Structural Dynamics

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Basic Concepts of Structural Dynamics/ -1

BASIC CONCEPTS OF STRUCTURAL DYNAMICS

D.K. Paul

Emeritus Fellow, Department of Earthquake Engg., IIT Roorkee, Roorkee, [email protected]

INTRODUCTION

Every structure vibrates under external excitation. The response mainly depends on its

mass, stiffness, damping and boundary conditions. All of these parameters can be

expressed by a single parameter frequency ‘ f ’or time period ‘T ’of vibration. A

structure may be idealized into single degree of freedom system (SDOFS) or a multi-

degree of freedom system (MDOFS). These idealized systems can then be analyzed

and its response to various excitations can be evaluated.

SINGLE DEGREE OF FREEDOM SYSTEM (SDOFS)

Figure 1 shows an idealization of a structure into spring-mass-dashpot system, which

can be idealized as a SDOFS.

x

k

Fig.1(a) Single degree of freedom (s.d.o.f) subjected to a force

k c

x m

x

kx

Fig.1(b) – Single Degree Freedom System

Restoring

force

Inertia Force

m x

)(t f

m m

External Applied

Load )(t f

(a) SDOFS (b) Free body diagram

c

xc

Damping

force




If x is the deflection of the mass then the spring/ restoring force will be xk acting

opposite to the motion producing a restoring force = xk . The damping force also acts

opposite to the motion and is assumed to be proportional to velocity of the moving mass

equal to xc , where c is the damping coefficient and x is the velocity of the vibrating

mass.

The inertia forces acting on the mass is the product of mass and absolute acceleration and

acts opposite to the motion.

Inertia force = xmdt

xd m

2

2

(1)

where, x is the absolute acceleration of the mass.

Equilibrium of forces gives the equation of motion of the system as follows:

)(t f xk xc xm (2)

For undamped free vibration, the damping and external force will vanish and the equation

can be expressed as:

0 xk xm (3)

Assuming pt b pt a x cossin as a solution to Eq.(3) gives undamped natural

frequency as:

m

k p rad/ sec or

pT

2 sec or

T f

1 c/ sec (4)

The constants band a are evaluated by applying the initial conditions i.e. at

oo x xand x xt ,0 and the solution can be expressed as

pt p

x pt x x o

o sincos

(5)

The damped free vibration can be obtained from

0 xk xc xm (6)

For an under damped system the solution of Eq.(6) can be expressed as

t pbt pae x t p )1(sin)1(cos 22 (7)




Where, is the damping ratio defined asccc / cc( is the coefficient of critical

damping); Constants a and b are obtained from the initial conditions as explained

above. The above solution is oscillatory and decays exponentially as shown in Fig.2.

Time

R e s p o n s e

Fig. 2 Undamped and damped response of a SDOFS

Table 1 gives the period of vibration of a 6 DOFS system (three translations and three

rotations of the roof slab) shown in Fig.1(a) for the given mass and stiffness.

Table 1-Time periods of vibration of structure (Fig.1(a)) and a comparison with

ETAB

Direction Mass(kN-sec

2/m)

Stiffness(kN/m)

Time period(sec)

Time period(sec)

(ETABS)

Translation in X direction

(horizontal) 6.00 71112.0 0.0600 0.0610

Tortional motion about Z-

axis 58.00 597349.5 0.0620 0.0690


(horizontal) 6.00 40000.0 0.0770 0.0790


(horizontal) 6.00 4000000.0 0.0077 0.0077

RESPONSE TO GROUND MOTION

The equation of motion of single degree of freedom system (SDOFS) subjected to

ground excitation (Fig.3) is expressed as:

k c

x m

x

z y

y

Fig.3 Single degree of freedom (s.d.o.f) subjected to support excitation

)(t x is the absolute motion of the

mass m

)(t z is the relative motion of the

mass m w.r.t. the fixed reference

)(t y is the absolute acceleration

of the ground




mx c x y k x y ( ) ( ) 0

mx c x y k x y ( ) ( ) 0 (8)

or m x y c x y k x y my( ) ( ) ( )

or mz cz kz my

z p z p z y 2 2 (9)

where, z = x - y is the relative motion of mass with respect to ground, x is the absolute

motion of mass with respect to fixed base and y is the absolute acceleration of ground,

m, k and c are the mass, stiffness and damping of the SDOFS system, pk

m is the

undamped natural frequency, c

mp2is the fraction of critical damping or damping

ratio. If the ground excitation is expressed as t y y sin0 then the response is worked

out as

z ty

pt( )

( ) ( )sin( )

0

2 2 2 2

1

1 2 (10)

Defining,2

p y z o st and the frequency ration, p

Therefore, the dynamic amplification w.r.t static deflection can be expressed as:

222 )2()1(

1

st

m

d z

z D (11)

Absolute acceleration of mass is given by

)2( 2

z p z p y z (12)

))(sin()2()1(

)2(10

222

2

0

t y

222

2

)2()1(

)2(1

y

y z Da

(13)

Figures 4 and 5 show the dynamic displacement and acceleration amplifications for

frequency ratios and damping ratios for sinusoidal excitation. Resonance




condition is observed when the excitation frequency matches with the natural

frequency of the system. In an earthquake, the damaging energy is carried by several

frequencies instead of only one frequency and therefore the motion is random in nature.

The response of the system to random excitation can be obtained using Green's

function, the solution of (9) gives,

p

p

d t pe y

p z d

t p

t

d

)(sin)(1 )(

0

(14)

d D

Fig. 4 – Dynamic displacement-magnification factor with damping and

frequency as parameters

D

Fig. 5 – Dynamic acceleration-magnification factor with damping and

frequency as parameters




Let

max

)(

0

* )(sin)(

d t pe y sd

t p

t

v (15)

Then, *

max

1

vd

d

s p

s z (relative max. displacement response) (16)

p p pd 21 (Since << 1) (17)

*

max s s z v (Relative max. velocity response) (18)

*

max va s p s x (19)

The detail of calculations for evaluating sv and sa can be found elsewhere.

d v s p s p (Pseudo-velocity) (20)

d a s p s p 2 (Pseudo-acceleration) (21)

Fig.6(a) Displacement, (b) Velocity and (c) Acceleration response spectra for

an earthquake ground motion

Period (sec)




A plot of maximum response quantity ( either relative displacement, maximum relative

velocity or absolute acceleration) against the period of vibration of structure of SDOFS

is defined as the Response Spectra of that response quantity. It can be obtained directly

using relations (16), (18) and (19). These plots are known as displacement, velocity and

acceleration response spectra respectively. Figure 5 shows the displacement, velocityand acceleration response spectra for El-Centro earthquake motion for 2% damping.

A response spectra vary much depends on the time history characteristics of ground

motion which largely depends upon on the local site condition. Prediction time history

of ground motion at a site is not possible as it is a random phenomenon. Therefore, an

average response spectra depending upon the site soil type is used as Design Response

Spectra as given in IS 1893-2002.

SELECTION AND DESIGN GROUND MOTIONS

Selection of ground motion acceleration time history at a site is required for linear or

nonlinear time history analysis of structure. This can done either modifying the existing

ground motion recorded at similar site condition or artificial simulation of spectra

compatible ground motion.

Modifying Existing Ground Motion

An actual recorded time history on a similar site conditions (near similar tectonic,

geological conditions, magnitude, epicentral distance and local site conditions) is usedfor structural analysis by modifying the record in two ways, such that, (i) the

predominant ground period of the recorded ground motion is matched with the

expected predominant ground motion of the site and (ii) the ground parameter (either

PGA, PGV, PGD or spectral ordinates) of the recorded ground motion is either scaled

up or scaled down depending upon the estimated ground parameter at the site. Where,

it is not possible to select actual recorded ground motions, spectra compatible ground

motion can be generated for analysis.

Spectrum Compatible Ground Motions

The ground acceleration time history is used as base excitation for dynamic analysis.

Spectrum compatible time histories can also be generated. A procedure for generation of

IS: 1893 (Part l)-2002 spectra compatible ground motion corresponding to hard soil

condition and Zone IV (0.24g) is presented. The WAVEGEN code is used for generation

of spectra compatible time histories. The computer code is based on the wavelet-based

algorithm presented by Mukherjee and Gupta (2002). The spectrum compatible time

histories are generated from the real earthquake, recorded at the stations having geology

similar to site condition. The five ground motions considered are from five earthquakes-

Northridge, El Centro, Uttarkashi, Kobe, and Chamoli. The compatible time history forone such Northridge earthquake is shown in Fig.7.




Northridge earthquake

time (sec)0 5 10 15 20

accelera

tion

(g)

-0.2

-0.10.0

0.1

0.2

Fig. 7 - Spectrum compatible ground acieration time history

A comparison of response spectra of five specified earthquakes with IS 1893:2002 (Part

1) with code based target design spectrum is shown Fig.7.

Fig 8 Comparison of response spectra

MULTI-DEGREE FREEDOM SYSTEM

Figure 8 shows a 2D frame of a four story building. The floors are considered rigid

and base is considered fixed.

For multi-degree freedom system, the equation of motion can be expressed as:

y I M z K z C z M (22)




where, K C M ,, are the mass, damping and stiffness matrices of the structure and

z is the vector of independent relative displacements. The undamped free vibration

of multi degree freedom system can be expressed as

4

f

4 x

3 f

3 x

2 f

2 x

1 f

1 x

y

/////////////////////////////////////////////////////////

Fig.9 Four storey building and deflected shape

0 z K z M (23)

Assume relative displacement vector )]([ y x z to be harmonic such as

ipt ea z (24)

where, a is constant and is the mode shape vector and is given by:

T n

T 21, (25)

Substitute (24) in (23),

02 K M p

Various forms of above equation are as follows:

or M p K 2 …(26)

or 21 p K M

or 2 p D …(27)

z




or 2

1 1

p M K

21

1

p

D …(28)

Both are of the form

A …(29)

Which represent an Eigen value problem. Since the solution converges to highest value of

and eigen vector which is the mode shape or the characteristic shape. The matrix

iteration of (29) will converge to the highest value of 2 p or the highest value of

frequency of vibration.

Thus when the mass moves in a vertical direction, measurement of the static deflection

st enable us to compute the period and frequency of vibration of the system. It is not

necessary that we know the mass m or the spring constant k .

We seek solution of

I p K M 21

… (30)

021 I p K M … (31)

For nontrivial solution,

021 I p K M (32)

Let z (33)

orr

r

i

n

r i

z )(

1

…(34)

is the normal or principle coordinates.

yC y

m

m

p r r

j j

m

j

r

j j

n

j

r r r

2)(

1

)(

12

(35)




d t p y

pm

m

r

t

or r

j j

n

j

r

j j

m

i

r

sin1

2)(

1

)(

1 (36)

d t p y p

m

m

z r

t

or r

j j

n

j

r

j j

n

j

r

in

r i

sin

1

2)(

1

)(

1

1

(37)

r d r

r r

vr

r

i

r

r

i S C S C

p z )(

1

)(

max

)( 1 … (38)

r

ar

r

i

r

i

S C x )(

max

)( .. .(39)

///////////////////////////////

Fig. 10 - Mode shapes

1 z

2 z

3 z

1t 2t 3t

Fig. 11 Combination of modal response




Figure 10 shows the modes shape of vibration and Fig. 11 shows the combination of modal

response. Only first few modes are significant and contribute to the total response. Higher

modes response is small. This mode superposition is only applicable to linear problems.

For nonlinear problem the response is obtained by direct time integration.

REFEENCES

Dowric, David (2011), Earthquake Resistant Design and Risk Reduction, John Wiley &

Sons, Inc., UK

Kramer, S.L. (2008), Geotechnical Earthquake Engineering, Published by Dorling

Kindersley (India) Pvt. Ltd.

Mukherjee, S. & Gupta, V. K. (2004). “Wavelet-based generation of spectrum-

compatible time-histories”. Soil Dynamics and Earthquake Engineering 22 (2002)

799 – 804

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