Module 6 (RESPONSE OF LINEAR ELASTIC S.D.O.F SYSTEMS TO EARTHQUAKE LOADING)

CE-409: Introduction to Structural Dynamics and Earthquake Engineering

MODULE 6

Prof. Dr. Akhtar Naeem Khan & Prof. Dr. Mohammad Javed [email protected] [email protected]

1

University of Engineering & Technology, Peshawar, Pakistan

RESPONSE OF LINEAR ELASTIC S.D.O.F SYSTEMS TO

EARTHQUAKE LOADING

CE-409: MODULE 6 (Fall-2013) 2

Earthquake Response of Linear System

In this lecture, we will study the earthquake response of linear SDOF systems subjected to earthquake excitations.

By definition, linear systems are elastic systems.

They are also referred to as linearly elastic systems to emphasize both properties.

fs fs

u

Linear elastic system

Non-linear elastic system

No energy is absorbed by systems

u

Elastic-perfectly plastic system (Elasto plastic system)

Non-linear inelastic system

Area enclosed by the curve = Energy absorbed by system

CE-409: MODULE 6 (Fall-2013) 33333

Consider a single story frame with lumped mass. Let the frame at the base displaces by an amount ug due to seismic waves. As a result

lumped mass at the top displaces by an amount ut ,such that:

Effective Earthquake Force

Where ug= Ground displacement. ut=Total displacement at the top end and u = Dynamic displacement of lumped mass at the top w.r.t shifted base.

gt uuu +=


The equation of motion for the frame subjected to the earthquake excitation can be derived by using the using dynamic equilibrium of forces as:

0fff SDI =++



Only the relative motion u between the mass and the base cause structural deformation which produces elastic and damping forces.

Thus for a linear system the inertial force fI is related to the acceleration of the mass by:

tu

kuf and ucf

;umf

s

t

D

I

===



By substituting the value of fI , the equation of motion become:

0kuucum t =++ 0kuuc)uum( g =+++ or

(t)umkuucum g −=++or

p(t)kuucum =++ Comparing with

(t)um(t)pp(t) geff −==


The term on the right-hand side of the equation may be regarded asthe Effective earthquake force.


Thus the ground motion can be replaced by the effective earthquake force (indicated by the subscript “eff”. Since this force is proportional to the mass, thus, by increasing the mass the structural designer increases the effective earthquake force


Effective earthquake force: horizontal ground motion

Base moving with

(t)um)t(p geff =

(t)ug


Strong motion record

Typical strong motion record


Strong motion recordStrong earthquakes can generally be classified into three groups:

1. Practically a single shock: Acceleration, velocity, and displacement records for one such motion are shown in figure. A motion of this type occurs only at short distances from the epicenter, only on firm ground, and only for shallow earthquakes.

CE-409: MODULE 6 (Fall-2013) 10

Strong motion recordA moderately long, extremely irregular motion : The record of the earthquake of El Centro, California in 1940, NS component exemplifies this type of motion. It is associated with moderate distances from the focus and occurs only on firm ground. On such ground, almost all the major earthquakes originating along the Circumpacific Belt are of this type.

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Strong motion recordA long ground motion exhibiting pronounced prevailing periods of vibration: A portion of the accelerogram obtained during the earthquake of 1989 in Loma Prieta is shown in figure to illustrate this type. Such motions result from the filtering of earthquakes of the preceding types through layers of soft soil within the range of linear or almost linear soil behavior and from the successive wave reflections at the interfaces of these layers.

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Strong ground motions recorded in various earthquakes

t

gu

Figure : Ground motions recorded during several earthquakes.

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Strong ground motion: Near source effect

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N-S component of horizontal ground acceleration recoded at El Centro, California during the Imperial Valley earthquake of 1940

Ground acceleration,

Ground velocity,

Ground displacement,

gu

gu

gu

Accelerogram used in these lectures

CE-409: MODULE 6 (Fall-2013)

Equation of motion for SDOF system subjected to EQ excitations

(t)uum

ku

m

cu (t)umkuucum gg −=++⇒−=++

( )nncr

ωm

k and 2mωcc Since === ζζ

(t)uuωu2ζu g2

nn −=++⇒ ω

≡

CE-409: MODULE 6 (Fall-2013) 16

Response quantities

Response is the structural system reaction to a demand coming from ground acceleration record

Thus a response quantity may be structural displacement, velocity, acceleration, internal shear, bending moment, axial force etc.

Sometime, the total acceleration, , of the mass would be needed if the structure is supporting sensitive equipment and the motion imparted to the equipment is to be determined.

tou

CE-409: MODULE 6 (Fall-2013) 17

Response quantities

Pounding damage, Hotel de carlo, Mexico city, 1985 earthquake

One of the important response quantity is total lateral displacement at the top end of structural system, , required to provide enough separation between adjacent buildings to prevent their pounding against each other during an earthquake

tou

CE-409: MODULE 6 (Fall-2013) 181818

(t)uuωu2ζu g2

nn −=++ ω

The time variation of ground displacement, from the given time variation of ground acceleration, can be determined by using any appropriate time stepping numerical method.

Closer the time interval, more accurate will be solution. Typically, the time interval is chosen to be 1/100 to 1/50 of a second, requiring 1500 to 3000 ordinates to describe the round motion of above given El- Centro ,1940, ground acceleration record having a duration of 30 sec.

Solution to equation of motion for SDOF system subjected to EQ excitation

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Structural disp., , due to ground acceleration,

0.319gugo =

g,ug

inu,

gu

2%ζ 0.5sec,Tn ==

SDOF system with

2%ζ 0.5sec,Tn

==

u

El Centro,1940, ground acceleration

Corresponding relative displacement at the top end of the SDOF frame

CE-409: MODULE 6 (Fall-2013) 20

(t)uuωu2ζu g

2

nn −=++ ω

The above given equation indicates that

Thus any two systems having the same values of Tn and ζ will have

the same deformation response u(t) even though one system may be more massive than the other or one may be stiffer than the other

),f(u n

ζT=

Influence of Tn and ζ on Peak displacement, uo , in a liner elastic SDOF system

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Effect of Tn on Deformation response history

0.319ggou =

g,ug

El Centro ground acceleration

Response of SDOF systems with different values of Tn to El Centro ground acceleration

In general, peak value of displacement at the top end of a SDOF increases with the increase in the time period of the system.

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Effect of Time Period

CE-409: MODULE 6 (Fall-2013) 232323

Effect of ζ on Deformation response history

In general, peak value of displacement at the top end of a SDOF increases with the decrease in the damping ratio of the system

0.319gugo =

g,ug

El Centro ground acceleration

Response of SDOF systems with different values of ζ to El Centro ground acceleration

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Approximate Periods of Vibration (ASCE 7-05)

Where h is the height of building in ft.

CE-409: MODULE 6 (Fall-2013) 25

Thus, structural systems with Tn=0.5sec, 1 and 2 sec may be considered

as 5, 10 and 20 story height buildings, respectively.

A building with 3 story height can be considered as Multi DOF system with at least 3 DOFs.

To keep the discussion simple at this stage, it will be a reasonable assumption to state that (out of 3 natural time periods of the 3 story building) we consider only fundamental natural time period (Tn=0.3 sec) to determine the response

quantities for the building.

Later on we will discuss how all 3 vibration modes (and the corresponding natural time periods) are calculated and are taken into account to find the total response of a building with DOF =3

Because the empirical period formula is based on measured response of buildings, it should not be used to estimate the period for other types of structure (bridges, dams, towers).

Approximate Periods of Vibration

CE-409: MODULE 6 (Fall-2013) 26

Response spectrum concept

A plot of the peak value of a response quantity as a function of the natural vibration period Tn of the system, or a related parameter such

as circular frequency ωn or cyclic frequency fn, is called the response

spectrum for that quantity.

Response is the structural system reaction to a demand coming from ground acceleration record (i.e. Accelerogram) and when the peak response commodities such as structural system displacement , velocity and acceleration are plotted against the structural system natural time period (or frequencies) will be called spectrum

( )ou ( )ou ( )otu

CE-409: MODULE 6 (Fall-2013) 27

Response spectrum concept

Peak values of response quantities and shape of response spectrum depends on the accelerogram

Each such plot is for SDOF system having a fixed damping ratio ζ, and several such plots for different values of ζ are included to cover the range of damping values encountered in actual structures.

The deformation response spectrum is a plot of uo against Tn for

fixed ζ. A similar plot for is the relative velocity response spectrum, and for is the total acceleration response spectrum.

out

ou

CE-409: MODULE 6 (Fall-2013) 28

Deformation response spectrum

Figure on next slide shows the procedure to determine the deformation response spectrum. The spectrum is developed for El Centro ground motions, as shown in part (a) of the figure.

The time variation of deformation induced by this ground motion in three SDF systems is presented in part (b) of the figure

The peak value of deformation D ≡ uo, determined for SDF

system with different Tn is determined and shown in part (c) of the

Figure

CE-409: MODULE 6 (Fall-2013) 292929

(a) El-centro ground acceleration; (b) Deformation response of three SDF systems with ζ=2% and Tn=0.5,1, and 2 sec; (c) Deformation response spectrum for ζ=2%

Construction of deformation response spectrum

CE-409: MODULE 6 (Fall-2013) 30

Pseudo-velocity response spectrum

Consider a quantity V for an SDF system with natural frequency ωn

related to its peak deformation D ≡ uo due to earthquake ground

motion:

DT

2πVDω

n

n==

The quantity V has the unit of velocity and is called relative pseudo- velocity or simply pseudo-velocity. The prefix pseudo is used because V is not equal to the peak velocity , although it has the correct units.

ou

CE-409: MODULE 6 (Fall-2013) 31

Pseudo-velocity response spectrum

T

2π DV

n

= D

V

2%=ζ

2%=ζ

Tn D V=D*2π/Tn

0.5 2.67 33.6

1.0 5.97 37.5

2.0 7.47 23.5

CE-409: MODULE 6 (Fall-2013) 32

Pseudo-acceleration response spectrum

DT

2πDωA

2

n

2n

==

D

A

2%=ζ

2%=ζ

Tn D A=D*(2π/Tn)2

0.52.67 1.09g

1.05.97 0.61g

2.07.47 0.191g

CE-409: MODULE 6 (Fall-2013) 33

Please note the following comments regarding pseudo commodities:

1. uo is same as D by definition.

2. Whereas is not taken as V, which by definition = ωnD

3. Similarly, is not taken as A which by definition= ωn2D

ou

otu

A caution about Pseudo responses

CE-409: MODULE 6 (Fall-2013) 34

Displacement Response Spectra for Different Damping values The higher the damping, the lower the relative displacement.

At a period of 2 sec, for example, going from zero to 5% damping reduces the displacement amplitude by a factor of two. While higher damping produces further decreases in displacement, there is a diminishing return.

The % reduction in displacement by going from 5 to 20% damping is much less that that for 0 to 5% damping.

Deformation response spectra f0r 1940 El-centro earthquake for different values of ζ

CE-409: MODULE 6 (Fall-2013) 35

Damping has a similar effect on pseudo acceleration. Note, however, that the pseudo acceleration at a (near) zero period is the same for all damping values.

Pseudo Acceleration Response Spectra for Different Damping Values

This value is always equal to the peak ground acceleration, 0.319g, for the ground motion in question. i.e. El-centro 1940 earthquake

Deformation response spectra f0r 1940 El-centro earthquake for different values of ζ

CE-409: MODULE 6 (Fall-2013) 36

Pseudo acceleration (A) Vs peak total acceleration

The term Pseudo shall not be conceived by its meaning (i.e. false as defined in English dictionaries). In fact it shall be taken as “an essence similar effect to their relevant commodities ”

It can be observed from below graph that pseudo acceleration , A , and peak value of true acceleration, have almost same values for systems with Tn≤ 10 sec and .

It is worth mentioning that for elastic system the ζ seldomly exceed 5% as such taking A same as negligible effect

tou

0.1 ζ ≤

( )tou

Tn- sec

tou

CE-409: MODULE 6 (Fall-2013) 37

Pseudo velocity (V) Vs peak system’s velocity ( )ou

As shown in below graph that for medium rise buildings (0.2≤ Tn ≤ 1 sec) as long as . Similarly (0.2≤ Tn ≤ 3 sec) for

ouV ≈0.1 ≤ζ

ouV ≈ouV ≈

ou85.0V ≈

ou85.0V ≈0.1 ≤ζ

CE-409: MODULE 6 (Fall-2013) 38

Combined D-V-A spectrum

The deformation, pseudo-velocity and pseduo-acceleration spectra are plotted for a wide and practical range of Tn and for a

particular value of ζ .

The above mentioned procedure is repeated for different values of ζ.

The results for different values of ζ over a wide range of Tn are

combined in a single diagram, called combined D-V-A diagram, as shown on next slide

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Use of D-V-A spectrum

Figure: Combined D-V-A response spectrum for El Centro ground motion; ζ = 2%.Note the values of D,V and A determined for a SDOF system with Tn=2 sec

Refer to slides 29, 31 and 32 for D,V and A, respectively

CE-409: MODULE 6 (Fall-2013) 404040

Combined D-V-A spectrum

Combined D-V-A response spectrum for El Centro ground motion; ζ = 0,5,10 and 20%

For a given earthquake, small variations in structural frequency (period) can produce significantly different results (See V value for Tn = 0.5 to3 sec for El-centro earthquake)

CE-409: MODULE 6 (Fall-2013) 41

Relation between peak Equivalent static force, fso , and Pseudo acceleration, A

oso k.uf =

.mωk since 2n=

).um.(ω

.m).uω(k.uf

o

2

n

o

2

noso

=

==⇒

m.Af so =⇒

CE-409: MODULE 6 (Fall-2013) 42

Peak Structural Response from the response spectrum

As already discussed on previous slide, peak value of the equivalent static force fso can be determined as:

mAkDf so ==The peak value of base shear, Vbo, from equilibrium of above given

diagram can be written as: m.A fV sobo ==

CE-409: MODULE 6 (Fall-2013) 43

.wg

A.A

g

wVor bo ==

Where w is the weight of the structure and g is the gravitational acceleration. When written in this form, A/g may be interpreted as the base shear coefficient or lateral force coefficient . It is used in building codes to represent the coefficient by which the structural weight is multiplied to obtain the base shear

Peak Structural Response from the response spectrum

CE-409: MODULE 6 (Fall-2013) 44

The frame for use in a building is to be located on sloping ground s shown in figure. The cross sections of the two columns are 10 in. square. Determine the base shears in the two columns at the instant of peak response due to the El Centro ground motion. Assume the damping ratio to be 5%. The beam is too stiffer than the columns and can be assumed to be rigid. Total weight at floor level = 10k

Problem M6.1

Solution

CE-409: MODULE 6 (Fall-2013) 45

5% Damped Elastic Displacement Response Spectrum for El Centro Ground Motion

uo is decreasing with increase in Tn

0.67″

Solution (contd….)

Computing the shear force at the base of the short and long columns.

( )( )

g76.0ft/sec 51.24

)12/67.0(/0.32A

D/T2A 76.0Du

2

2

2

no

===⇒

=⇒′′==

ππ

Comments:Although both columns go through equal deformation, however, the stiffer column carries a greater force than the flexible column. The lateral force is distributed to the elements in proportion to their relative stiffnesses. Sometimes this basic principle , if not recognized in building design, lead to unanticipated damage of the stiffer elements.

CE-409: MODULE 6 (Fall-2013) 46

Response Spectrum normalized with peak ground parameters thereby giving the amplification magnitudes for D,V and A e.g., for a system with Tn = 0.5 sec and ζ =0.05. The amplification factors for D,V and A are αD 0.2 , ≈ αV 1.9and ≈ αA 2.3, respectively≈

Velocity amplification factor

Acceleration amplification factor

Displacement amplification factor

D= ugo for Tn>15 sec

gou A ≈

u= -ug

u= 0Very rigid systems Very flexible systems

CE-409: MODULE 6 (Fall-2013) 47

Spectral regions in Response Spectrum

)TT(for u *const. A or

TTfor const. u

A

cngo

cn

go

<=

<≈

i.e. A directly varies with PGA. Therefore region from Tn = 0 to Tc is defined Acceleration sensitive region.

Same logic apply in defining velocity sensitive and displacement sensitive regions

CE-409: MODULE 6 (Fall-2013) 48

Design SpectrumResponse spectrum cannot be used for the design of new structures, or the seismic safety evaluation of existing structures due to the following reasons:

Response spectrum for a ground motion recorded during the past is inappropriate for future design or evaluation.

The response spectrum is not smooth and jagged, specially for lightly damped structures.

The response spectrum for different ground motions recorded in the past at the same site are not only jagged but the peaks and valley are not necessarily at the same periods. This can be seen from the figure given on next slide where the response spectra for ground motions recorded at the same site during past three earthquakes are plotted

CE-409: MODULE 6 (Fall-2013) 494949

Figure: Response spectra for the N-S component of ground motions recorded at the imperial valley Irrigation district substation, El Centro, California, during earthquakes

of May 18,1949;Feb 9,1956;and April 8,1968; ζ = 2%.

CE-409: MODULE 6 (Fall-2013) 50

Design Spectrum

Due to the inappropriateness of response spectrum as stated on previous slide, the majority of earthquake design spectra are obtained by averaging a set of response spectra for ground motion recorded at the site the past earthquakes.

If nothing have been recoded at the site, the design spectrum should be based on ground motions recorded at other sites under similar conditions such as magnitude of the earthquake, the distance of the site from causative fault, the fault mechanism, the geology of the travel path of seismic waves from the source to the site, and the local soil conditions at the site.

CE-409: MODULE 6 (Fall-2013) 51

Design Spectrum

For practical applications, design spectra are presented as smooth curves or straight lines.

Smoothing is carried , using statistical analysis, out to eliminate the peaks and valleys in the response spectra that are not desirable for design. For this purpose statistical analysis of response spectra is carried out for the ensemble of ground motions.

Each ground motion, for statistical analysis is normalized (scaled up or down) so that all ground motions have the same peak ground acceleration, say ;other basis for normalization can be chosen.gou

CE-409: MODULE 6 (Fall-2013) 52

Construction of Design Spectrum

Researchers have developed procedures to construct such design spectra from ground motion parameters. One such procedure is illustrated in given figure.

The recommended period values Ta = 1/33 sec, Tb = 1/8 sec, Te = 10

sec, and Tf = 33 sec, and the

amplification factors αA, αV , and αD

for the three spectral regions (given table on next slide), were developed by the statistical analysis of a larger ensemble of ground motions recorded on firm ground (rock, soft rock, and competent sediments).

CE-409: MODULE 6 (Fall-2013) 53

Amplification factors for construction of Design Spectrum

CE-409: MODULE 6 (Fall-2013) 54

Construction of Design Spectrum (firm soil)

We will now develop the 84.1 percentile design spectrum

for ζ=5%

For convenience, a peak ground acceleration is selected; the resulting spectrum can be scaled by η to obtain the design spectrum corresponding to

The typical values of

, recommended for firm ground, are used. For , these ratios give

g1ugo =

ηgugo =

6u

u*u and in./sec/g 48u

u

go2

gogo

go

go ==

g1ugo =in 36ugo = and in/sec 48ugo =

CE-409: MODULE 6 (Fall-2013) 55

Using the values on previous slide and values given in table 6.9.2 (slide 53) for 84.1 percentile and ζ =5%, the Pseudo-velocity design spectrum can be dawn as shown in Figure 6.9.4

3.2v =α

71.2A =α 01.2D =α


CE-409: MODULE 6 (Fall-2013) 56

Displacement and Pseudo-acceleration design spectra can be drawn from pseudo-velocity design spectrum using the relations being already discussed and reproduced here for the convenience:

The Pseudo-acceleration and displacement design spectra drawn by using above given equation are drawn in Figures 6.9.5 and 6.9.6 on next two slides

nn

n

n

T

2πV.VωA

V2π

T

ω

VD

==

==


CE-409: MODULE 6 (Fall-2013) 57

Construction of Design Spectrum (firm soil)Acceleration sensitive region

Velocity sensitive region Displacement sensitive region

CE-409: MODULE 6 (Fall-2013) 58


CE-409: MODULE 6 (Fall-2013) 59

Design Spectrum for various values of ζ

Pseudo- velocity design spectrum for ground motions with

; ζ = 1,2,5,10 and 20 %.

in. 36u and in/sec, 48u,1u gogogo === g

CE-409: MODULE 6 (Fall-2013) 606060

Pseudo- acceleration design spectrum (84.1 th percentile) drawn on log scale for ground motions with ; ζ = 1,2,5,10 and 20 %.in. 36u and in/sec, 48u,1u gogogo === g


CE-409: MODULE 6 (Fall-2013) 6161616161

Figure: Pseudo- acceleration design spectrum (84.1 th percentile) drawn on linear scale for ground motions with ;

ζ = 1,2,5,10 and 20 %.

in. 36u and in/sec, 48u,1u gogogo === g


CE-409: MODULE 6 (Fall-2013) 62

Envelope Design spectrumFor some sites a design spectrum is the envelope of two differentelastic design spectra as shown below

Site

Nearby fault producing moderate EQ

Far away fault producing large EQ

Site

Nearby fault producing moderate EQ

Far away fault producing large EQ

CE-409: MODULE 6 (Fall-2013) 63

(a) A full water tank is supported on an 80-ft-high cantilever tower. It is idealized as an SDF system with weight w = 100 kips, lateral stiffness k = 4 kips/in., and damping ratio ζ = 5%. The tower supporting the tank is to be designed for ground motion characterized by the design spectrum of Fig. 6.9.5 scaled to 0.5g peak ground acceleration. Determine the design values of lateral deformation and base shear.(b) The deformation computed for the system in part (a) seemed excessive to the structural designer, who decided to stiffen the tower by increasing its size. Determine the design values of deformation and base shear for the modified system if its lateral stiffness is 8 kips/in.; assume that the damping ratio is still 5%. Comment on how stiffening the system has affected the design requirements. What is the disadvantage of stiffening the system?

Problem M6.2

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Solve following exercise problems (Chopra’s book, second or

third edition)

1. Problem 6.10

2. Problem 6.15

Further problem for practice:

6.12 to 6.14,6.16 and 6.17

Home Assignment M6