wall Structure

N2

mod1

N2

uni1

FEMA

mod2

FEMA

uni2

ATC

403

Triser-

vices4

MPA

Mode I5

BSL6

Priestley

8

T(s) 0.39

0.43 0.32 0.35 0.38 0.38 0.32 - 0.78

cy 0.26 0.23 0.19 0.25 0.23 0.24 0.20 - -

Sh 0.00 0.00 0.02 0.02 0.03 0.00 0.00 - -

D F 1.34 1.00 1.30 1.20 1.34 1.34 1.34 1.34 1.34

Limit State 2

T - - - - 0.38 0.38 0.32 - -

cy - - - - 0.23 0.25 0.22 - -

Sh - - - - 0.02 0.00 0.00 - -

D F - - - - 1.34 1.34 1.34 - -

Response spectrum

PGA=0.4g

PGA=0.2g

PGA=0.4g

PGA=0.2g

PGA=0.4g

PGA=0.2g

PGA=0.4g

PGA=0.2g

Time- history mean spectrum Time- history mean + StDev spectrum Resulting spectrum EC8 Elastic spectrum

Lateral Load Pattern

Frame

1

2

3

4

5

-0.5 -0.4 -0.3 -0.2 -0.1 0

Story Force / Base Force

Modal Ai BSL Uniform

Wall

1

2

3

4

5

-0.5 -0.4 -0.3 -0.2 -0.1 0

Story Force / Base Force

Modal Ai BSL Uniform

Results Displacements

Top Displacement (Drift) - Frame

0 0.5 1 1.5 2

Mean

M+StDev

Mean Synth

M+Stdev Synth

N2 mod

N2 uni

FEMA mod

FEMA uni

Chopra

ATC 40

TriServices

BSL

Priestley

% H

Sa 2Sa

Maximum Drift - Frame

0 0.5 1 1.5 2

Mean

M+StDev

Mean Synth

M+Stdev Synth

N2 mod

N2 uni

FEMA mod

FEMA uni

Chopra

ATC 40

TriServices

BSL

Priestley

% Hs

Sa 2Sa

Results – Maximum Plastic Rotations

Column Plastic Rotations - Frame

0 0.005 0.01 0.015

Mean

M+StDev

Mean Synth

M+Stdev Synth

N2 mod

N2 uni

FEMA mod

FEMA uni

Chopra

ATC 40

TriServices

BSL

Priestley

Fardis

rad

Sa 2Sa

Beam Plastic Negative Rotations - Frame

0 0.005 0.01 0.015 0.02

Mean

M+StDev

Mean Synth

M+Stdev Synth

N2 mod

N2 uni

FEMA mod

FEMA uni

Chopra

ATC 40

TriServices

BSL

Priestley

Fardis

rad

Sa 2Sa

Results Displacements

Top Displacement (Drift) - Wall

0 0.5 1 1.5 2

Mean

M+StDev

Mean Synth

M+StDev Synth

N2 mod

N2 uni

FEMA mod

FEMA uni

Chopra

ATC 40

TriServices

BSL

Priestley

% H

Sa 2Sa

Maximum Drift - Wall

0 0.5 1 1.5 2

Mean

M+StDev

Mean Synth

M+StDev Synth

N2 mod

N2 uni

FEMA mod

FEMA uni

Chopra

ATC 40

TriServices

BSL

Priestley

% H

Sa 2Sa

Results – Maximum Plastic Rotations

Maximum Plastic Rotation - Wall

0 0.002 0.004 0.006 0.008 0.01 0.012

Mean

M+StDev

Mean Synth

M+StDev Synth

N2 mod

N2 uni

FEMA mod

FEMA uni

Chopra

ATC 40

TriServices

BSL

Priestley

rad

Sa 2Sa Wall Plastic Rotation Reduction

Wall + Frame

0 20 40 60 80 100

Mean

Mean Synth

N2 mod

N2 uni

FEMA mod

FEMA uni

Chopra

ATC 40

TriServices

BSL

diff %

Sa 2Sa

Features

Procedure Analysis Spectrum

Iteration Consistency of

MDOFSDOF

Graphic

Presentation

ATC 40 Pushover Equiv. El. Yes Yes (1st Mode) Yes

TriServices Pushover Equiv. El. Yes Yes (1st Mode) Yes

FEMA Pushover Inelastic Yes1

No No

BSL Pushover Equiv. El. Yes No Yes

N2 Pushover Inelastic No Yes Yes

Yield Spectra Pushover Inelastic No NA Yes

Chopra &

Goel

Pushover

(Several)

Inelastic Yes1

Yes (El. Modal

Shapes)

No/Yes2

Priestley (A) -

Equiv. El. Yes/No3

No No

Fardis &

Panagiotakos (B)

Linear Equal Disp. No Yes (Elastic) No

(A) - Predetermined plastic mechanism and displacement shape. Appropriate for regular structures.

(B) - Fundamental period T > Tc. Predetermined elastic displacement shape (global plastic mechanism). 1 Due to bilinear idealization 2 For each mode

3 For new structures (direct displacement-based design)

• The procedures generally yield results of adequate accuracy (particularly for T>Tc domain)

• Global quantities (top displacements, maximum drifts) more accurate then local ones (plastic rotations)

• Demand Spectrum – the most important parameter

• Final results show low sensitivity to characteristics of SDOF equivalent system

• BSL 2000 – most accurate results

• MPA without further simplifications – significantly more complex

• Main dilemma – inelastic demand spectra or equivalent elastic?

Conclusions