Modal Time History Analysis by Mostafa Tazarv

8/10/2019 Modal Time History Analysis by Mostafa Tazarv

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Linear Time History Analysisof MDOF Structure by ModeSuperposition MethodusingNewmarks Method

Carleton University

Mostafa Tazarv

Graduate StudentVersion 1.0


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ModalTimeHistoryAnalysis MostafaTazarv

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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 Nemarks method and Wilson- method. Modal response should be assembled toobtain 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.

Time History Response by Newmark Method: NM

I tried to write a self-explanatory m-file. Therefore, I just copy 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 100100). However, you want to do themodal 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 twocomponents where Fis spatial distribution of load on DOFs andf(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 () and

eigenvaluses ( in which frequencies are sorted from smallest to greatest ( f1< f2< f3


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function [disp,EF]=NM(n,nom,dt,F,ft,M,K,zet,omega2,phi)

INPUTS:

% n: Number of Stories or generally, Number fo DOFs

% nom: Number of Modes that you want to consider in analysis; (nom


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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 Fig. 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

)/(

0.50.5-000000

0.5-1.00.5-00000

00.5-1.00.5-0000

000.5-1.00.5-000

0000.5-1.00.5-00

00000.5-1.00.50-0

000000.5-1.51.0-

0000001.0-2.0

109 mNK

=

)/.(

40000000

04000000

00400000

00080000

00008000

0000080000000080

00000008

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.


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Figure 2- Mode Shapes of First Two Modes of Shear Building (Normalized to Mass)

=

0.24290.2132-

0.18490.2062-

0.08280.1923-

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


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ModalTi

As m

2Y nn +&&

half-sine

is illustr

follows:functi

Displace

that disp

shown t

eHistoryA

entioned2YY nnnn +

&

wave with

ted in Fig

on [dis

ment and el

lacement a

em in mm a

Figure

alysis

before, t

)(.. tfFTn=

he period o

re 3. All r

,EF]=NM

astic force t

d elastic fo

nd kN.

- Decomposi

o solve

, load shou

f 4.93 a

quired data

8,2,0.0

ime history

ce respons

g of Loads i

equation

d be in the

nd 0

has been

5,F,ft,

of some st

will be in

to Two Com

of m

form of

0 0 75

rovided so

,K,0,om

ries are sho

m and N, r

onents (dt fo

tion of

.

00 0 0 0

far to do

ga2,phi

wn in Figu

spectively.

r f(t) is 0.005

MostafaT

6|P

each

. Here,

(kN). This

odal analy

es 4 and 5.

However, I

sec)

azarv

a g e

ode,

is a

load

is as

Note

have


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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


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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, 0.17106 mm4 and 200103 MPa, respectively. Element self-weight and rotational

mass of nodes are neglected. Lumped mass matrix was derived by applied load in the dimension

of 88 (gravity and lateral direction) (Table 1). Global stiffness matrix will be 1212 (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 88 (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


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Table 2- Condensed Stiffness Matrix without P-Effect 88 (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 ()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+9.4734e0000000

005+6.5069e000000

005+4.7739e00000

005+2.046e0000

005+1.3905e000

204.4400

79.0140

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 since we want to use only these

modes.

Sym.


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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,

whereis ground acceleration normalized to g.is plotted in figure 8. .. 1 1 1 12001 1 1 1 1 1 1 1

Figure 8- Ground Acceleration of Elcentro N-S Component in g (f(t))


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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

pl ot ( el cent r o( : , 1) , di sp( : , 3) *1000) ; % Change uni t f r om m t o mmpl ot ( el centr o( : , 1) , EF( : , 3) ) ;

Reference:

1. Humar J. L., Dynamic of Structures, Prentice Hall, 1990

2.

Chopra A., Dynamic of Structures, Prentice Hall, 19953. MATLAB, The MathWorks Inc., 2009

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Modal Time History Analysis by Mostafa Tazarv