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Linear Time History Analysis
of MDOF Structure by Mode
Superposition Method using Newmark’s β Method
Carleton University
© Mostafa Tazarv
Graduate Student
Version 1.1 (April 2011)
Cite as: Tazarv, M., “Linear Time History Analysis of MDOF Structures by
Mode Superposition Method”, available at: http://alum.sharif.edu/~tazarv/
Modal Time History Analysis © Mostafa Tazarv
2 | P a g e
Introduction
Structural dynamic is a mandatory graduate level course for structural/earthquake engineering
student all around the world. One of the most important topics of this course is to solve modal
equations of motion of a Multi Degree of Freedom (MDOF) structure by a numerical method
such as Nemark’s β method and Wilson-Ѳ method. Modal response should be assembled to
obtain each DOF response. Applied load can be a base excitation earthquake or time-dependent
loads on stories.
Here, I will introduce a MATLAB function which can do a time history analysis of an n-DOF
structure with a certain Number of Modes (nom). Then, I will show how to use this function with
two examples, one excited by half-cycle harmonic on two stories and another excited by Elcentro
earthquake. This function will also be verified by an example detailed in “Dynamic of Structure”
by Chopra.
Time History Response by Newmark Method: NM
I tried to write a self-explanatory m-file. Therefore, I’ll explain a bit more the important part of
the function called “NM” in this section. A new feature is to give you the option to specify the
number of modes you want to consider in mode superposition analysis. For example, there is a
100-DOF structure (the size of mass and stiffness matrixes are 100�100). However, you want to
do the modal analysis only for first 10 modes not all the modes which is very common in real
situation.
In this function to solve equation of motions for different modes, Newmark Linear Method has
been used. We can decompose time-dependent applied force, ���� � �. ���, into two
components where F is spatial distribution of load on DOFs and f(t) is time-variant component of
load. By modifying inputs F (a vector) and f(t), you can analyze the structure for either
earthquake (base-seismic-excitation) or time-dependent load applied to different stories. I will
show it in examples later. That's your responsibility to organize eigenvectors (Φ) and
eigenvaluses (�� in which frequencies are sorted from smallest to greatest ( f1< f2< f3<…). If
you want to use MATLAB built-in function to find Φ (phi) and �(omega2), you should be
careful about abovementioned arrangement.
Preferred unit is SI (N, kg, s). Specifically for this function, stiffness (K) is in N/m, mass (M) is
in kg, force (F) is in N, Gravitational Constant (g) is 9.810 m/s2 and Ground Acceleration (Ag)
should be normalized to g. However, you are free to choose your unit.
Modal Time History Analysis © Mostafa Tazarv
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function [disp,EF]=NM(n,nom,dt,F,ft,M,K,zet,omega2,phi)
% INPUTS:% INPUTS:% INPUTS:% INPUTS:
% nnnn: Number of Stories or generally, Number fo DOFs
% nomnomnomnom: Number of Modes that you want to consider in analysis; (nom <= n)
% dtdtdtdt: Time Interval (Sampling Time) of force or earthquake
% FFFF: Spatial Load Distribution Vector; in the case of base seismic
% exitation F=-g*M*[1 1 1 ... 1]'; (A vector)
% ftftftft: Time-Variant Component of Load (A Vector); in the
% case of base seismic exitation ft=Ag (Ground Acceleration in g)
% MMMM: Mass Matrix of Stories (n x n)
% KKKK: Stiffness Matrix of Stories (n x n)
% zetzetzetzet: Damping Ratio in percent (%); e.g. 5
% omega2omega2omega2omega2: Eigen-Values Matrix (nom x nom); e.g. [phi,omega2]=eig(K,M);
% phiphiphiphi: Eigen-Vectors (nom x nom); e.g. [phi,omega2]=eig(K,M);
% OUTPUTS (Story Response):% OUTPUTS (Story Response):% OUTPUTS (Story Response):% OUTPUTS (Story Response):
% dispdispdispdisp: Displacement Time History of Stories (or DOFs)
% EFEFEFEF: Elastic Force Time History of Stories (or DOFs)
As long as the size of stiffness matrix and mass matrix are equal, there will be no restriction on this
function.
Modal Time History Analysis © Mostafa Tazarv
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Example 1: Time-Dependent Load on a Shear Building with 8-DOF
It is desired to analyze a shear building detailed in Figure 1 with Mode Superposition Method
only with first 2 modes in an undamped condition (zet=0). Load considered for this example is
time-dependent half-sine impulse force on two stories (Figure 3). For a shear frame, it is easy to
derive stiffness and mass matrixes which are shown as follows:
Figure 1- Shear Building
−
−+−
−+
=
nk
k
kkkk
kkk
K
...00
......0
0
00
3
3322
221
,
=
nm
m
m
M
...00
......00
000
000
2
1
)/(
0.5 0.5- 0 0 0 0 0 0
0.5- 1.0 0.5- 0 0 0 0 0
0 0.5- 1.0 0.5- 0 0 0 0
0 0 0.5- 1.0 0.5- 0 0 0
0 0 0 0.5- 1.0 0.5- 0 0
0 0 0 0 0.5- 1.0 0.50- 0
0 0 0 0 0 0.5- 1.5 1.0-
0 0 0 0 0 0 1.0- 2.0
109 mNK
×=
)/.(
4 0 0 0 0 0 0 0
0 4 0 0 0 0 0 0
0 0 4 0 0 0 0 0
0 0 0 8 0 0 0 0
0 0 0 0 8 0 0 0
0 0 0 0 0 8 0 0
0 0 0 0 0 0 8 0
0 0 0 0 0 0 0 8
10 26 msNkgM =
×=
Derived mass and stiffness matrixes can be used as inputs of eigen-problem and modal analysis.
For this section, only 2 modes are desired. Then, eigenvalues and eigenvectors have been
calculated. Figure 2 illustrates modal shape of first two modes normalized to mass.
Modal Time History Analysis © Mostafa Tazarv
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Figure 2- Mode Shapes of First Two Modes of Shear Building (Normalized to Mass)
×=Φ −
0.2429 0.2132-
0.1849 0.2062-
0.0828 0.1923-
0.0391- 0.1720-
0.1423- 0.1403-
0.1776- 0.0994-
0.1280- 0.0518-
0.0727- 0.0264-
103
� � �4.1408 00 29.8429�
NOTE: � should be a square matrix in the size of nom � nom
Modal Time History Analysis © Mostafa Tazarv
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As mentioned before, to solve equation of motion of each mode,
n
T
nnnnnnnMtfFYYY /)(..2 2 ϕωωξ =++ &&& , load should be in the form of ���� � �. ���. Here, ���
is a half-sine wave with the period of 4.93 ��� and � � �0 0 0 � 75 100 0 0 0�� (kN). This
load is illustrated in Figure 3. All required data has been provided so far to do modal analysis as
follows:
function [disp,EF]=NM(8,2,0.005,F,ft,M,K,0,omega2,phi)
Displacement and elastic force time history of some stories are shown in Figures 4 and 5. Note
that displacement and elastic force response will be in m and N, respectively. However, I have
shown them in mm and kN.
Figure 3- Decomposing of Loads into Two Components (dt for f(t) is 0.005 sec)
Modal Time History Analysis © Mostafa Tazarv
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Figure 4- Displacement Time history of Some Stories considering Two Modes
Figure 5- Elastic Force Time history of Some Stories considering Two Modes
Modal Time History Analysis © Mostafa Tazarv
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Example 2: A MDOF Frame Structure under Elcentro Earthquake
A frame structure is shown in Figure 6. It is desired to find lateral displacement and elastic force
time history of node 3 (in X direction) under Elcentro earthquake N-S component (ground
motion record is available online: http://www.vibrationdata.com/elcentro.htm) considering first 3
modes with 5% damping ratio.
Section area, second moment of inertia and elastic modulus of all elements are same which are
650 mm2, 0.17�10
6 mm
4 and 200�10
3 MPa, respectively. Element self-weight and rotational
mass of nodes are neglected. Lumped mass matrix was derived by applied load in the dimension
of 8�8 (gravity and lateral direction) (Table 1). Global stiffness matrix will be 12�12 (12-DOF)
for this 2D frame. By means of static condensation, we can properly eliminate mass-less DOFs
in stiffness matrix which is summarized in Table 2.
Figure 6- Frame Shape and Loading
Table 1- Mass Matrix 8�8 (Unit: kg) Node 2 Node 3 Node 4 Node 5
Ux(r1) Uy(r2) Ux(r3) Uy(r4) Ux(r5) Uy(r6) Ux(r7) Uy(r8)
122.3242 0 0 0 0 0 0 0
0 122.3242 0 0 0 0 0 0
0 0 122.324159 0 0 0 0 0
0 0 0 122.3242 0 0 0 0
0 0 0 0 122.3242 0 0 0
0 0 0 0 0 122.3242 0 0
0 0 0 0 0 0 122.3242 0
0 0 0 0 0 0 0 122.3242
Modal Time History Analysis © Mostafa Tazarv
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Table 2- Condensed Stiffness Matrix without P-∆ Effect 8�8 (Unit: N, mm) Node 2 Node 3 Node 4 Node 5
Ux(r1) Uy(r2) Ux(r3) Uy(r4) Ux(r5) Uy(r6) Ux(r7) Uy(r8)
15333.84 15317.5 -15323.8 -15314.9 -0.32002 -2.90886 0.556449 0.32002
15317.5 58655.22 -15319.4 -15323.4 0.184047 1.672916 -0.32002 -0.18405
-15323.8 -15319.4 58655.22 15317.92 -43333.1 1.672916 -0.32002 -0.18405
-15314.9 -15323.4 15317.92 15329.35 -1.67292 -7.65064 2.90886 1.672916
-0.32002 0.184047 -43333.1 -1.67292 58655.22 -15317.9 -15323.8 15319.41
-2.90886 1.672916 1.672916 -7.65064 -15317.9 15329.35 15314.92 -15323.4
0.556449 -0.32002 -0.32002 2.90886 -15323.8 15314.92 15333.84 -15317.5
0.32002 -0.18405 -0.18405 1.672916 15319.41 -15323.4 -15317.5 58655.22
Note: Change the UNIT to N/m
Table 3- Mode Shape (Φ)
Mode1 Mode2 Mode3 Mode4 Mode5 Mode6 Mode7 Mode8
0.723607 -30.639 -3.42432 4.09675 -1.00044 0.484086 -1.00004 -0.25215
4.35E-05 -0.00632 -0.00081 -2.6451 1.366837 1.877068 -2.19516 -0.40269
1 1 1 1 1 1 1 1
-0.27623 -31.6572 -4.4256 -4.09384 0.999798 -0.48417 1.000152 0.252098
1 0.998295 1.000088 -0.99981 1.000012 -1 1.000012 -1
0.334075 -32.4779 4.254245 -4.09407 -0.99977 -0.48415 -1.00015 0.252112
0.665804 33.45968 -3.25315 -4.09688 -1.00031 -0.4841 -1.00001 0.252139
-3.02E-05 -0.00607 0.000737 -2.6453 -1.36677 1.877095 2.195148 -0.40269
Note: Phi is normalized to be unity in the lateral direction of node 3
� �
005+9.4734e 0 0 0 0 0 0 0
005+6.5069e 0 0 0 0 0 0
005+4.7739e 0 0 0 0 0
005+2.046e 0 0 0 0
005+1.3905e 0 0 0
204.44 0 0
79.014 0
19.586
(rad/s)2
Dynamic properties are shown above. First 4 mode shapes are also plotted in Figure 7. We
will use only first 3 mode shapes and natural frequencies. Note that since stability condition
for numerical solution will not meet in modes above three, we have to consider only three
modes (higher modes have very small periods in this problem).
Sym.
Modal Time History Analysis © Mostafa Tazarv
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Figure 7- Mode Shape of first 4 modes
In the case of earthquake, � � ��. . �1 1 1… �� where g=9.81m/s2. Also, ��� � "#���
where "#��� is ground acceleration normalized to g. ��� is plotted in Figure 8.
� � ��. . �1 1 1… �� � �1200�1 1 1 1 1 1 1 1��
Figure 8- Ground Acceleration of N-S Component of Elcentro Earthquake in g (f(t))
Modal Time History Analysis © Mostafa Tazarv
11 | P a g e
Recall NM Function:
function [disp,EF]=NM(8,3,0.02,F,ft,M,K,5,omega2,phi)
Responses are plotted in Figure 9 for node 3 in X-direction with 5% damping ratio.
Figure 9- Response of Node 3 of Frame Structure in X-Dir. to Elcentro N-S Component
Hints:
plot(elcentro(:,1),disp(:,3)*1000); % Change unit from m to mm plot(elcentro(:,1),EF(:,3));
Modal Time History Analysis © Mostafa Tazarv
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Verification:
In this section accuracy of this function will be investigated by comparing the results with the
example 13.2.6 detailed on page 485 of “Dynamic of Structures, Theory and Application to
Earthquake Engineering” (Chopra, 1995) which is a five-story shear frame.
System properties. The lumped mass mj = m = 100 kips/g at each floor (g=386.22 in/s2), the
lateral stiffness of each story is kj = k = 31.54 kips/in., and the height of each story is 12 ft. The
damping ratio for all natural modes is $%= 5%. The mass matrix M, stiffness matrix K, natural
frequencies, and natural modes (normalized to 1 at roof) of this system are as follow:
)/.(
0.2589 0 0 0 0
0 0.2589 0 0 0
0 0 0.2589 0 0
0 0 0 0.2589 0
0 0 0 0 0.2589
2 inskipsM
=
)/(
31.5400 31.5400- 0 0 0
31.5400- 63.0800 31.5400- 0 0
0 31.5400- 63.0800 31.5400- 0
0 0 31.5400- 63.0800 31.5400-
0 0 0 31.5400- 63.0800
inkipsK
=
=Φ
1.0000 1.0000 1.0000 1.0000 1.0000
2.6825- 1.8308- 0.7154- 0.3097 0.9190
3.5133 0.5211 1.2036- 0.5944- 0.7635
3.2287- 1.3979 0.3728 1.0882- 0.5462
1.9190 1.6825- 1.3097 0.8308- 0.2846
22)/(
448.6144 0 0 0 0
0 344.8605 0 0 0
0 0 208.9717 0 0
0 0 0 84.0919 0
0 0 0 0 9.8694
srad
=ω
For the given K and M, the natural periods are Tn = 2.0, 0.6852, 0.4346, 0.3383, and 0.2966 sec.
Modal Time History Analysis © Mostafa Tazarv
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Chopra used all 5 modes in modal analysis. Excitation is Elcentro earthquake detailed in
example 2 of this script. Therefore, ��� � "#��� and
� � ��. . �1 1 1… �� � �99.9924 � �1 1 1 1 1 ��
Desired data are available now and then we can recall NM
function [disp,EF]=NM(5,5,0.02,F,ft,M,K,5,omega2,phi)
Figure 10 illustrates a comparison of the roof displacement, roof elastic force and base shear with
this function and Chopra’s solution set. Minor differences may refer to the different types of
analytical algorithms as well as accuracy in digits in current MATLAB version (R2009a)
comparing to Chopra’s software about a decade ago.
a) b)
Figure 10- Comparison of Responses between (a) MN Function and (b) Chopra’s Solution (Chopra 1995, pp
485)
Reference
1. Humar J. L., “Dynamic of Structures”, Prentice Hall, 1990
2. Chopra A. K., “Dynamic of Structures, Theory and Application to Earthquake Engineering”, Prentice
Hall, 1995
3. MATLAB, The MathWorks Inc., 2009