Structural Engineering DivisionDepartment of Civil Engineering

Indian Institute of Technology MadrasIndian Institute of Technology MadrasChennai, India.Chennai, India.

Prof. A. Meher PrasadProf. A. Meher Prasad

Department of Civil EngineeringDepartment of Civil EngineeringIndian Institute of Technology MadrasIndian Institute of Technology Madras

email: prasadam@iitm.ac.in

( )x t

( ) ( )andy t y t

Base - Excited SDF Linear Systems

The response of the viscously damped linear (system) oscillator shown in the sketch will now be investigated for an excitation of the base.

Displacement of the base at any time ‘t’ will be denoted by y(t) and the associated velocity and acceleration will be denoted by respectively. The exciting motion is considered to be known and unaffected by the motion of the oscillator itself. As before, the absolute displacement of the mass will be denoted by X and associated velocity and acceleration by and Both X and Y are measured from the static equilibrium, and both they and their derivatives are considered to be positive when directed to the right.

P(t)

x(t)

m

k

c

y(t)

( )x t

The equation of motion for the system is obtained as usual by considering the equilibrium of forces acting on the mass. These forces include the spring force, damping force and D’Alembert inertia force all of each are directed to the left. Equilibrium requires that

Equilibrium requires that (B1)

The equation can now be written either in terms of the absolute displacement x, as (B2) or in terms of relative displacement, or spring deformation, u = x - y (B3) (B4)Upon dividing through by ‘m’ and introducing the quantity ‘p’ and , equation (B2) and (B4) can also be written as

(B5)and (B6)

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

)()( tyctykkxxcxm

)(tymkuucum

)(22 22 typypxpxpx

yupupu 22

The choice between Eqn.(B2) and (B4) or between Eqn.(B5) and (B6) in a given problem depends on how the ground motion is specified and what response quantity x or u is interested in.

For example, if the ground motion is specified as an acceleration history and we are interested in resulting spring force, ku the Eqn. (B4) and (B6) would probably be the most convenient.

On the other hand, if we are interested in absolute displacement and both are specified, Eqn. (B2) and (B5) would be the most convenient to use. Clearly, once either x or u has been determined the other may be computed from Eqn. (B3) 

x

( ) ( )andy t y t

 

For undamped systems, c = 0, Eqn. (B5) reduces to,

(B7)

This equation is same as the differential equation governing the motion of a fixed base system subjected to a force for which the associated static displacement

(B8)

Absolute Displacement of Undamped System

)(22 typxpxm

0

0

( ) cos sin ( )sin ( ( )t

o

vx t x pt pt p y p t d

p

)()( tytxst

The solution of equation (B7) can therefore be obtained from the solution for the force-excited system considered previously, simply by replacing in the latter solution xst by y (t).

In terms of Duhamel’s integral, the solution may be written as

(B9)

It follows further that the response spectra for the fixed-base, force-excited systems presented previously can also be interpreted as spectra for the absolute displacement of base-excited systems.

It is only necessary to replace the quantity (xst)o in the expression for the amplification factor by the peak value of the base displacement, yo .

In the other words, the spectral ordinates should be interpreted to be ratio of .The histories of the base motion and the excited force must naturally be the same in the two cases.

ostxx )(max

ostxx )(max

Example: A vehicle, idealized as a SDF undamped system moved at a speed of 20 m/s over an irregular rigid pavement. The shape of the irregularity is a half sine wave and its peak value is. Prior to crossing the irregularity, the vehicle is considered to have no vertical motion. If the natural frequency of the vehicle is f = 2 cps, what would be the maximum vehicle displacement of the vehicle for (a) L=1.5 m (b) L=6 m (c) L=24 m.

As it crosses the irregularity, the vehicle is subjected to a base motion, the displacement history of which is a half sine pulse. The maximum displacement of the mass may then be from the response spectrum presented before.

v

L

yo

Noting that the duration of the pulse, t1, is given by

t1=L/V

and that, V=20.1 m/s, the values of frequency parameter, ft1, for

the three cases are

for (a)

for (b)

for (c)The corresponding values of are , for (a) for (b)

for (c)

maxX

max 00.6X y

max 01.7X y

max 01.1X y

1 0.15ft

1 0.6ft

1 2.4ft

If the irregularity were a full sine wave and L the length of each half wave, the resulting displacements could be determined from the spectrum given before. The result in this case are as follows

 

for (a) L=1.5 m,

for (b) L= 6 m ,

for (c) L=24 m ,

 

max 00.58X y

max 03.23X y

max 01.2X y

The analogy referred to in the preceeding section is valid only for undamped systems. It can be used as an approximation for damped systems only when the damping is small.

However, for the special case of a sinusoidal base excitation;

the RHS of equation (B5) reduces to ,

where is a phase angle defined by ,

Absolute Displacement of Damped System

0( ) siny t y wt

2

2 2 2 20 0

2( ) [sin 2 cos ] 1 4 sinp y y p y t t p y t

p p p

tan 2 2p

In this case, the solution for steady state response may be written by analogy to the solution given by equation 69 for the corresponding force exited system.

It is only necessary to interpret the quantity (xst)0 in the later solution

as,  

This leads to   

2

2 2 20 01 4 1 4y y

p

2 2

2 2 2 20

1 4( )

(1 ) 4

x t

y

The deformation, u , of the base-excited systems can also be obtained from the equivalent force excited, fixed-base system.

Comparison of Eqn. (B4) and Eqn. (B1) the force P(t) for the force excited system can be taken as, P(t)= - my(t). Then the two equations will be similar.

The initial conditions of u for the base excited systems are the same as those on force excited systems, the solutions of the differential equations will also be the same.

The desired deformation, u, will be equal to the displacement X,of the force excited system shown:

Spring deformation of systems subjected to Base excitation

..

- my(t)m

k

c

x

..

In the analysis of fixed base system,extensive use was made of the concept of instantaneous amplification factor,defined as

0

( )( )

( )st

x tIAF

x

It is desirable to evaluate at this stage the counter part of this factor for the base-excited system.

Noting that ,0 0 0

0 2( )st

P m y yx

k k p

where y0, is the absolute maximum acceleration of the base motion,we conclude that,

..st y

)t(up

)x(

)t(x)IAF(

0

2

0

(B12)

.

y.

x

The solution is obtained from the information presented before,making use of Eqn. B12.

For,

Example : Evaluate the deformation of a SDOF undamped system subjected to a rectangular pulse of amplitude and duration t1. Assume that the initial values of y and and of x and are zero.Thus initial values of u and are zero

..

y0

.

u

222 11

0

2

1

tt(psin

ptsin

y

)t(up,tt ..

ptcosy

)t(up,tt ..

1

0

2

1

The acceleration,velocity and displacement histories of the base motion considered in this solution are shown.

This type of base excitation is of interest in the design of equipment in moving vehicles,but is clearly of no interest in the design of structures subjected to ground motions.

.

( )y t

t1

..

( )y t)t(y

t1t1

For an arbitrary base motion, the deformation of an undamped system can be expressed in terms of Duhamel’s integral as follows.

0

0

1( ) cos sin ( )sin ( ( )

t

o

uu t u pt pt y p t d

p p

(B13)

Pseudo - acceleration

The quantity in equation B12 ,which has units of acceleration,will be referred to as the instantaneous pseudo acceleration of the system,and will be denoted by A(t).

)t(up2

)t(up)t(A 2 (B14)

Thus ,equation B12 can also be written as,

..0

0

( ) ( )( )

( )st

x t A tIAF

x y

(B15)

Referring now to Eqn. B1,it can readily be shown that,for undamped systems,the pseudo-acceleration,A(t) is also equal to the absolute acceleration of the mass, .

However, this identity is not valid for damped systems,and A(t) should be looked upon merely as an alternate measure of the spring deformation,u(t).

..

x

Spectral Quantities

The absolute maximum value of u,without regards to sign will be referred to as the spectral value of u and will be denoted by U.

The absolute maximum value of A(t),without regards to sign,will be referred to as the pseudo-acceleration of the system,and will be denoted by A, thus

UpA 2The product of the mass m and the pseudo-acceleration,A represents the maximum spring force, Qmax, indeed

(B16)

mAUmpUm

kmkUQmax 2 (B17)

This may also be viewed as the equivalent lateral static force which produces the same effects as the maximum effects by the ground shaking.

It is sometimes convenient to express Qmax in the form ,

CWQmax (B18)

Where W = mg is the weight of the system. The quantity C is the so called lateral force coefficient, which represents the number of times the system must be capable of supporting its weight in the direction of motion.

From Eqn.B17 and B18 it follows that, C=A/g (B19)

Another useful measure of the maximum deformation, U is the pseudo velocity of the system,defined as, V = p U (B20)

The maximum strain energy stored in the spring can be expressed in terms of V as follows:

Emax = (1/2) (kU) U = (1/2) m(pU)2 = (1/2)mV2 (B21)

Under certain conditions, that we need not go into here,V is identical to ,or approximately equal to the maximum values of the relative velocity of the mass and the bays,U and the two quantities can be used interchangeably.However this is not true in general,and care should be excercised in replacing one for the other.

Deformation spectra

1.Obtained from results already presented

2.Presentation of results in alternate forms

(a) In terms of U

(b) In terms of V

(c) In terms of A

3.Tripartite Logarithmic Plot

extreme right;It approaches U=y0 at extreme left; a value of ..

yA 0

It exhibits a hump on either side of the nearly horizontal central portion;and attains maximum values of U,V and A which may be materially greater than the values of

. ..

0 0 0, ,y y and y

It is assumed that the acceleration trace of the ground motion,and hence the associated velocity and displacement traces, are smooth continuous functions.

The high-frequency limit of the response spectrum for discontinuous acceleration inputs may be significantly higher than the value referred to above,and the information presented should not be applied to such inputs.

The effect of discontinuous acceleration inputs is considered later.

General form of spectrum

Spectral regions

The characteristics of the ground motion which control the deformation of SDF systems are different for different systems and excitations.

The characteristics can be defined by reference to the response spectrum for the particular ground motion under consideration.

Spectra for maximum and minimum accelerations of the mass (undamped elastic systems subjected to a Half cycle Acceleration pulse)

Spectra for maximum and minimum acceleration of the mass(undamped Elastic systems subjected to a versed-sine velocity pulse)

Deformation spectra for undamped elastic systems subjected to a versed-sine velocity pulse

‘B’ Level Earthquake (=10% ; μ=1.0)

Deformation spectrum for undamped Elastic systems subjected to a half-sine acceleration pulse

Logarithmic plot of Deformation Spectra

Advantages:

• The response spectrum can be approximated more readily and accurately in terms of all three quantities rather than in terms of a single quantity and an arithmetic plot.

• In certain regions of the spectrum the spectral deformations can more conveniently be expressed indirectly in terms of V or A rather than directly in terms of U. All these values can be read off directly from the logarithmic plot.

It is convenient to display the spectra or a log-log paper, with the abscissa representing the natural frequency of the system,f, (or some dimensionless measure of it) and the ordinate representing the pseudo velocity ,V (in a dimensional or dimensionless form).

On such a plot ,diagonal lines extending upward from left to right represent constant values of U, and diagonal lines extending downward from left to right represent constant values of A. From a single plot of this type it is thus possible to read the values of all three quantities.

0y0

..

y0

.

yV Log scale

Natural Frequency F (Log scale)

0D0V

0A

Displacement sensitive

Velocity sensitive

Acceleration sensitive

General form of spectrum

Logarithmic plot of Deformation Spectra

Deformation Spectra for Half-cycle Acceleration pulse:

This class of excitation is associated with a finite terminal velocity and with a displacement that increases linearly after the end of the pulse.

Although it is of no interest in study of ground shock and earthquakes ,being the simplest form of acceleration diagram possible ,it is desirable to investigate its effect.

When plotted on a logarithmic paper, the spectrum for the half sine acceleration pulse approaches asymptotically on the left the value.

oV y

This result follows from the following expression presented earlier for fixed base systems subjected to an impulsive force,

max

IX

mp

where 1

0

( )t

I P t dt

Letting and and noting that ( ) ( )P t m y t maxX U1

0

( )t

oy t dt y

we obtain,

oy( This result can also be determined by considering the effect of an instantaneous velocity change, ,i.e. an acceleration pulse of finite magnitude but zero duration. The response of the system in this case is given by,

( ) cos sinoo

uu t u pt pt

p

Considering that the system is initially at rest, we conclude that,

0o o ou u y and

where, ( ) sinoyu t ptp

The maximum value of u(t), without regards to signs, is

oo

yU V y

p

or )

oo

yU V y

p

or

(This result can also be determined by considering the effect of as instantaneous velocity change, i.e an acceleration pulse of finite magnitude but zero duration.the response of the system in this case is given by

ptsinp

uptcosu)t(u

.

0

0

Considering that the system is initially at rest,we conclude that ,

000 0..

yuandu

where, ptsinp

u)t(u

.

0

The maximum value of u(t),without regards to signs,is

00

..

yVorp

yU

0

.

y

Example:

For a SDF undamped system with a natural frequency,f=2cps,evaluate the maximum value of the deformation,U when subjected to the half sine acceleration pulse. Assume that ,t1=0.1sec. Evaluate also the equivalent lateral force coefficient C, and the maximum spring force,Q0

ft1= 2 x 0.1 = 0.2

From the spectrum, .

0V y

0 0.5y g

Alternatively,one can start reading the value from the spectrum proceeding this may, we find that

50

0

.y

A..

..

y

A

0

Therefore..

1 0

2

...

1 00 1

0

2 2 12 0.1 9.81

21 0.1 1

9.81 0.0242 2

242 2 8 0.52

8 0.1 0.5 0.4

0.4

fU f y

U

t yy t gA fVC

g g g g g

Q CW W

Accordingly

0

..

02 2 2 2

10.8 2 0.4

0.4

0.8 0.8 0.5 9.810.024

4 2

gAC

g g

Q W

yAand U m

p p

The value of and as read from the spectrum are

approximate. The exact value of determined is

0.7. This leads to

.

y

V

0

..

y

A

0

..

y

A

0

00.385 0.385 0.025C Q W and U

If the duration of the pulse were f1=0.75sec instead of 0.1sec , the results would be as follows

.therefore,and..ft 5175021

..

0

0

2 2 2

1.5

1.5 0.50.75

0.75

1.5 0.5 9.810.047

4 2

A

y

A gC

g g

Q W

AU m

p

If the duration of the pulse were t1,as in the first case, but the natural frequency of the system were 15cps instead of 2cps, the results would have been as follows: ft1=15 * 0.1=1.5

Therefore,

and

22 2

1.5 0.5 1.5 0.75

0.75

0.75 9.810.00082

4 15

A AC

y g

Q W

AU m

p

• Plot spectra for inputs considered in the illustrative example and compare

• The spectrum for the longer pulse will be shifted upward and to the left by a

factor of 0.75/0.10 = 7.5

Same as in both cases

f

V

For t1=0.75sec

For t1=0.1sec

..

0y

0y

0y

May be determined from the spectrum by interpreting as

When displayed on a logarithmic paper with the ordinate representing V and the abscissa f, this spectrum may be approximated as follows:

max

0st

x A

x y

Design Spectrum

(Log scale)

(Log scale)

1.5=

Refer to spectrum for

Note the following

• At extreme right . Explain why?

• Frequency value behind which is given by ftra= 1.5

• The peak value of A=2x1.6 Explain why.

In general for pulses of the same shape and duration with different peak values

• If duration on materially different

0yA

Deformation Spectra for Half-Cycle Velocity Pulses

0

0yA

0y

20

1

( )j

n

j

A y

be conservative. Improved estimate may be obtained by considering relative durations of the individual pulses and superposing the peak component effects.The peak value of V is about 1.6 yo

It can be shown that the absolute maximum value of the amplification factor for a system subjected to a velocity trace of a given shape is approximately the same as the absolute maximum value of for an acceleration input of the same shape.

This relationship is exact when the maximum response is attained following application of the pulse. But it is valid approximately even when the peak responses occur in the forced vibration era.

The maximum value of U is yo and the spectrum is bounded on the left by the diagonal line U =yo

0 0A y0 0V y

It should be clear that,

(a) The left-hand, inclined portion of the spectrum to displacement sensitive.

(b) The middle, nearly horizontal region of the spectrum is governed by the peak value of the velocity trace. It is insensitive to the shape of the pulse which can more clearly be seen in the acceleration trace.

(c) The right hand portion is clearly depended on the detailed features of the acceleration trace of the ground motion. In all cases, the limiting value of on the right is equal to These limits appear different in the figure because of the way in which the results have been normalised.

Note that the abscissa is non-dimensionalised and the ordinate with respect to the total duration of the pulse and the ordinate with respect to the maximum ground velocity. It follows that to smaller values of corresponds to larger values of peak acceleration 1 / dt t 0y

Design spectrum for the absolute maximum deformation of systems subjected to a half cycle velocity pulse(undamped elastic systems;continuous input acceleration functions)

Design Rules

Deformation spectra for undamped elastic systems subjected to skewed versed-sine velocity pulses

See spectrum for undamped systems, =0, on the next page

Note that:

(a) The RHS of the spectrum is as would be expected from the remarks

already made.

(b) Peak value of V is approximately 3.2 yo. This would be expected, as the velocity trace of the ground motion, has two identical pulses.

(c) At the extreme left and of the spectrum, U=y0. The system in this case is extremely flexible and the ground displacements is literally absorbed by the spring.

Deformation Spectra for Half-cycle Displacement Pulse

However the spectrum is no longer bounded on the left by the line U= yo, but exhibits a hump with peak value of U0=1.6 y0

It can be shown that the peak value of U/y0 for a system subjected to a displacement trace is approximately the same as the peak value of V/y0, induced by a velocity input of the same shape.

Further more the peak value of U occurs at the same value of the dimensionless frequency parameter,f1 as the peak value of V. However it is necessary to interpret t1 as the duration of the displacement pulse, rather than of that of velocity pulse.

Design Rules

Design spectrum for maximum deformation of systems subjected to a half cycle displacement pulse

Deformation spectra for damped elastic systems subjected to a half cycle displacement pulse

Deformation spectra for full cycle Displacement pulse

The spectra on the next page are for the following full cycle displacement pulse :

t

yo

yo

t

0.618yo

yo

.

.

t

0.618yo

0.618yo

..

..

yo

..

yo

..

0.94f1f1 0.94f1

y..

y.

y

(a) The part on the extreme left for which U=yo .This corresponds to the first maximum,which occurs at approximately the instant that y attains its first maximum.

(b) The smooth transition curve which defines the second maximum. This maximum occurs approximately at the instant that y(t) attains its second extremum, and is numerously greater than the peak value of the second pulse of the contribution of the first pulse.

(c) The hump on the left, which corresponds to the maximum that occurs after termination of the pulse

As would be expected ,the maximum value of U in this case is approximately 3.2 yo .Furthermore, the left hand portion of the spectrum consists of three rather than two distinct parts:

Deformation spectra for elastic systems with viscous damping

• General form of spectrum is as shown in next slide

(a) It approaches V= y0 at the extreme left; value of at the

extreme right ; it exhibits a hump on either side of the nearly

horizontal central portion; and attains maximum values of U, V and

A, which may be materially greater than the values of

respectively.

0A y

0 0 0,y y and y

(b) It is assumed that the acceleration force of the ground motion,

and hence the associated velocity and displacement

forces, are smooth continuous functions.

Generation of results

(c) The high frequency limit of the response spectrum for discontinuous

acceleration inputs may be significantly higher than the value referred

to above, and the information presented should not be applied to such

inputs.

The effect of discontinuous acceleration input is considered later .

General form of spectrum

Acceleration spectra for elastic system - El Centro Earthquake

SDF systems with 10% damping subjected to El centro record

Building Code

Natural period,secs

Bas

e sh

ear

coef

ficie

nt, C

The characteristics of the ground motion which control the deformation of SOF systems are different for different systems and excitations. The characteristics can be defined by reference to the response spectrum for the particular ground motion under consideration .

Spectral Regions

Systems the natural frequency of which corresponds to the Inclined left-hand portion of the spectrum are defined as low-frequency systems :

systems with natural frequencies corresponding to the nearly horizontal control region will be referred to as a medium-frequency systems ; and

systems with natural frequencies corresponding to the inclined right handed portion will be referred to as high-frequency systems.

Minor differences in these characteristics may have a significant effect on

the magnitude of the deformation induced.

Low frequency systems are displacement sensitive in the sense that their

maximum deformation is controlled by the characteristics of the

displacement trace of the ground motion and are insensitive to the

characteristics of an associated velocity and displacement trace:

Ground motions with significantly different acceleration and velocity traces out comparable displacement traces induce comparable maximum

deformations in such systems.

• The boundaries of the various frequency regions are different for different excitations and, for an excitation of a particular form, they are a function of the duration of the motion. It

follows that a system of a given natural frequency may be displacement sensitive, velocity sensitive or acceleration sensitive depending on the characteristics of the excitation to which it is subjected

Effect of Discontinuous Acceleration Pulses

The high frequency limit of the deformation spectrum is sensitive to whether the acceleration force of the ground motion is a continuous or discontinuous diagram. The limiting value given priority applies only to continuous functions

The sensitivity of the high-frequency region to the detailed characteristics of the acceleration input may be appreciated by reference to the spectra given in the following these pages.

0A y

These spectra provide further confirmation to the statement made

previously to the effect that low-frequency and medium-frequency

systems are insensitive to the characteristics of the acceleration force

of the ground motion.

Deformation spectra for damped elastic systems subjected to a full cycle displacement pulse

Compound PulsesEarthquake RecordsEureka recordEl-centro record

Design Spectrum

Minimum number of parameters required to characterize the design ground motionMax values ofThe predominant frequency (or deviation) of the dominant pulses in

The degree of periodicity for (the number of dominant pulses in) each diagram.Dependences of these characteristics on

Local soil conditionsEpicentral distance andSeverity of ground shaking

, and y y y

, and y y y

Application to Complex Ground Motions

Effect of damping:

• Effect is different in different frequency ranges

• Effect is negligible in the extremely low frequency regime (U=y0) and extreme high frequency ranges (A = y0).

u+ p2u = y0(t)

low frequency u=y(t) u0 =y0

high frequency p2u=A(t)=y(t) A =y0

....

....

..

..

Eureka, California earthquake of Dec 21,1954 S 11o E component.

Eureka Quake

Elcentro ,California Earthquake of May 18,1940,N-S component

Undamped Natural Frequency, f, cps

V

p

seud

o ve

loci

tyY

c M

axim

um G

roun

d V

eloc

ity=

Further discussion of Design Response Spectra

The specification of the design spectrum by the procedure that has been described involves the following basic steps:

1. Estimating the maximum values of the ground acceleration, ground velocity and ground displacement. The relationship between y0, y0, y0 is normally based on a statistical study of existing earthquake records. In the Newmark – Blume – Kapur paper (“Seismic Design spectra for Nuclear Power Plants”, Jr. of Power Division, ASCE, Nov 1973, pp 287-303) the following relationship is used.

0.3 : 7.32 m/sec : 1g for rock

0.9 : 14.6 m/sec : 1g for

. ..

2. Estimating the maximum spectral amplification factors, αD, αV, αA ; for the various parts of the spectrum.

Again these may be based on statistical studies of the respective spectra corresponding to existing earthquake records.

The results will be a function not only of the damping forces of the system but also of the cumulative probability level considered.

Damping %critical

One sigma (84.1%) Median (50%)

αD αV αA αD αV αA

0.5 3.04 3.84 5.10 2.01 2.59 3.65

1 2.73 3.38 4.38 1.82 2.31 3.21

2 2.42 2.92 3.66 1.63 2.03 2.74

3 2.24 2.64 3.24 1.52 1.86 2.46

5 2.01 2.30 2.71 1.39 1.65 2.12

7 1.85 2.08 2.36 1.29 1.51 1.89

10 1.69 1.84 1.99 1.20 1.37 1.64

20 1.38 1.37 1.26 1.01 1.08 1.17

Following are the values proposed in a recent unpublished paper by Newmark & Hall for horizontal motions:

Example: Determine the response spectrum for a design earthquake with . Take use the amplification factors given in the preceding page. Take the knee of amplified constant acceleration part of the spectrum at 8 cps and the point beyond which at 25 cps

Note: In the spectra recommended in the Newmark-Blume-Kapur paper, the line de slope upward to the left and the line of slopes further downward to the right.

0A y

00.3 30 c / sec and 25 cey y m y m 0.05

Y=0.00127A=0.3gC=0.3Q=0.3N

yo =30cm/s

30 x2.3 = 69cm/s

30

x2.3

= 6

9cm

/s

30

8cps