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CHAPTER 6 EARTHQUAKE RESPONSE OF LINEAR SYSTEMS

One of the most important applications of theory of

structural dynamics is in analyzing the response of structures to ground shaking caused by an earthquake. This chapter deals with linear systems, which are elastic systems, so we will refer to them by linearly elastic systems.

Earthquake Excitation

For engineering, the time variation of ground acceleration is the most useful way of defining the shaking of the ground during an earthquake. Ground acceleration appears on the right hand side of equation of motion when a SDF system is subjected to ground excitation.

A basic instrument, called strong-motion accelerograph, records three components of motions including two horizontal components and one vertical component. Acceleration is vector quantity in 3-D space that varies with time. It requires three components to define a vector in 3-D space.

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Instrument does not record continuously all the time. It is triggered to start recording when the first waves of earthquake arrive because there may not be any strong ground motion for months.

Basic element of an accelerograph is a transducer

element, which is an SDF mass-spring-damper system. It is characterized by its natural frequency nf and viscous damping ratio ζ ; typically nf =25 Hz and ζ =60% for modern analog accelerograph and nf =50 Hz and ζ =70% for modern digital accelerograph. These transducer parameters allow the instruments to record ground acceleration containing frequency up to 60% of nf without excessive distortion.

Before 1990s, strong recorded ground motions were very

rare because numbers of instruments were limited and they were not located near the origin of earthquakes. After 1994 Northridge, California; 1995 Kobe, Japan; 1999 Turkey and Taiwan earthquakes, hundreds of strong motion records became available.

Earthquake ground motions are irregular in nature and

they vary widely in terms of amplitude, duration, frequency content, and wave form. The peak acceleration can be more than 1g and duration of the strong phase may be as short as a few seconds or as long as a few minutes.

The factors that affect characteristics of ground motion at

a location are 1) Source (magnitude, fault mechanism) 2) Path (distance from epicenter, geology, direction) 3) Site (soil condition at the location considered)

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El Centro Ground Motion

The earthquake ground motion in the north-south component recorded at the El Centro station during Imperial Valley earthquake on May 18, 1940 will be used for discussion in this chapter.

Acceleration is specified at very small time interval

(0.005-0.02 sec) to capture the highly irregular variation with time. The velocity and displacement time history were obtained by integrating the acceleration time history.

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Equation of Motion For a linear SDF system subjected to ground motion excitation ( )gu t , the motion of the mass is governed by the equation

( )gmu cu ku mu t+ + = −

Divide by m on both sides

( )22 n n gu u u u tζω ω+ + = −

For a given ground acceleration ( )gu t , the response ( )u t depends on the natural frequency nω (or period nT ) and damping ratio ζ of the system. Thus, two systems with the same nT and ζ will have the same response ( )u t even if one of them has more mass and is stiffer than the other.

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Response History

For a given ground motion ( )gu t , the deformation response of an SDF system depends on the natural frequency and damping ratio. The response ( )u t of an SDF system can be determined from numerical procedure discussed in the previous chapter.

Observe the responses ( )u t of three SDF systems with

the same damping ratio 2%, but different natural period of vibration. The time required for an SDF system to complete a cycle of vibration when subjected to earthquake ground motion is very close to its natural period of vibration.

For these cases, the one with longest natural period has

the largest peak deformation. This trend is not necessarily true over the entire range of periods.

Compare another three SDF systems, all with the same

natural period of vibration of 2 sec but different damping ratio of 0, 2, and 5%. The time required for an SDF system to complete a cycle of vibration is similar for all three cases, but the amplitude always decreases as damping ratio increases. This trend is consistent with the study of response to harmonic and pulse excitations.

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Once the deformation response history has been

evaluated by dynamic analysis of structure, the internal forces can be determined by static analysis of structure at each time instant. By the concept of equivalent static force Sf , which is related to the earthquake force specified in building code,

( )Sf ku t=

where k is the lateral stiffness of the frame. 2nk mω=

( ) ( )2S nf m u t mA tω= =

where ( ) ( )2

nA t u tω=

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( )A t is called pseudo-acceleration. The equivalent static force equal mass time pseudo-acceleration, not the total acceleration ( )tu t . ( )A t is obtained by multiplying ( )u t by

( )22 2 /n nTω π= .

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Internal Forces

For a one-story frame, the internal force can be determined at any time instant by static analysis of the structure subjected to the equivalent static lateral force ( )Sf t at the same time instant.

Base shear ( ) ( ) ( )b SV t f t mA t= = Overturning moment ( ) ( ) ( )b S bM t hf t hV t= = where h is the height of the structure.

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Response Spectrum Concept First introduced by M. A. Biot (1932), the response

spectrum method is a central concept in earthquake engineering. It provides a convenient mean to summarize the peak response of all possible SDF systems to a particular component of ground motion.

A plot of peak value of a response quantity as a function

the natural vibration period nT or frequency nω or cyclic frequency nf is called “response spectrum.” Each plot is for a fixed damping ratio ζ . Engineers prefer to use nT rather than

nω because nT is more familiar. A variety of response quantity can be defined.

Deformation response spectrum ( ) ( ), max , ,o n ntu T u t Tζ ζ≡

Relative velocity response spectrum ( ) ( ), max , ,o n ntu T u t Tζ ζ≡

Acceleration response spectrum ( ) ( ), max , ,t to n nt

u T u t Tζ ζ≡

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Deformation, Pseudo-Velocity, and Pseudo-Acceleration Response Spectra

Previously we learned that only deformation ( )u t is needed to compute internal forces. Pseudo-velocity and pseudo-acceleration are discussed because they are useful in studying characteristic of response spectra, constructing design spectra, and relating structural dynamics to building codes. Deformation response spectra

The peak value of deformation time history response of an SDF system with natural period nT due to a particular ground excitation can be plotted as a point on the deformation response spectra. The peak value of deformation is denoted by

( )maxo t

D u u t= = If many of such analyses are repeated for many SDF

systems with a fixed damping ratio ζ but different natural periods nT , the deformation response spectra can be constructed for the range of nT considered. Similar spectrum for other value of damping ratio can be constructed in a similar manner.

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Pseudo-velocity response spectrum

Consider a quantity V for an SDF system with natural frequency nω which has the peak deformation D

2n

n

V D DTπ

ω= =

The quantity V has units of velocity. It is related to the peak value of strain energy soE by

2

2somVE =

derived from ( )22 2 2/

2 2 2 2no

so

k Vku kD mVEω

= = = =

V is called the peak relative pseudo-velocity or peak pseudo-velocity. It is not the same as peak relative velocity ou . Pseudo-velocity response spectrum is a plot of V as a function of the natural period nT or natural frequency nω of the system.

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Pseudo-acceleration response spectrum

22 2n

n

A D DTπω

⎛ ⎞= = ⎜ ⎟

⎝ ⎠

The quantity A has units of acceleration and is related to

the peak value of base shear boV as

bo SoAV f mA wg

= = =

where w is the weight of the structure and g is the gravitational acceleration.

/A g may be interpreted as the base shear coefficient or lateral force coefficient. It is used in building codes to represent the coefficient that is multiplied to the weight to obtain base shear force.

The pseudo-acceleration A is different from the total acceleration tu so it is called “pseudo.”

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Combined D-V-A spectrum

Because the three quantities are related through

nn

A V Dωω

= = or 22

n

n

TA V D

Tπ

π= =

They can be combined in one plot and three different quantities can be read from three different axes. Such a four-way plot in logarithmic scale is called the “tripartite plot.”

V can be read from the vertical axis, while A and D can be read from diagonal axes.

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Combined D-V-A response spectra for El Centro ground motion are plotted for damping ratio = 0, 2, 5, 10, and 20% to cover the range of damping ratio in practically real structures. The response spectra are plotted for a wide range of natural period nT from 0.02 to 50 sec.

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Peak structural response from the response spectrum If the response spectrum is available for a given ground motion, the peak value of deformation and internal forces can be readily determined.

2 2n n

oT T

u D V Aπ π

2⎛ ⎞= = = ⎜ ⎟⎝ ⎠

Sof kD mA= =

No further dynamic is required. Only static analysis of structure subjected to the equivalent static force provides the peak values of internal forces during response to the given earthquake.

boV kD mA= =

bo boM hV=

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Response Spectrum Characteristics

We now study the important characteristic of response spectrum of earthquake ground motions. The response spectrum for north-south component of El Centro ground motion for damping ratio = 0, 2, 5, and 10% is plotted together with the ground motion parameters gou , gou , and gou .

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To show the relationship between response spectrum and ground motion parameters gou , gou , and gou , the response spectrum for El Centro ground motion is plotted in term of normalized response quantities / goD u , / goV u , and / goA u .

And the curve for damping ratio=5% only is shown together with the idealized smooth multi-linear curve, which can be separated into 7 parts between points a, b, c, d, e, and f.

, , , , , anda b c d e fT T T T T T may be different for other ground motions

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• For systems with very short period, n aT T< =0.035 sec, A

approaches gou for all damping ratio and D is very small. • This trend is understood based on physical reasoning that a

very short period system is very rigid and moves together with the ground.

• From the response history of an SDF system with 0.02secnT = due to El Centro ground motion, we can

observe that the deformation is very small and total acceleration is approximately equal to the ground acceleration and equal to negative of pseudo-acceleration.

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From equation of motion, because the system is very rigid, 0u → and 0u → u 2 nuζω+ ( )2

n gu u tω+ = − ( ) ( )gA t u t= − and ( ) ( ) ( )t

gu t u t u t= + ( )gu t • For very long period system, 15secn fT T> = , D approached

gou for all damping ratio and A is very small. • This case is concerned with very flexible systems where the

mass essentially stays still while the ground at the base moves rapidly.

• The deformation equal to ground displacement with opposite sign ( ) ( )gu t u t= − . The total acceleration is close to zero 0tu .

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Therefore, from above observation, a response spectrum can be divided into three regions: 1) For the very short period region, n aT T< , A is equal to gou .

For b n cT T T< < , A may be idealized as constant at a value equal to gou amplified by a factor depending on ζ . This region ( n cT T< ) is called “acceleration-sensitive region.”

2) For the very long period region, n dT T> , D equals to gou . For d n eT T T< < , D may be idealized as constant at a value equal to gou amplified by a factor depending on ζ . This region ( n dT T> ) is called “displacement-sensitive region.”

3) For intermediate region, c n dT T T< < , V may be idealized as constant at a value equal to gou amplified by a factor depending on ζ . This region is called “velocity-sensitive region.”

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Effect of Damping on Response Spectrum • The effects of damping vary with spectral regions.

Damping is most influential in the velocity-sensitive region.

• Effect of damping is stronger for smaller damping values.

• If damping ratio increases from 0 to 2%, reduction of response is greater than when damping ratio increases from 10 to12%.

• If the ground motion is nearly harmonic, e.g. Mexico city 1985, the effect of damping will be larger than short duration ground motion like pulse.

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Elastic Design Spectrum

Ground motions at the same location during different earthquake events can be significantly different.

Factors includes magnitude of earthquake, fault mechanism, distance of site from earthquake fault, geology and travel path of seismic waves, and the local soil conditions.

The response spectrum to be used for structural design

should not be for a particular ground motion but rather represent possible ground motions based on statistics of many ground motions.

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The average of many response spectra is the “mean response spectrum,” whereas the mean-plus-one-standard-deviation response spectrum can also be evaluated.

These response spectra are smoother than a response spectrum for an individual ground motion.

Researchers have developed procedures to construct a smooth design spectrum using ground motion parameters gou ,

gou , and gou , and amplification factors , , andA V Dα α α in different spectral regions based on statistics.

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Comparison of Design and Response Spectra