American University of Beirut
Faculty of Engineering & Architecture
Department of Civil and Environmental Engineering
CIVE – 616-EARTHQUAKE ENGINEERING
“HOMEWORK (2)”
Prepared by:
Ali El-Hajj
ID # 201820917
27th Feb. 2018
Problem 1: Free Vibration Response
1) The free vibration response - displacment time series - of a single-degree-offreedom (SDOF) system is given by the following equation:
u(t) = exp (-Ʒ ωn t) [u (0) cos (ωDt) +
ú (0) + Ʒ 𝜔𝑛 𝑢 (0)
𝜔𝐷
sin (ωDt)]
The MATLAB function called “FREE_VIBRATION.m” is defined to take
some of the input parameters of the above equation such as u (0), ú (0), Ʒ, and
Tn, then calculates the output time vector (t) at discrete time steps dt =0.02
secs, and the displacment vector u(t) corresponding to each time step.




u (0): initial displacment of the system
ú (0): initial velocity of the system
Ʒ: damping ratio
Tn: natural period of the system
2) The script presented in MATLAB file “FREE_VIBRTION_RUN.m” is used
to run the function defined in part (1), that computes and plots the free
vibration response for systems having the initial conditions u (0) =0, ú (0)
=20cm/sec, and Tn=0.5 sec for different damping ratios. The results for
damping ratios (Ʒ) of 0, 0.02, and 0.05 are presented in
Figure 3
Figure 1, Figure 2,
and
respectively.
3) The script presented in MATLAB file “FREE_VIBRTION_RUN.m” is used
to run the function defined in part (1) that computes and plots the free
vibration response for systems having the initial conditions u (0) =0, ú (0)
=20cm/sec, and Tn=2 sec for different damping ratios. The results for
damping ratios (Ʒ) of 0, 0.02, and 0.05 are presented in
Figure 6
respectively.
2
Figure 4, Figure 5,
and
Figure 1
Figure 2
3
Figure 3
Figure 4
4
Figure 5
Figure 6
5

By comparing figures 1, 2 and 3 for constant period of 0.5s, as the damping ratio
increases, the rate at which the motion decay decreases. The same applies for the
comparison of figures 4, 5, and 6 for constant period of 2sec.

By comparing figures of different periods, as the period increase, the max displacment
response increases.
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Problem 2: Numerical Methods
1) When acceleration time series is given in the form of a defined function,
the velocity time series would be obtained by integrating the
acceleration function over the time of the series if it had an exact
mathematical solution. The deformation time series would also be
obtained in a similar procedure by integrating the velocity time series
function.
Since it is difficult, or even not possible, to define the ground
acceleration time series due to earthquake as definite mathematical
function, thus approximate numerical integration methods must be used
for deriving the velocity and displacment time series, given the ground
acceleration.
The MATLAB function “GROUND_Acc_Vel_Disp.m” is defined to
read the input quantities of earthquake ground acceleration over
specified time step dtg for a given earthquake from a text file. The
ground acceleration time series must be saved in the text file in the
name of “filename.txt” so that the mentioned function reads the input
from the text file and uses “cumsum” function for numerical
integration to compute the velocity time series, and then the
displacement time series for a given earthquake ground acceleration record.
The script presented in MATLAB file “GROUND_Acc_Vel_Disp
_PLOTS_PEAKS.m” is used to run the above function and plot the
ground acceleration, velocity and displacement time series of the Napa
2014 Earthquake. The results are presented in Figure 7, Figure 8, and Figure
9
respectively.
7
Figure 7
Figure 8
8
Figure 9
2) The MATLAB function called “PG_VALUES.m” is defined to take
the input of ground acceleration, velocity and displacement time series,
and compute the absolute maximum value of each that is the peakground-acceleration (PGA), peak-ground-velocity (PGV), peakground-displacment (PGD).
The script presented in MATLAB file “GROUND_Acc_Vel_Disp
_PLOTS_PEAKS.m” also gives the peak ground motion parameters
for the Napa 2014 Earthquake that are:
 PGA = 418.0450 cm/sec2
 PGV = 81.9091 cm/sec
 PGD = 33.9694 cm
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3) The equation of motion for a system subjected to ground motion
excitation is given by:
𝐶
𝐾
m ű(t) + 𝑚 ú(t) + 𝑚 u(t) = -m űg(t)
(Equation 1)
ű(t) + 2 Ʒ ωn ú(t) + ωn2 u(t) = -űg(t)
(Equation 2)
The solution for this equation can be obtained using the central
difference method which is based on a finite difference approximation
of the time derivatives of displacement (i.e., velocity and acceleration).
Taking constant time steps ∆t, the central difference expressions for
velocity and acceleration at time i are [1]:
𝑢(𝑖+1) −𝑢(𝑖−1)
úi =
űi =
(Equation 3)
2∆𝑡
𝑢(𝑖+1) −2 𝑢(𝑖) + 𝑢(𝑖−1)
(Equation 4)
(∆𝑡)2
REF _Ref507427060 \h \* MERGEFORMAT úi =
𝑢(𝑖+1) −𝑢(𝑖−1)
2∆𝑡
𝑢(𝑖+1)−2 𝑢(𝑖)+ 𝑢(𝑖−1)(∆𝑡)2
(Equation 4) in
(Equation 2):
𝑢(𝑖+1) −2 𝑢(𝑖) + 𝑢(𝑖−1)
(∆𝑡)2
𝑢(𝑖+1) −𝑢(𝑖−1)
+ 2 Ʒ ωn [
2∆𝑡
] + ωn2 u(i) = - űg(i)
Rearranging terms in (Equation 5) gives:
Ҏ
𝒖(𝒊+𝟏) = Ҟ
1
Ʒ 𝜔𝑛

Ҟ= [ (∆𝑡)2 +

Ҏ = -ű𝑔(𝑖) – a 𝑢(𝑖−1) – b 𝑢(𝑖)

a = [ (∆𝑡)2 -

b= [𝜔𝑛 2 −
1
∆𝑡
Ʒ 𝜔𝑛
∆𝑡
]
]
2
]
(∆𝑡)2
Initial conditions for time step i=0

ű0 = -ű𝑔(0) – 2 Ʒ 𝜔𝑛 ú0 - 𝜔𝑛 2 𝑢(0)
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(Equation 5)

𝑢−1 = 𝑢0 - ∆t(ú0 ) +
(∆𝑡)2
2
(ű0 )
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The MATLAB function called “Deformation.m” is defined to read the
input of an earthquake ground acceleration time series from a text file
as described before, in addition to the input of specified structure’s
natural period (Tn), specified damping ratio (Ʒ) or (Z), as defined in the
MATLAB library, initial velocity of the system (V0), and initial
displacement of the system (U0). This function outputs and plots the
deformation time series of the system having the specified parameters,
initial conditions and subjected to the given ground motion.
The script presented in MATLAB file “Deformation_RUN” is used to
compute and plot the deformation time series of the system. The results
of systems, having same Ʒ=0.05, ú0 = 0, and 𝑈0 = 0, but with varying
natural periods (Tn= 0.28, 0.57, 1.2,2.5) and subjected to same
earthquake ground motion recorded during Napa 2014 Earthquake , are
plotted in
Figure 10, Figure 11, Figure 12,
and
Figure 13
respectively. The
results of the maximum deformation of each system with different
values of (Tn) are also presented in Table 1.
Table 1
Tn (sec)
0.28
0.57
1.2
2.5
D (cm)
1.1667
6.1274
47.2712
48.05
12
Figure 10
Figure 11
13
Figure 12
Figure 13
14
4) The MATLAB function called “Response_SPECTRA.m” is defined to
take the input of a vector of damping ratios (Ʒ) and a vector of natural
periods (Tn) for systems of interest or over specified range, in addition
to the initial conditions of these systems, u (0), and ú (0). This function
calls the previously defined function “Deformation.m”
to compute
the deformation response for each of the specified periods and damping
ratios. It outputs then a vector of deformation of systems as function of
periods, which can be used to pot the deformation response
spectrum(D). Furthermore, it computes the vectors for plotting velocity
response and acceleration response by multiplying each deformation
response of each system by 𝜔𝑛 and 𝜔𝑛 2 respectively.
5) The script presented in MATLAB file “RESPONSE_SPECTRA
_RUN.m” is used to compute and plot deformation and pseudoacceleration response spectra corresponding to ground motion of Napa
2014 Earthquake. The initial displacment and velocities are u (0) =0,
and ú (0) =0. The spectra are plotted for values of natural periods
between 0 and 10 secs, with 0.1sec increment, and for damping ratios
of 0, 0.02, and 0.05 and are presented in Ошибка! Источник ссылки
не найден. and Ошибка! Источник ссылки не найден..
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Figure 14
Figure 15
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6) Earthquake name:
Imperial
Earthquake date:
6/6/1938
Earthquake magnitude:
5
Station name:
El Centro Array #9
Station Rupture distance:
34.98 Km
𝑉𝑠30:
213.44 (m/sec2)
Record Sequence number:
4
Horizontal-1 Acc. Filename is RSN4_IMPVALL.BG_B-ELC000.txt
Figure 16
17
Figure 17
Figure 18
18
Figure 19
Figure 20
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Problem 3: Earthquake response
1)
(adapted from Chopra, p.9)
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(a) If the cross section of the beam is much larger than that of the columns, the
stiffness of the SDOF system may be approximated by (Equation 1.3.2)
K=
254
∗104 ]
12
[64∗109 ]
24∗[2∗105 ]∗[
= 24,414.063 N/mm = 24,414,063 N/m
𝑚
5000
𝐾
24,414,063
Tn = 2𝜋√ = 2𝜋√
= 0.09 sec
Ʒ = 0.05
Using the response spectra as plotted in problem 2.5 and presented in Figure 21
and
Figure 22
at time step = 0.05 sec and with ground acceleration scaled to
0.5g:
A = 580 cm/sec2 = 5.8 m/sec2
D = 0.00125 m
Base shear fs0 = mA = 29 KN
Design moment Mb0 = m*A*h = h*fs0 = 116 KN.m
21
Figure 21
Figure 22
22
(b) If the cross section of the beam is smaller than that of the columns, the
stiffness of the SDOF system may be approximated by (Equation 1.3.3)
K=
254
∗104 ]
12
[64∗109 ]
6∗[2∗105 ]∗[
= 6,103.516 N/mm = 6,103,516 N/m
𝑚
5000
𝐾
6,103,516
Tn = 2𝜋√ = 2𝜋√
= 0.18 sec
Ʒ = 0.05
Using the response spectra as plotted in problem 2.5 and presented in Figure 21
and
Figure 22
at time step = 0.05 sec and with ground acceleration scaled to
0.5g:
A = 8.8 m/sec2
D = 0.0065 m
Base shear fs0 = mA = 44 KN
Design moment Mb0 = m*A*h = h*fs0 = 176 KN.m
2) When the rigidity of the beam increase, the stiffness of the frame increases,
thus deformation response is reduced, and design forces are increased. On the
other hand, as the rigidity of the beam decrease, the stiffness of the frame
decrease, thus deformation response is increased, and design forces are
reduced.
23
References
Dynamics of Structures: Theory and Applications to Earthquak Engineering, by A.K. Chopra, Fourth
Edition, Pearson Prentice Hall, 2014
https://ngawest2.berkeley.edu/
Sigmon, Kermit, and Timothy A. Davis. MATLAB primer. CRC Press, 2004.
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