Quan Gu

(古 泉)

分析非线性结构与土体系的一种实用耦合

子结构方法

A new practical coupling substructure

method for seismic analysis of SSI systems

• Outline

Background 1

Applications 3

4 Conclusion

2 A coupling method

地震引起结构破坏

数值方法：适用于复杂结构的非线性问题，土-结构的接触面和边

界问题，但计算量大，耗费时间多，也存在一定的局限性

研究背景

目的意义

基于子结构法提出一种简单、实用的计算方法——数值解与解析解耦合的新方法

背景

目的

用有限元程序OpenSees模拟

复杂结构的非线性行为，解析解分析土的行为，降低运算成本，提高计算效率

意义

所提的SSI耦

合计算方法和部分研究成果为工程设计人员提供参考

解析方法：只适用于简单的线弹性结构、刚性基础和等效线弹性的半无限大地基组成的体系，具有较大的局限性

Analytical solution 1

2 Coupling analytical & numerical methods

Coupling method 2

( )T g B Ru u u H u

0 g Bu u u

Equations of motion for SSI with SDOF structure

0T R Rmu cu ku

0( ) ( ) ( ) 0S B g R g BH t m u u m u u u H

0( ) ( ) ( ) 0S T R g BM t I I Hm u u u H

HHK

HHC

MMKMMC

H

H

m

gu

guBu

Ru

0u

[1]( ) ( )( ) ( )

( ) ( )( ) ( )

S SHH HM

MH MMS S

H uK KGa

K KM a

[1] Luco, Wong. Seismic response of foundations embedded in a layered half-space[J].

Earthquake Engineering and Structure Dynamic，1987，15(3):233-247.

• Coupling numerical & analytical methods

Analytical solution 1 (SDOF)

2 2 2 2

1

1 1 1 1 1

22 2 2 20 0

2 2

1 1 1 1 1 1

22 2

2

1 1 1

1 ( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) (1 ) ( ) ( ) ( ) ( ) ( ) (1 ) ( )

( ) ( ) ( ) ( )

R B g

R HH B HM g

R HH

i U U H U

m mGa GaU K U K H U

m m mH m

GaU K

mH

32 20

2 2 2

1 1 1

( ) ( ) (1 ) ( ) ( ) ( ) ( )TB MM g

I I GaU K H U

H m mH

Matlab

( );RU ( );BU ( )H Get freq. responses：

Fourier

Analytical solution 1 (SDOF)

• Coupling numerical & analytical methods

(c) Relative movements to foundation (b) Rigid movements by foundation；

(a) Total disp. of SSI system

gu

gugu

Analytical solution 1 MDOF

T Rˆ u u u

0 1 0 114 1

0 1 2

0 1 2 0

ˆ [ , , , ,

, , ( ) , , ,

( ) , , , , ]

B B B B

B B B B B

B B B B

u H a u H

a u H H a

u H H a u

u

1 2 3 4

5 6 7 8 9

10 11 12

[ , , , ,

, , , , ,

, , ,0,0]

R R R RR

R R R R R

R R R

u u u u

u u u u u

u u u

u

Fourier

13u

14u

4u5u

6u

10u

11u

12u

1u2u

3u

7u8u

9u

Equations of motion of MDOF SSI system

T R R( ) ( ) ( ) ( )t t t t Mu Cu Ku F

2 ( ) ( ) ( ) ( )i MU CU KU P

( ) ( ) i tt e dt

U u

( )( )( ) ( )

( )( ) ( ) ( )

SBHH HM

SMH MM B

HuK K

GaMK K a

a

Combing

Analytical solution 1 MDOF

Analytical solution 1

2 Coupling analytical & numerical methods

Coupling method 2

guF

MSu

S

(a)

(2) ( ) ( )S u ψ u

FBuM

B

(b)

Coupling methods 2

1

(2)( ) ( ) ( ) ( )1 1

( ) ( )/ /( ) ( ) ( ) ( )

S SHH HM HH HM

s s

S SHM MM HM MM

F FK K C C

M a M aK K C CGa Ga

u C F

S g u u u

(1)

S g u u u

OpenSees Analytical

substructure

Integration

using CS [1]

[1] Gu Q., Ozcelik O.*, 2011, “Integrating OpenSees with other software -- with application to coupling problems

in civil engineering”, Structural Engineering Mechanics, An International Journal. Volume 40, (1）.

About OpenSees

OpenSees (Open System for Earthquake Engineering

Simulation) is a C++ based open source finite element

software, increasingly widely used in earthquake

engineering.

It is developed by Pacific Earthquake Engineering

Research Center (PEER) since1997, co-developed by

UC Berkeley, UCLA, UCSD, Stanford and more than 10

other universities，representing the most frontier

research results in earthquake engineering in US.

OpenSees is platform of NEESgrid.

OpenSees is one of the main integration platforms by

China Major Research Plan Project.

Advantages of OpenSees

Open source, code co-developed and shared by academic society, enabling

much easier collaboration, keeping integrating most advanced research

results.

Strong nonlinearity，including structural and soil nonlinear models, and lot

of nonlinear algorithms.

Advanced OOP framework based on C++, and easier parallel computation.

Sensitivity, reliability and optimization.

High performance computing, like Open Science Grid 、TerraGrid

Academic society made of professors and students from all over the world,

like wiki, forum, workshop, OpenSees day, and training.

May collaborating with other platforms using OpenFresco, CS techniques etc.

Find minimum: 2( )[ ( ) ( )]W

V H C

0; 0p qa b

V V

1 2, , ma a a0 1 2, , , kb b b b；

1 2

0 1 2

1 2

1 2

( )( )( )

( ) 1

n

n

m

m

b b z b z b z

a z a z a z

UH

F

Fourier

iz e

Time domain recursive method[2]：

1 2

0 1 2

( ) ( 1) ( 2) ( )

( ) ( 1) ( 2) ( )

m

n

t a t a t a t m

b t b t b t b t n

u u u u

F F F F

[2] Safak. Time-domain representation of frequency-dependent foundation impedance

functions[J]. Soil Dynamics and Earthquake Engineering(26), 65–70, 2006.

( ) ( ) ( ) U H F

Coupling method 2

Frequency dependent compliance functions |C

MH|

Frequency [Hz]

0 50 1000.005

0.01

0.015

0.02

0.025Analytical

Estimated

Frequency [Hz]

0 50 100

0.16

0.18

0.2

0.22

0.24

|CM

M|

|CH

H|

Frequency [Hz]

0 20 40 60 80 1000.05

0.1

0.15

0.2

0.25Analytical

Estimated

Frequency [Hz]

0 50 1000.005

0.01

0.015

0.02

0.025

|CH

M|

Flowchart of the new SSI methods (implicit and explicit)

Coupling method 2

SSI systems with MDOF structure 4

SSI systems with SDOF structure 3

Applications 3

Millikan library 5

No. Case 1

No. Case 2

Time(s) damping Time(s) Damping

1 0.0156 Rayleigh

2 0.0078 Rayleigh

0.0156 Stiffness 0.0078 Stiffness

No. Case 3

No. Case 4

Time(s) damping Time(s) Damping

3 0.0039 Rayleigh

4 0.0025 Rayleigh

0.0039 Stiffness 0.0025 Stiffness

Rayleigh damping （mass+stiffness）

stiffness damping

Time step 0.0156秒（1/64） 0.0078秒(1/128) 0.0039秒(1/256) 0.0025秒(1/400) 、 、 、

SDOF structure – soil systems

linear elastic cases

Nor

mal

ized

Dis

p. R

espo

nses

Frequency [Hz]

1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

30

35Analytical - Abs. Disp

Analytical - Horizontal

Analytical - Rotation

Coupling - Abs. Disp

Coupling - Horizontal

Coupling - Rotation

(a)

(1) Case 1 (0.0156 s (1/64))

Displacement frequency responses

Nor

mal

ized

Dis

p. R

espo

nses

Frequency [Hz]

1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

30

35Analytical - Abs. Disp

Analytical - Horizontal

Analytical - Rotation

Coupling - Abs. Disp

Coupling - Horizontal

Coupling - Rotation

(a)

Nor

mal

ized

Dis

p. R

espo

nses

Frequency [Hz]

1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

30

35Analytical - Abs. Disp

Analytical - Horizontal

Analytical - Rotation

Coupling - Abs. Disp

Coupling - Horizontal

Coupling - Rotation

(b)

(3) Case 3 (0.0039 s (1/256)) 1.76Hz

2.6%

15.85%

Best step size 0.0039 s (1/256) ，using Rayleigh damping

1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

30

35

Frequency[Hz]

1.76Hz

(1) Case 2 (0.0078 s (1/128))

(a) Rayleigh damping (b) stiffness damping

Norm

aliz

ed D

isp. R

esp.

Frequency [Hz]

1 2 3 40

1

2

3

4

Curv. [1/m]

Mo

m.

[kN

*m]

(d)

-5 0 5

x 10-3

-1

-0.5

0

0.5

1x 10

9

No

rma

lize

d D

isp

. R

esp

.

Frequency [Hz]

1 2 3 40

5

10

15

Curv. [1/m]

Mo

m.

[kN

*m]

(c)

-2 -1 0 1 2

x 10-3

-1

-0.5

0

0.5

1x 10

9

No

rma

lize

d D

isp

. R

esp

.

Frequency [Hz]

1 2 3 40

5

10

15

20

25

-1 -0.5 0 0.5 1

x 10-4

-5

0

5

10x 10

7

Curv. [1/m]

Mo

m.

[kN

*m]

(b)

SDOF structure–soil systems: nonlinear structures

1.76Hz

No

rma

lize

d D

isp

. R

esp

.

Frequency [Hz]

1 2 3 40

5

10

15

20

25

30

-1 -0.5 0 0.5 1

x 10-4

-5

0

5

10x 10

7

Curv. [1/m]M

om

. [k

N*m

]

(a)

* Abs. Dips

Soil Horizontal

Soil Rocking

* Abs. Dips

Soil Horizontal

Soil Rocking

Disp. freq. resps with varying input amplitudes(0.001m，0.005m,0.01m,0.05m）

1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

30

Frequency[Hz]

Norm

alized D

isp. R

esponses Impl. Abs. disp.

Impl. -Soil Horizontal

Impl. -Soil Rocking

Expl. Abs. disp.

Expl. -Soil Horizontal

Expl. -Soil Rocking

1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

30

Frequency[Hz]

implicit vs explicit methods

dt=0.0078s dt=0.0039s

SDOF structure–soil systems: nonlinear structures

Disp. freq. responses with different input amplitudes(0.001m，0.005m）

Curvature [1/m]

Mo

men

t [k

N*

m]

-4 -2 0 2 4 6 8

x 10-4

-1

-0.5

0

0.5

1

1.5x 10

9

Mom

ent

[kN

*m]

Curvature [1/m]

-10 -5 0 5

x 10-4

-1.5

-1

-0.5

0

0.5

1

1.5x 10

9

(a) (b)

Mom

ent

[kN

*m]

Curvature [1/m]

-6 -4 -2 0 2 4

x 10-4

-1.5

-1

-0.5

0

0.5

1x 10

9

(a)

Curvature [1/m]

Mom

ent

[kN

*m]

-1 0 1 2 3

x 10-3

-1.5

-1

-0.5

0

0.5

1

1.5x 10

9

(b)

(a) Considering SSI (b) Not considering SSI

El Centro

San Fernando

Nonlinear

structural responses

—— considering SSI v.s. no SSI

0 5 10 15 20 25 30-2

0

2

Accl.[m

/s2]

0 5 10 15 20 25 30-0.1

-0.05

0

0.05

Foun. H

or.

D

isp.[m

]

0 5 10 15 20 25 30

-0.2

0

0.2

Time[sec]

Str

u. A

bs.

Dis

p.[m

]

Impl. Considering SSI

Expl. Considering SSI

Not considering SSI

Nonlinear

structural responses

—— considering SSI v.s. no SSI

SSI systems with MDOF structure 4

SSI systems with SDOF structure 3

Applications 3

Millikan library 5

Linear MDOF structure – soil systems

Rayleigh damping

Explicit methods

• Application

(1) Case 1 (0.0078 s (1/128))

The best step size is 0.0039 s，with Rayleigh damping

(2) Case 2 (0.0039 s (1/256))

1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

Frequency[Hz]

Norm

aliz

ed D

isp. R

esponses

Analytical - Abs. Disp

Analytical -Soil Horizontal

Analytical -Soil Rocking

Expl. Coup. - Abs. Disp

Expl. Coup. -Soil Horizontal

Expl. Coup. -Soil Rocking

1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

Frequency[Hz]

< 5.0%

Nonlinear structure N

orm

aliz

ed D

isp

. R

esp

.

Frequency [Hz]1 2 3 4

0

5

10

15

20

25

30

Curv. [1/m]

Mo

m.

[kN

*m]

* Abs. Dips

Soil Horizontal

Soil Rocking

* Abs. Dips

Soil Horizontal

Soil Rocking

(a)

-2 -1 0 1 2

x 10-4

-2

-1

0

1

2x 10

8

Norm

aliz

ed D

isp. R

esp

.

Frequency [Hz]

1 2 3 40

5

10

15

20

25

30

Curv. [1/m]

Mo

m.

[kN

*m]

(b)

-1 -0.5 0 0.5 1

x 10-4

-5

0

5

10x 10

7

No

rma

lize

d D

isp

. R

esp

.

Frequency [Hz]

1 2 3 40

5

10

15

Curv. [1/m]

Mo

m.

[kN

*m]

(c)

-2 -1 0 1 2

x 10-3

-1

-0.5

0

0.5

1x 10

9

No

rma

lize

d D

isp

. R

esp

.

Frequency [Hz]

1 2 3 40

1

2

3

4

Curv. [1/m]

Mo

m.

[kN

*m]

-5 0 5

x 10-3

-1

-0.5

0

0.5

1x 10

9

(d)

Disp. freq. resps with varying input amplitudes(0.001m，0.005m,0.01m,0.05m）

Mo

men

t [k

N*

m]

Curvature [1/m]

-6 -4 -2 0 2 4 6 8

x 10-3

-6

-4

-2

0

2

4

6x 10

8

(a)

Curvature [1/m]

Mo

men

t [k

N*

m]

-4 -2 0 2 4 6 8

x 10-3

-6

-4

-2

0

2

4

6x 10

8

(b)

Curvature [1/m]

Mom

ent

[kN

*m

]

-1 -0.5 0 0.5 1 1.5

x 10-3

-4

-2

0

2

4

6x 10

8

(b)

Mom

ent

[kN

*m

]

Curvature [1/m]

-1 -0.5 0 0.5 1 1.5

x 10-3

-6

-4

-2

0

2

4

6x 10

8

(a)

(a) Considering SSI (b) Not considering SSI

Column in

the first story

El Centro earthquake

Column in

second story

—— considering SSI v.s. no SSI

Nonlinear structural resps

0 5 10 15 20 25 30

-2

0

2

Accel.[m

/s2]

0 5 10 15 20 25 30-1

-0.5

0

Foun. H

or.

D

isp.[m

]

0 5 10 15 20 25 30-1

-0.5

0

Str

u. 1F

A

bs. D

isp.[m

]

0 5 10 15 20 25 30-1

-0.5

0

Time[sec]

Str

u. 2F

A

bs. D

isp.[m

]

Implicit Considering SSI

Explicit Considering SSI

Not considering SSI

—— considering SSI v.s. no SSI

SSI systems with MDOF structure 4

SSI systems with SDOF structure 3

Applications 3

Millikan library 5

Millikan library

• Applications

m9.43

m39

m7.34

m5.30

m2.26

m7.17

m4.13

m1.9

m9.4

m01

2

3

4

5

6

7

8

9

m9.21

R

m3.4B

Displacement envelops of different floors（/m）

Flo

ors

Disp envelop/m

0

2

4

6

8

10

12

0 0.05 0.1 0.15 0.2 0.25 0.3

SSI

No SSI

Column moment-curvature responses

Mo

men

t [k

N*

m]

Curvature [1/m]

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

8

(a)

Curvature [1/m]

Mo

men

t [k

N*

m]

-0.02 -0.01 0 0.01 0.02 0.03-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

8

(b)

Mom

ent

[kN

*m

]

Curvature [1/m]

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02-2

-1.5

-1

-0.5

0

0.5

1

1.5x 10

8

(a)

Curvature [1/m]

Mom

ent

[kN

*m

]

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

8

(b)

(a) Considering SSI (b) No SSI

—— SSI v.s. no SSI

First floor

Second floor

Nonlinear structural resps

0 5 10 15 20 25 30-0.5

0

0.5

Foun. H

or.

D

isp.[m

]

Impl. Considering SSI

Expl. Considering SSI

Not considering SSI

0 5 10 15 20 25 30-0.5

0

0.5

Str

u. T

op

Dis

p.[m

]

0 5 10 15 20 25 30-20

-10

0

10

Time[sec]

Str

u. T

op

Accel.[m

/s2]

—— SSI v.s. no SSI

Conclusions 4

A novel coupling method for nonlinear SSI analysis is

presented, with both implicit and explicit methods developed.

Frequency dependent compliance functions are represented in

time domain by a discrete recursive soil filter method.

Analytical solution to MDOF linear structure-rigid foundation-

linear half space soil systems is derived in freq. domain.

A comprehensive study to the method is made by

SDOF/MDOF structure-soil systems and a Millikan Library

SSI example.

• Conclusions

Ref: Huang, Ozcelik and Gu, 2014, A Practical and Efficient Coupling Method

for Large Scale Soil-Structure Interaction Problems, SDEE, 2014, under review.

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