Earth 2011-lec-06

2.4. MDOF Ground Excitation

EARTHQUAKE ENGINEERING

2.4.1. MDOF Equation of Motion

2.4.2. MDOF Free Vibrations

2.4.3. MDOF Response to Earthquakes

2.4.4. MDOF Modal Analysis

2.3.2. Response to General Dynamic Loading

2.3.2. Forced Vibrations General Loading

Common Types of Dynamic Loads

Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr2

PeriodicSinusoidal


Common Types of Dynamic LoadsPeriodic

Sinusoidal

Other




Non Periodic

Impulse

Sinusoidal

Other




Non Periodic

Impulse

Sinusoidal

Explosion

Other




Non Periodic

Impulse

Sinusoidal

Explosion

Other


Earthquake


Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr

Response to General Dynamic Loading)(tFkxxcxm

F(t) is given as a relation between time and Force

0 1002.66 0.002 772.421 0.004 582.664 0.006 427.027 0.008 300.089 0.01 197.234 0.012 114.537 0.014 48.6668 0.016 -3.20412 0.018 -43.4678 0.02 -74.1465 0.022 -96.9462 0.024 -113.303 0.026 -124.424 0.028 -131.319 0.03 -134.835 0.032 -135.674 0.034 -134.422



Response to General Dynamic LoadingThe solution is carried out using different numerical Integration techniques as

Numerical Evaluation of DuHamel Integral

Newmark - Method

Wilson - Method

Central Difference Method



Incremental Equation of Motion

Subtracting the Equation of Motion at times t and t + t the resulting Incremental equation of motion can be derived as

)(tFxkxcxm

)(tFkxxcxm ttt )( ttFkxxcxm tttttt


Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr

Newmark - Method (Linear Acceleration)- Given : m, c, k, xo, vo, ao, Fi- Select t

- Calculate where

k

tc2

tm4k 2eff

- For each step :- Calculate F where

i1i FFF

- Calculate where iieff am2vc2tm4FF

iieff am2vc2tm4FF

- Calculate x where x = / iieff am2vc2tm4FF

xktc2

tm4k 2eff

xktc2

tm4k 2eff

- Calculate v where iv2xt

2v

- Calculate a where ii2 a2vt

4xt4a

- Calculate xi+1, vi+1 and ai+1 wherexi+1= xi+ x, vi+1= vi+ v and ai+1 = ai+ a



Response to Impact

0 0.5 1 1.5 2 2.5

-8

-6

-4

-2

0

2

4

6

8

10

x 10-3 Deflection History

Time [sec]

Dis

plac

emen

t

0 0.5 1 1.5 2 2.50

100

200

300

400

500

600Load Record

Time [sec]

Load

kN



Response to Impact

0 0.5 1 1.5 2 2.5

-8

-6

-4

-2

0

2

4

6

8

10

x 10-3 Deflection History

Time [sec]

Dis

plac

emen

t

0 0.5 1 1.5 2 2.50

100

200

300

400

500

600Load Record

Time [sec]

Load

kN

2.3.3. Response to Ground Acceleration


Response to Ground ExcitationEquation of Motion

gttt xmkxxcxm

gxm Is the Load Equivalent to ground acceleration



Response to General Ground Excitation

0 5 10 15 20 25 30 35 40 45 50

-0.01

-0.005

0

0.005

0.01

0.015Deflection History

Time [sec]D

ispl

acem

ent

0 5 10 15 20 25 30 35 40 45 50

-0.1

-0.05

0

0.05

0.1

0.15Earthquake Record

Time [sec]

Acc

eler

atio

n (g

)



F1

F2

F3

F4

x4

x3

x2

x1F1

F2

F3

F4

m1

m2

m3

m4

k1

k2

k3

k4



nmm

mm

M3

2

1

][

}{}]{[}]{[}]{[ FxKxCxM

Mass, Damping, and Stiffness Matrices According to the Number of Degrees of Freedom

][],[],[ KCM

44

4433

3322

221

][

kkkkkk

kkkkkkk

K



4

3

2

1

}{

xxxx

x

4

3

2

1

}{

xxxx

x

4

3

2

1

}{

FFFF

F

}{}]{[}]{[}]{[ FxKxCxM Acceleration, Velocity, Displacement and Load Vectors According to the Number of Degrees of Freedom

}{},{},{},{ Fxxx

4

3

2

1

}{

xxxx

x

2.4.2. MDOF Free Vibrations


{X} ={f} (An cos wnt + Bn sin wnt)

{X} = - w2 {f} (An cos wnt + Bn sin wnt)..

- [M] w2 {f} +[K] {f} = 0

([K]- [M] w2 ) {f} = 0 Eigen Value problem

| [K]- [M] w2 | = 0

Free Vibrations of MDOF

}0{}]{[}]{[ xKxM

2.4.3. MDOF Response to Earthquakes


Response of MDOF to Ground motion

gxMxKxCxM }1]{[}]{[}]{[}]{[

The Same Numerical Techniques are used to determine the response of MDOF Structures to General Dynamic Loads

2.4.4. MDOF Mode Superposition


Mode Shapes are orthogonal with respect to the mass and stiffness matrices

Tj

Ti M }]{[}{ ff <> 0 For i=j

= 0 For ij

<> 0 For i=j = 0 For ij

Tj

Ti K }]{[}{ ff



yx }{}{ f

Mode Superposition aims at uncoupling of the Equation of Motion (For each DOF)

Substitute by

}{}}{]{[}}{]{[}}{]{[ FyKyCyM fff

}{}{}}{]{[}{}}{]{[}{}}{]{[}{ FyKyCyM TTTT fffffff

Which is the uncoupled Equation of Motion of the MDOF System which can be solved separately for each DOF and combined again

}{}{}]{ˆ[}]{ˆ[}]{ˆ[ FyKyCyM Tf



Which is the Single Normalized Equation of Motion of DOF Number

iTi

Ti

iiii MFyyy

}]{[}{}{}{}{}{2}{ 2

fffww

iiiii Fyyy }ˆ{}{}{2}{ 2 ww

i

Ti

iiii MFyyy ˆ

}{}{}{}{2}{ 2 fww

}{}{}]{ˆ[}]{ˆ[}]{ˆ[ FyKyCyM Tf

For Each DOF i



Is the participation Factor for modal analysis

}{}]{[}{}]{[}{}{}{2}{ 2

gi

Ti

Ti

iiii uM

IMyyy ff

fww

iTi

Ti

i MIM}]{[}{}]{[}{

fff

}{}{}{2}{ 2giiiii uyyy ww

iTi

Ti

iiii MFyyy

}]{[}{}{}{}{}{2}{ 2

fffww

For Earthquake response

2.4. MDOF Ground Excitation


Questions

Discuss the free vibration Equation of Motion of Multi DOF Systems and the meaning of Natural periods and Mode shapes

What is the Computational merit of mode superposition method

Discuss the Meaning of the Participation Factor in Modal Analysis



Questions

Discuss the Techniques of Numerical Integration of The Dynamic Equation of Motion

What are the main categories of structures regarding to Damping

Use the MATLAB Segment defined to determine the response of the Structure defined in the previous lecture Questions to “Al Aqaba” Earthquake given

Education

Earth 2011-lec-06