Improved Modal Pushover Analysis for Multi-span Continuous Bridge Structures

The Eighteenth KKCNN Symposium on Civil Engineering-KAIST28 December 18-20, 2005, Taiwan

Improved Modal Pushover Analysis for Multi-span Continuous Bridge Structures

Hyo-Gyoung Kwak¹ and * Seong Jin Hong²

1, 2Department of Civil and Environmental Engineering, KAIST, Daejeon 305-701, Korea khg@kaist.ac.kr, ccalong97@naver.com

ABSTRACT

A simple but effective analysis procedure to estimate seismic capacities of multi-span continuous bridge structures is proposed on the basis of nonlinear modal pushover analysis considering all the dynamic modes of structure. Unlike previous studies, the proposed method eliminates the coupling effects induced from the direct application of modal decomposition by introducing an identical stiffness ratio and an approximate elastic deformed shape. Finally, in order to establish the validity and applicability of the proposed method, correlation studies between new simplified analysis and nonlinear time history analysis are conducted for a three span continuous bridge.

INTRODUCTION

Under a near-field strong ground motion, a structure exhibits nonlinear behavior and large plastic deformations are resulted. Especially, large deformations in the primary members of a structure may lead to the total collapse of a structure. Therefore, an exact evaluation of seismic deformation has been regarded as of great importance recently. To estimate the inelastic behavior of a structure, the ATC-40 and FEMA-274 documents are introducing simplified nonlinear analysis procedures based on the capacity spectrum method. To apply one of these methods, a capacity curve of a structure, converted to the A-D format spectrum from the force versus displacement curve, needs to be established. Since the nonlinear dynamic response of a structure has been evaluated without any nonlinear time-history analysis by matching the two curves of the capacity diagram and inelastic demand diagram developed by Chopra and Goel, the objective of this paper is concentrated on the introduction of an improved method to construct the capacity diagram of multi-span continuous bridges.

MODAL PUSHOVER ANALYSIS

For the simplified nonlinear dynamic analysis of building structures, modal pushover

1 Professor

2 Graduate Student

analysis was introduced by Chopra and Goel. In this method, the seismic capacity of a structure is determined by a pushover analysis using the inertia force assumed to be distributed in proportional to each mode.

The dynamic equation of a MDOF structure subject to ground motion ug(t) can be represented by Eq. (1), and Eq. (2) can also be derived by modal decomposition. To solve this equation, F_sn/L_n- D_n relation must be predefined.

..

[ ]{ } [ ]{ } {m u^.. + c u^. + f (u, u)_s ^. }= −[ ]{ }m 1 u t_g( ) (1)

.. . ..

2 ^sn ( )

n n n n g

n

D D F u t

ζ ω L

+ + = − where

. .

( , ) { } { (^T , )}

sn n n n s n n

F D D = φ f D D (2)

If pushover analysis is performed with n^th-mode inertia force distribution {sn*}=[m]{φn}, it is assumed that the system exhibits the corresponding n^th-mode response {un}={φn}qn,

where {φn} is the n^th-mode shape of a system. In advance, the Vbn-urn pushover curve obtained from the nonlinear static analysis of a structure subject to lateraly distributed forces {s_n*}, can be converted to the Fsn/Ln-Dn relation according to the following relations:

*

bn bn sn

n

n n n n

V V F

A =M = L = L

Γ and _n ^rn

n rn

D u

= φ

Γ (3)

,where A_n=F_sn/L_n and D_n mean the normalized base shear forces and roof displacements.

Solutions of the dynamic equation (2) make it possible to calculate the roof displacement of the structure as follows:

( )

2

0 0 0 0

1

{ } { }

N

r rn n n n n

n

U u D

=

=

∑

^u = Γ ^{φ or}

^{{ } { } { }}

1 1

N N

n n

n n

φ D

= =

=

∑

ⁿ =

∑

Γ ⁿ

U u (4)

In spite of its simplicity, modal pushover analysis still has several defects in the direct application to bridge structures. Firstly, this method ignores the coupling effects of each mode caused by inelastic structural responses. Secondly, all the capacity curves show difference at every pier as well as at every mode, and it may result in an increase of analysis steps. Finally, if an unsymmetrical bridge structure is pushed over with the distributed load {s_n*}, some piers may show undesirable capacity curves, which cannot be defined. Therefore, it might be impossible to estimate seismic capacity of unsymmetrical bridge structures by using classical modal pushover analysis.

ADVANCED PUSHOVER ANALYSIS

The application of modal decomposition to inelastic structural behavior is valid only if the elastic mode shape {φn} of a structure maintains without any change even after yielding of structure. Hence, an advanced pushover analysis method, which can consider higher modes as well as a fundamental mode of bridge structures while maintaining this precondition, is proposed.

Identical stiffness ratio for each mode

Mass matrix, stiffness matrix, natural frequency and elastic mode shape of an elastic system are defined as [m], [k0], ωo and {φo}, respectively. And [k(t)] and ω are the stiffness matrix and natural frequency of an inelastic system.

(

[ ( )]k t −ω²[ ]m

){ }

φ =α( ) [t

(

k⁰]−ω0²[ ]m

){ }

φ =0 ^∴

{ } { }

^{φ = φ0} (5) Then, only in the case of [k(t)]=α(t)[k0], the mode shape of an inelastic system {φ} is maintained as {φo}. On condition that the elastic mode shape {φo} is unchanged, coupling effects do not occur since the nondiagonal terms would be eliminated.

Approximate elastic deformed shape

To maintain the elastic mode shape in inelastic states as well as in elastic states, the result of pushover analysis should be corrected. The final corrected approximate elastic deformed shape, ua,r, can be represented as follows:

, ,

a r e r

u =βu where , ²

1

; ( )

m

r e r

r

Min u u

β β

=

⎧ ⎛ − ⎞⎫

⎨ ⎜ ⎟⎬

⎝ ⎠

⎩

∑

⎭ (6) Where ur is the final displacement of pier r, which can be obtained by pushover analysis, ue,r is the critical elastic deformation of pier r and m is the number of piers. Keeping the elastic mode shape unchanged through this correction, stiffness ratio, α, becomes identical not only for each mode but also for ever pier of a bridge structure.

Distributed load for pushover analysis

Advanced pushover analysis aims for getting satisfactory results through only one pushover analysis which corresponds to those of the many static analyses in accordance with every mode. Therefore, for only one pushover analysis procedure, a new distributed load {P}

is proposed as Eq. (7). Herein, Ai0 and γ represents the i^th mode pseudo acceleration of an elastic structure and the load factor, respectively.

(

0

)

1

{ } [ ]{ }

N

i i i

i

γ A φ

=

=

∑

Γ

P m (7)

Advanced pushover analysis procedure

Both pushover and capacity curves are idealized as bilinear stiffness degrading model. To solve Eq. (2), Eq. (7) is substituted into F_sn/L_n and the arranged result of F_sn/L_n is shown in Eq.

(8). And D_ny can be described as Eq. (9) with the initial yielding load factor, γy, since elastic stiffness is ωn2

in F_sn/L_n-D_n relation.

0

0

{ } { }^T { }^T [ ]{ }

sn n n n n n

n

n n n

F A

L L L A

γ ^Γ γ

= φ P = φ m φ =

(8)

0 2

n y

ny n

D A γ

= ω (9) As shown in Fig. 1, on the basis that γ0 and α are the final load factor and the stiffness ratio, the final n^th mode displacement of SDOF system, D_n0, can be represented as Eq. (10).

Then, the final corrected approximate elastic deformed shape, ua,r0 can be expressed as Eq.

(11). Consequently, stiffness ratio α can be induced as Eq. (12) by arranging Eq. (11).

0 0 0

0 2 2

( )

n y n y

n

n n

A A

D γ γ γ

ω αω

= + − (10)

0 0 0 0 0

, 0 2 2 2

1 1

( ) ( )

N N

n y n y n y rn n

a r rn n ry

n n n

n n

A A A

u φ γ γ γ u γ γ φ

ω αω α ω

= =

⎧ ⎛ − ⎞⎫ − ⎛ Γ ⎞

=

∑

⎨⎩ Γ ⎜⎝ + ⎟⎠⎬⎭= +

∑

⎜⎝ ⎟⎠ (11)

0 , 0

y ry

a r ry y

u

u u

γ γ

α γ

= − ×

− ⁰₂

1 N

n y

ry rn n

n n

u A γ

φ ω

=

⎛ ⎛ ⎞ ⎞

= Γ

⎜ ⎜ ⎟ ⎟

⎜ ⎝ ⎠ ⎟

⎝^Q

∑

⎠ (12)

ury u_r0 u_a,r0 P_ry

P_r0 P_r

pushover correction

ury u_r0 u_a,r0 P_ry

P_r0 P_r

pushover correction

A_n0γ₀ F_sn/L_n

2 2

0 y 0( 0 y)

n n

n n

A γ A γ γ

ω αω

⎛ + − ⎞

⎜ ⎟

⎝ ⎠

2 0 y

n

An γ ω

⎛ ⎞

⎜ ⎟

⎝ ⎠

ω_n²

A_n0γ_y αω_n²

D_n A_n0γ₀

F_sn/L_n

2 2

0 y 0( 0 y)

n n

n n

A γ A γ γ

ω αω

⎛ + − ⎞

⎜ ⎟

⎝ ⎠

2 0 y

n

An γ ω

⎛ ⎞

⎜ ⎟

⎝ ⎠

ω_n²

A_n0γ_y αω_n²

D_n

(a) Pushover curve subjected to {P} (b) A-D format capacity curve Fig. 1 transformation of A-D format capacity curve from pushover curve

Through the procedure explained above, the desired Fsn/Ln-Dn relation can be acquired. As a result, the response of SDOF system can be calculated by using Eq. (2) and the F_sn/L_n-D_n relation. Therefore, it is possible to estimate the seismic deformations of a bridge structure, without performing rigorous nonlinear time history analysis.

NUMERICAL APPLICATION

Through the proposed advanced pushover analysis method, the seismic responses of a bridge model B22 are estimated under 7 artificial ground motions of PGA 1.0G:

California(1933 S07E), El Centro(1940 S00E), Mexico City(1985 SCTS00E), Taft(1952 EW), Northridge(1994 CHAN3), San Fernando(1971 N76W) and San Francisco(1957 S09E). The only transverse displacements are considered since failure behaviors are mainly caused by the inelastic deformation in the transverse direction. The results obtained by the new method are verified by comparing them to the real behaviors of the bridge, which is simulated by nonlinear time history analysis.

14

6.5 3

0.3 3

2.5 0.3 0.25

Unit :m 14

6.5 3

0.3 3

2.5 0.3 0.25

Unit :m

Unit :m

10 5

5 Pier 1 Pier 2

Pier 3

4@50=200 Unit :m

10 5

5 Pier 1 Pier 2

Pier 3 4@50=200

^{0.8 2.9 0.8}

0.8 1.4 0.8

0.8 2.9 0.8

0.8 1.4 0.8

(a) Geometrical shape properties (b) Cross section properties Fig. 2 Target bridge model B22

Target bridge model

The detailed properties of the target bridge model are shown in Fig. 2. Nonlinear plastic

hinge element is adopted for the lower part of each pier; EI=39805.6 MN-m, κy=0.0036 and α=0.0001. Except the plastic hinges, other elements are considered as retaining elastic characteristics. Damping ratio of the bridge model is defined as 5%, based on the rayleigh damping theory. Pushover analysis and nonlinear time history analysis are conducted by using OpenSees 1.6.2.(The PEER Center, 2005)

Fsn/Ln-Dn relation

Pushover analysis of the bridge B22 subjected to {P} is performed and the results are idealized as bilinear stiffness degrading model. As shown in Fig. 3, the pushover response of pier 1 was converted to A-D format capacity spectrum of each mode. All the capacity curves can be easily obtained by only one pushover procedure with the distributed load {P}.

Although Fig. 3 presents the only curves from pier 1, the exactly same capacity curves as Fig.

3(b) are able to be plotted by converting pushover responses of other piers.

0.0 0.1 0.2 0.3 0.4

0 10 20 30 40 50 60

After correction Fpier 1(MN)

U_{pier 1}(m)

Before correction

0.0 0.2 0.4 0.6

0 2 4 6 8 10

Dn(m) Fsn/Ln(m/s2 )

1st mode 2nd mode 3rd mode

α=0.407

(a) Pier 1 pushover curve subjected to {P} (b) A-D capacity curve of each mode Fig. 3 Transformation of B22 A-D format capacity curves

Time history analysis

Using the Fsn/Ln-Dn relation acquired in Fig. 3, time history analysis of inelastic SDOF dynamic equation, Eq. (2), was conducted under the 7 artificial ground motions. From Dn(t) and Eq. (4), time history displacements of piers are estimated through advanced pushover analysis (NEW) and OpenSees (NHA), respectively. The displacements of piers under artificial Northridge ground motion are plotted in Fig. 4.

0 5 10 15 20 25 30

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

Pier 1 displacement (m)

Tim e (sec)

NHA NEW

(a) Displacement of pier 1

0 5 10 15 20 25 30

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

Tim e (sec)

Pier 2 displacement (m)

NHA NEW

0 5 10 15 20 25 30

-0.4 -0.2 0.0 0.2 0.4

NHA NEW

Pier 3 displacement (m)

Tim e (sec)

(b) Displacement of pier 2 (c) Displacement of pier 3 Fig. 4 Time history displacement of B22 (Northridge Art. EQ)

Error rates (mean root square error) of the evaluated maximum displacement at each pier under the 7 ground motions are displayed in Fig. 5. The error rate is calculated as Eq. (13), where N is the number of the ground motions. The largest error is caused at pier 1 of which length is the shortest one and the estimations of the maximum displacements at the longest pier 3 result in the smallest error rate. The overall error rates show tendency not to exceed by 6%. Namely, although the analytical model is an unsymmetrical bridge structure, advanced pushover analysis provides reasonable prediction about the behavior of the structure.

2

n n

1

( ) ( )

1

( )

N

n n

NEW NHA

MRSE N = NHA

⎧ − ⎫

= ⎨ ⎬

⎩ ⎭

∑

(13)

Pier 1 Pier 2 Pier 3

0 2 4 6 8 10

Elem ent

Error rate at each B22 pier (%)

Fig. 5 Error rate of maximum displacement predicted by advanced pushover analysis CONCLUSIONS

This study proposed a new simplified analysis procedure for seismic performance evaluation of bridge structures, which can take into account all the dynamic modes. Through an introduction of the identical stiffness ratio α, and correction to approximate elastic deformed shape ua,r, the coupling effects are eliminated. In advance, the use of an appropriate distributed load {P} makes it possible to predict the dynamic responses for all kinds of bridge structures with a simple static analysis procedure. In addition to all these advantages, the advanced pushover analysis method introduced in this paper can more effectively predict seismic deformations of bridge structures.

REFERENCE

Applied Technology Council, “Seismic evaluation and retrofit of concrete buildings”, Report ATC 40, November, 1996

Federal Emergency Management Agency, “NEHRP guidelines for the seismic rehabilitation of buildings”, FEMA 273; and “NHERP commentary on the guidelines for the seismic rehabilitation of buildings, FEMA 274. October, 1997

A. K. Chopra, R. K. Goel, “Capacity-demand-diagram methods for estimating seismic deformation of inelastic structures: SDF systems”, Report No. PEER-1999/02, Pacific Earthquake Research Center, University of California, Berkeley, 1999

Krawinkler H., Seneviratna G.D.P.K., “Pros and cons of a pushover analysis of seismic performance evaluation”, Engineering Structures, Vol.20, 452~464, 1998

A. K. Chopra, R. K. Goel, “A modal pushover analysis procedure for estimating seismic demands for buildings”, Earthquake engineering and structural dynamics, Vol.31, 561~582, 2002.