3.3 numerical analysis of seismic response of sdof...

1

Chapter 3

Seismic Responses of SDOF and MDOF

2015/10/20

熊海贝，同济大学土木工程学院

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1

2015/10/20


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Outline

3.1 Free Vibration of SDOF Systems

3.2 Forced Vibration of SDOF Systems

3.3 Numerical Analysis of Seismic Response of SDOF

3.4 Response Spectrum of SDOF

3.5 Response of Nonlinear SDOF Systems （*）

3.6 Free Vibration of MDOF Systems

3.7 Response Specturm Method of MDOF

3.8 Earthquake Action and Responses of MDOF

3.9 Time History Method of MDOF

Dynamics

gxmkxxcxm

Then gxxxx 22

dtsinextx t

t

g

2

02

11

1

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Base excitation (Earthquakes)

x ground

mass k

c

gx

From the moment, (k-1)

Derivate the moment, k

Method:

linear acceleration, Newmark-β, and Wilson-θ method

to solve the differential issue.

-1 -1 -1( ), ( ), ( )k k kx t x t x t

( ), ( ), ( )k k kx t x t x t

( ), ( ), ( )x t x t x t

-1 -1 -1( ), ( ), ( )k k kx t x t x t

( ), ( ), ( )k k kx t x t x t

t

-1kt kt

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Assumption: linear acceleration increase

No. 5 No. 6

2

No. 7 No. 8

No. 9 No. 10

No. 11


A response spectrum is simply a plot of the peak or

steady-state response (displacement, velocity or

acceleration) of a series of oscillators of varying natural

frequency, that are forced into motion by the same base

vibration or shock.

The resulting plot can be used to pick off the response of

any linear system, given its natural frequency of oscillation.

One such use is in assessing the peak response of a

SDOF to earthquakes.

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3


Time [sec]

4038363432302826242220181614121086420

Accele

ration [cm

/sec2] 40

20

0

-20

-40

Time [sec]

50484644424038363432302826242220181614121086420

Accele

ratio

n [g]

40

20

0

-20

-40

Time [sec]

363432302826242220181614121086420

Accele

ration [cm

/sec2] 40

20

0

-20

-40

combined to produce a design spectrum


Design spectrum in Chinese code

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GB 50011-2010）

α- T Curve

α is defined as the ratio of the horizontal seismic action to

the gravity of a SDOF elastic system.

GαGg

S＝

g

SmgmSF aa

a

kx

S

g

x

g

Sα

g

aga

max

max

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1. Seismic Effect Coefficient, α

(地震加速度影响系数) 2. Earthquake Coefficient

（地震系数）

There exists some corresponding relation between

earthquake intensity and earthquake coefficient.

Intensity 6 7(7.5) 8(8.5) 9

k 0.05 0.1(0.15) 0.2(0.30) 0.4

Acc. 0.05g 0.1g(0.15g) 0.2g(0.3g) 0.4g

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g

xk

g max

3. Dynamic Coefficient

（动力系数）

It is the ratio of maximum absolute acceleration to

maximum ground motion acceleration for SDOF system,

reflecting how much the maximum absolute acceleration is

amplified due to dynamic effect.

maxg

a

x

S

＝

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Seismic Effect Coefficient

18

αmax

Intensity 6 7 8 9

Frequently

Occurred Eq.

0.04 0.08（0.12） 0.16（0.24） 0.32

Rare Occurred

Eq.

—— 0.50（0.72） 0.90（1.20） 1.40

1. Seismic Effect Coefficient, α

(地震加速度影响系数)

4

4. Site Predominant Period 场地卓越周期 Tg

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Earthquake wave is influenced with soil profile:

• Active faults nearby,

• Soil dynamic response

• Regional geology influence

• Liquefaction

• Landslide

• Subsidence (下陷性地震)

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Site Category

Site Group

I II III IV

Group 1 0.25 0.35 0.45 0.65

Group 2 0.30 0.40 0.55 0.75

Group 3 0.35 0.45 0.65 0.90

Site Predominant Period Tg （s）

4. Site Predominant Period, Tg

To express site characteristics and the distance to epicanter,

It is divided to three groups, Group 1 to 3 for near-earthquake

(近震) to far-earthquake （远震）.

Based on Chinese Seismic Code (GB 50011-2010),

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63.0

05.09.0

324

5.002.001

ς

6.18.00

5.0012

Attenuation index number

declined slope coefficient

Damping adjusting coefficient

• Response spectra are very useful tools

If we know a single story building’s natural period, and the

building construction site, we can find the seismic effective

coefficient to determine the equivalent static force (earthquake

load), the we can combine the Eq. load with other loads. Then we

can check the system if it meets the requirements.

If we can resolve a MDOF to a family of equivalent SDOF

system, then we can calculate multi-story building’s Eq. loads using

Response Spectra of SDOF.


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1. Nonlinearity of Materials

2. Motion Equations of SDOF Nonlinear

System (K(t))

3. Solutions of Nonlinear Motion Equations

4. Hysteresis (滞回) Model

( ; skeleton curve （骨架曲线）

and hysteretic curve （滞回曲线）)

F,M,


5

Hysteretic curve


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Skeleton curve

Hysteretic curve

Loading and un-loading is in different way


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tk

27

Then,

sf x k x x＝

gmx+cx+k x x mx t ＝－


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But, at moment, t, if is very small, during tk-1 to tk ,

is a constant value.

1kg1k1k1k1k tx＝－mtxtxktxctxm

kgkkkk tx＝－mtxtxktxctxm

• if,

k-1kk-1kk-1k txtx＝x，Δtxtx＝x，ΔtxtΔx＝x

1kkk-1gkgg t，Δt＝ttxtx＝xΔ


t

tk

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gxmΔkΔΔxcΔxmΔ

Based on ΔtxΔ

2

1Δtx＝xΔ 1k

22

1k1k ΔtxΔ6

1Δtx

2

1ΔtxΔx＝

We can get, Δtxx2Δt

3Δx

Δt

6＝xΔ 1k1k2

Then,

Δtx

2

1x3

Δt

3ΔΔΔtx＝xΔ 1k1k1k

Δtx2

1x3Δx

Δt

3＝ 1k1k

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Dynamic Incremental Equilibrium:

1-k1k1k2x3cx

t

c3txx2

t

m3x

t

6m （

g1k xmxktx

2

1 ＝－）

txx2t

3mxmxk

t

3c

t

6m 1-k1-kg2

＝－

tx

2

1x3c 1-k1-k

Pxk ＝Simplified as,

Equivalent Stiffness Equivalent Restoring Force

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6

0

pxpy

K1

K2

K12

pK3

Idealized three-lines semi-degradation restoring model

(理想的“半退化三线型”恢复力模型)

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3.5 Response of Nonlinear SDOF Systems （*） 3.6 Free Vibration of MDOF Systems

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time, sec 0.0 0.2 0.4 0.6 0.8

Vibration Period TW g

K 2

Single-degree-of-freedom oscillators

W

K T

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First-

Mode

Shape

Third-

Mode

Shape

Second-

Mode

Shape

• Linear response can be viewed in terms of individual

modal responses.

Idealized

Model

Actual Building

• Multi-story buildings can be idealized and analyzed

as multi-degree-of-freedom systems.

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Multi-degree-of-freedom systems

Systems having n degrees of freedom will have:

n modes of vibration;

n natural frequencies.

Response will, in general, be some linear

combination of response in these modes

Common approach is to treat the n-DOF system as

n SDOF systems instead

The basic period is fundamental period.


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Fundamental Vibration Period for Buildings

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7

Formulation of the Equation of Motion and Selection of the Dynamic Degrees of Freedom

Assumption:

• Beams and floors are concentrated

to a mass particle

• Axial deformations of the beams

and columns are neglected

• Axial load on columns are neglected


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.. ..

11 1

1 11 12 1 1 2 2 1

[ ( ) ( )]

( ) [ ( ) ( )]

gI

S S S

f m x t x t

f f f K x t K x t x t

.. ..

11 1 1 1 1 1 2 2 2 1( ) ( ) ( ) ( ) ( ) 0gI Sf f m x t m x t K x t K x t K x t

.. ..

11 1 2 1 2 2 1( ) ( ) ( ) ( ) ( )gm x t K K x t K x t m x t

.. ..

22 2 2 2 2 2 1( ) ( ) [ ( ) ( )] 0gI Sf f m x t m x t K x t x t

.. ..

22 2 2 2 1 2( ) ( ) ( ) ( )gm x t K x t K x t m x t


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.. ..

1 11 1 2 2 1

.. ..2 2 2 22

2

( )0 0( ) ( )

0 0( )( ) ( )

g

g

x tm K K K mx t x t

m K K mx tx t x t

Define:

1

2

1 2 2

2 2

1

2

0=

0

( )( )

( )

mM

m

K K KK

K K

x tx t

x t

..

.. 1

..

2

( )( )

( )

1

1

x tx t

x t

I

.. ..

( ) ( ) ( )gM x t K x t M I x t


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MDOF:

Raleigh damping

.. . ..

( ) ( ) ( ) ( )gM x t C x t K x t M I x t

0 1C a M a K

1 2 1 2 2 10 2 2

2 1

2 ( )

2 2 1 1

1 2 2

2 1

2( )


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Modal Analysis

Overview of the method:

The equations of motions, when transformed to modal coordinates, become uncoupled.

The response in each mode can be computed independently of the other modes by solving an SDOF system with the vibration properties of that mode.

Modal responses are combined to obtain the total response.


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思路 42

• 由几何坐标（节点位移）转换到广义坐标（固有振型的振幅）的转换来完成。

• 由于固有振型的正交性，运动方程可解耦，从而使多自由度方程变为多个单自由度方程进行求解。

• 振型叠加后在转换到几何坐标。

将多自由度体系以一种方法，转换到单自由度体系，然

后用单自由度体系的方法求解单自由度的振型和反应，

然后想办法叠加，再转换到几何坐标系中。

转来转去

8

gM x t C x t K x t M I x t ＝－

坐标转换（Rayleigh-Ritz法）

x t = q tX令:

Geometry (几何坐标) modal coordinates (广义坐标)

x t = tX q

x t = tX q


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1

2

1

( )

( )

M t M( )

( )

M t

T T

j j i ni

n

T

jj j

q t

q t

X X q X X X Xq t

q t

X X q

g

M t C t K q t

M I x t

T T T

j j j

T

j

X X q X X q X X

X

＝－

M 0 K =0T T

j ji iX X X X 和

Orthogonality of Modes


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2K q t K q t K q tT T T

j j jj j jj jX X X X X X

2

1 0 0 1( K ) q t ( ) q tT T

j jj j jX M X X M X

Divided by MT

j jX X

2 2

0 1 gt ( + ) t q t - x tj j j j j jq q ＝


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modal equations Express the j mode contribute to general response.

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Mode Participant Coefficient 振型参与系数

n

1i

n

1i

2

jiijii

j

T

j

T

j

j XmXmXMX

IMX

＝＝

＝＝


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Modal Analysis

We define:

Generalized modal mass:

Generalized modal stiffness:

Generalized modal damping:

Generalized modal forces:

n

T

nnM m

n

T

nnK k

n

T

nnC c

)()( ttP T

nn p


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The modal equations will be uncoupled and reduces to:

Or in matrix form:

M is the diagonal matrix of the generalized modal mass

K is the diagonal matrix of the generalized modal

stiffness

C is diagonal matrix of the generalized modal damping

P(t) is diagonal matrix of the generalized modal forces

)(tPqKqCqM nnnnnnn

P(t)KuuCuM


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9

Divide the uncoupled equations by :

Solve the above equation for the modal coordinate

similar to an SDF system.

nM

n

nnnnnnn

M

tPqqq

)(2 2

)(tqn


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Modal Analysis

Displacements:

Contribution of the nth mode to the displacement is:

Combining these modal contributions gives the total displacement:

N

n

N

n

nnn tqtut1 1

)()()( u

)()( tqt nnn u

)(tu


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Modal Analysis: Summary

Define the structural properties:

Determine the mass matrix m and the stiffness matrix k.

Determine the modal damping ratios

Determine the natural frequencies and mode shapes .

Compute the response in each mode by uncoupling the equations of motion and solving for modal coordinates .

n

n

)(tqn


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Modal Analysis: Summary (cont’d)

Compute n-modal displacements un(t) .

Compute the element forces associated with the n-modal displacements rn(t) .

Combine the contributions of all the modes to determine the total response.


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Homework Review: SDOF

Rreview：MDOF

T

jm = X M X2

j j


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3.3 numerical analysis of seismic response of sdof...

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