Earth 2011-lec-06
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Transcript of 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
2.3.2. Forced Vibrations General Loading
Common Types of Dynamic LoadsPeriodic
Sinusoidal
Other
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr3
2.3.2. Forced Vibrations General Loading
Common Types of Dynamic LoadsPeriodic
Non Periodic
Impulse
Sinusoidal
Other
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr4
2.3.2. Forced Vibrations General Loading
Common Types of Dynamic LoadsPeriodic
Non Periodic
Impulse
Sinusoidal
Explosion
Other
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr5
2.3.2. Forced Vibrations General Loading
Common Types of Dynamic LoadsPeriodic
Non Periodic
Impulse
Sinusoidal
Explosion
Other
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr6
Earthquake
2.3.2. Forced Vibrations General Loading
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
2.3.2. Forced Vibrations General Loading
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr8
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
2.3.2. Forced Vibrations General Loading
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr9
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
2.3.2. Forced Vibrations General Loading
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
2.3.2. Forced Vibrations General Loading
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr11
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
2.3.2. Forced Vibrations General Loading
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr12
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
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr13
Response to Ground ExcitationEquation of Motion
gttt xmkxxcxm
gxm Is the Load Equivalent to ground acceleration
2.3.3. Response to Ground Acceleration
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr14
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
)
2.4.1. MDOF Equation of Motion
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr1515
F1
F2
F3
F4
x4
x3
x2
x1F1
F2
F3
F4
m1
m2
m3
m4
k1
k2
k3
k4
2.4.1. MDOF Equation of Motion
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr16
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
2.4.1. MDOF Equation of Motion
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr17
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
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr18
{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
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr19
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
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr20
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
2.4.4. MDOF Mode Superposition
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr21
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
2.4.4. MDOF Mode Superposition
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr22
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
2.4.4. MDOF Mode Superposition
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr23
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
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr24
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
2.3.3. Response to Ground Acceleration
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr25
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