CE658 Earthquake Engineering

June 20, EPOKA University, Tirana, ALBANIA.

Response spectrum analysis and design response spectra

Laidon Zekaj1

1 Department of Civil Engineering, EPOKA University, Albania

ABSTRACT

One of the most important issues of earthquake engineering is to assess the response of

structures due to the ground shaking caused by earthquakes. The most common representation

of the seismic action on structures in different codes is achieved through the response

spectrum analysis (RSA).

This paper deals with the explanation of the response spectrum concept, how it is

constructed and used in order to determine the peak responses of the structures directly from

the response spectrum.

Further, is discussed in general the response spectrum method of analysis. In order to

give a more precise view of the response spectrum analysis and its application it is presented

how this method is implemented by Eurocode 8 and the main points where one should focus

when using RSA.

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1. INTRODUCTION

The most useful way of describing the ground shaking during an earthquake for

engineering purposes is the plot of ground acceleration versus time like the one shown in

figure 1 below.

To provide the information shown in figure 1, we need a basic instrument called

accelerograph. Such equipment makes it possible to measure all of the three components of

ground shaking during an earthquake.

Since it is impossible to know where the next earthquake will take place and it is not

logical to install these instruments everywhere because of their installation and maintenance

cost, it is only probable to obtain such records only in strong – shaking regions.

As we will see in the forthcoming paragraphs, the accelerograms provide the basis of

the response spectrum construction.

2. RESPONSE QUANTITIES AND RESPONSE HISTORY

In structural engineering it is very important and useful to know the deformation of a

structure because the latter is directly related to the internal forces of structural elements such

as the bending moment, shear force and axial force. If we take a look to the equation of

motion of a single degree of freedom system (SDOF) (Equation 1), it is obvious that the

displacement of a system depends only on the natural period nT of the system and its damping

Figure 1. Typical earthquake accelerogram

Gro

und

acce

lera

tion,

per

cent

of g

ravi

ty

Time, sec

3

ratioζ . Such thing can be noticed too from the Figure 2 which implies that a large period

results in bigger deformation and in contrast a bigger damping ration results in a smaller

deformation of the structure.

( )22 n n gu u u u tζω ω+ + = − (1)

3. THE NOTION OF RESPONSE SPECTRUM

The concept of the earthquake response spectrum was first introduced in 1932 and has

become in now days a very important concept in earthquake engineering. As Chopra defines

in his book "Dynamics of Structures", a plot of the peak value of a response quantity as a

function of the natural vibration period nT of the system, or a related parameter such as

circular frequency nω , is called the response spectrum for that quantity.

Figure 2. Deformation response of single degree of freedom systems

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To provide the response spectrum we need several plots of SDOF system with constant

value of the damping ratio ζ as shown in Figure 3 below.

Depending on the quantity we want to use it is possible to have three kinds of response

spectrums:

a. Deformation response spectrum

b. Velocity response spectrum

c. Acceleration response spectrum

4. CONSTRUCTION OF RESPONSE SPECTRUM

For this purpose we will take in consideration the deformation responses of the three

SDOF introduced in Figure 3 above. For all of these SDOF the ground motion is the same and

the damping ratio as we can see is set to 2%. Only the frequency of vibration (in this case

represented by the period nT ) is changed.

Let us focus on the first SDOF with 0.5 secnT = , the computed peak displacement is

0u = 2.67 inches. The second system has a period of 1.0 secnT = and the peak displacement is

0u = 5.97 inches, in continuity the third one shows 2.0 secnT = and peak displacement of

0u = 7.47 inches. Each of these peak responses provides one point in the displacement

response spectra plot. In order to obtain this plot it is needed to repeat the process over a range

of nT while keeping fixed the value of damping ratio ζ (for this case 2%ζ = ). The

deformation response spectrum provided by this procedure is shown below in Figure 4.

Figure 3. Acceleration response spectrum

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After we have obtained the deformation response spectra, we can easily get the pseudo –

velocity response spectra and the pseudo – acceleration response spectra by using respectively

formula (2) and formula (3) given below

2n

n

V D DTπω= = (2)

22 2n

n

A D DTπω

⎛ ⎞= = ⎜ ⎟

⎝ ⎠ (3)

Given that each of the spectra above are related to each other by the formula (2) and (3)

we can say that they represent the same information but using different response quantities, so

if we have one the response spectrums we can derive the other two. But for design purposes

Figure 4. Response spectra (a) deformation response spectrum (b) pseudo-velocity response spectrum (c) pseudo-acceleration response spectrum

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the shape of the spectrum must approximated and for this purpose we need all the three

response spectrums and this is achieved through the D-V-A spectrum which brings up a

combined plot showing all three spectral quantities. This type of plot was first introduced by

Velestos and Newmark in 1960. This type of combined plot can be constructed on a four way

logarithmic paper and it is shown below in figure 5. As it is shown in the figure, for a given

value of natural period nT the values of spectral displacement D and spectral acceleration A

can be read from the diagonal axes.

A response spectrum should cover a wide range of natural vibration periods and a

practical range of ζ = 0 to 20% in order to provide us the peak responses of all possible

structures. A response spectrum for 0.05...20nT = sec and ζ = 0 to 20% is represented in

figure 6.

5. ELASTIC DESIGN SPECTRUM

The design spectrum has to complete some certain requirements because it will be used

for the design of new structures or the seismic retrofitting of existing structures, to resist

further earthquakes. We cannot use the response spectrum of a past earthquake to design a

new building in a given site because the response spectrum for another ground shaking in the

same site doesn’t have the same jaggedness and the peaks and valleys are not at the same

Figure 5. Combined D-V-A response spectrum, damping ration 2%ζ =

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periods. So the design spectrum should consist of a set of smooth curves. The design

spectrum should be a representative of all past recorded earthquake ground motions.

The derivation of design spectrum is based on statistical analysis of the normalized

response spectra like the one in figure 6.

Figure 5. Combined D-V-A response spectrum for El Centro groundmotion, damping ration 0, 2,5,10 and 20%ζ =

Figure 6. Response spectrum for El Centro ground motion withnormalized scales / , / , /g g gD u V u A u

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As it is obvious for each natural period there are many spectral values as the number of

ground motion records. Through the statistical analysis of these data it is provided the

probability distribution for the spectral ordinate, its mean value and its standard deviation . If

we connect all mean values we get the mean response spectrum as well if we connect all mean

plus one standard deviation values we get the mean plus one standard deviation spectrum as it

is presented in figure 7 below. From the figure it can be noticed that such type of smooth

curves spectrum can be more easily idealized by straight lines as we mentioned in the

paragraph below.

For this purpose there have been developed methods to construct design spectra

directly from ground motion parameters and this procedure is presented in figure 8. The

values of recommended periods , , and a b e fT T T T and the amplification factors for the three

spectral regions are retrieved by earlier analysis of a large number of ground motions

recorded in different soil types. The amplification factors are introduced in table 1 in the next

page.

Figure 7. Mean and mean +1σ spectra with probability distributions forV at nT = 0.25, 1 and 4 sec; 5%ζ = Dashed lines show an idealized designspectrum. (After Chopra "Dynamics of Structures")

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Source: Chopra "Dynamics of structures"

The elastic design spectra can be used to estimate the deformations of systems and design

forces too in order that these systems remain elastic.

Once the design spectrum is drawn in a four way log plot, the normalized acceleration

response spectrum can be obtained in an ordinary plot. A typical plot of the normalized

pseudo acceleration spectrum (derived from a log plot) as given in the codes of practice is

shown in Figure 9. It is seen from the figure that spectral accelerations (Sa) for a soft soil

profile are more compared with those of a hard soil profile at periods of more than 0.5 s. This

is the case because the amplified factors , and A V Dα α α substantially change with the soil

conditions.

Figure 8. Construction of elastic design spectrum(Chopra "Dynamics of Structures")

Table 1. Amplification factors; Elastic design spectra

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6. RESPONSE SPECTRUM ANALYSIS

6.1. Peak modal responses

The design of structures is generally based on the values of peak forces of deformations

over an earthquake – induced response. As we saw from the paragraphs above it was possible

for single degree of freedom systems to get these peak responses from the response spectrum

of a specified ground motion. This method can be applied also for multi degree of freedom

systems but the results will not be exact but accurate enough for structural design purposes.

The peak response of a MDF corresponding to the n-th natural mode can be retrieved

from the earthquake response spectrum according to the formula (4).

st

no n nr r A= (4)

where

nor - n-th contribution to any response quantity

stnr - modal static response

nA - pseudo acceleration corresponding to natural period nT and damping ration nζ

Figure 9. Design pseudo – acceleration spectrum

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6.2. Modal combination rules

In order to determine the peak value ( )maxo tr r t≡ one needs to combine the peak

modal responses ( )1, 2,...,nor n N= . Actually this is impossible, because as we can see from

figure 10, the modal responses ntr reach their peaks at different time instants and the

combined response ( )r t attains its peak at yet different time.

It is clear that, when using a response spectrum the information regarding to the time instants

at which peak modal responses occur is not available, such thing leads to the need of some

assumptions which consist of three rules of modal combinations that will be introduced

below.

• ABSSUM – Absolute sum modal combination rule

1

N

o non

r r=

≤∑ (5)

Actually this rule is too conservative and it is not popular in design of structures.

Figure 10. Base shear and fifth-story shear; modal contributions ( )bnV t and

5 ( )nV t and total responses, ( )bV t and 5 ( )V t (After Chopra)

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• SRSS – Square root of sum of squares

1/2

2

1

N

o non

r r=

⎛ ⎞⎜ ⎟⎝ ⎠∑ (6)

This modal combination rule applies well to structures with well separated natural periods.

• CQC – Complete quadratic combination

1/2

1 1

N N

o in io noi n

r r rρ= =

⎛ ⎞⎜ ⎟⎝ ⎠∑∑ (7)

This rule for modal combination can be applied to wider range of structures because it goes

beyond the limitations of SRSS rule. The correlation coefficient inρ can be calculated

according to Der Kiureghian by the formula below

( )( ) ( )

2 3/2

2 22 2

8 1

1 4 1in in

in

in in in

ζ β βρ

β ζ β β

+=

− + + (8)

Where iin

n

ωβω

=

6.3. Comments

The response spectrum analysis is a procedure to determine the response of a systems

subjected to earthquake excitations but it reduces to several steps of static analysis of a system

subjected to static forces which provides the static modal response stnr that is multiplied by

the spectral ordinate nA to retrieve the peak modal response nor as it can be seen from

equation (4). As long as the response spectrum analysis uses the dynamic properties of a

structure such as natural period, modal damping ratios and natural modes it is considered as a

dynamic procedure.

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6.4. Step by step procedure to complete a response spectrum analysis

1. Define the structural properties

a. Determine the mass and stiffness matrices

b. Estimate the modal damping ratios

2. Determine the natural frequencies nω and natural modes nφ of vibration

3. Compute the peak response in the n-th mode by the following steps, to be repeated for

all modes, n=1,2,…,N

a. Corresponding to the natural period nT and damping ration nζ , read nD and nA ,

the deformation and pseudo – acceleration, from the earthquake response

spectrum or design spectrum

b. Compute the lateral displacements and rotations of the floors

c. Compute the equivalent static forces: lateral forces ynf and torques nfθ

d. Compute the story forces – shear, torque and overturning moments – and element

forces – bending moments and shears – by three dimensional static analysis of the

structure subjected to external forces ynf and nfθ .

4. Determine an estimate for the peak value r of any response quantity by combining the

peak modal values rn using one of the rules presented above.

7. RESPONSE SPECTRUM ANALYSIS IN EUROCODE 8

7.1. Overview

The seismic design of new buildings according to contemporary building codes

including EC8 is force – based. It counts for linear elastic analysis and uses 5% damped

elastic spectrum divided by a factor q called "behavior factor" that counts mainly for

ductility and energy dissipation capacity and also for overstrength. RSA is exactly called

in EC8 "modal response spectrum analysis" and in contrast with US codes, EC8 accepts

this as the reference method for the design of new buildings and fully respects its rules

and results. The pseudo – acceleration response spectrum ( )aS T is normally used but if

spectral displacements are of interest (e.g. for displacement – based assessment or design)

they can be derived from ( )aS T as explained in the paragraphs above.

The EC8 spectra includes ranges of :

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• constant spectral pseudo – accelerations for natural periods between BT and CT

• constant pseudo – velocity for natural periods between CT and DT

• constant pseudo – displacements for periods longer than DT

Also in EC8 the elastic response spectrum is taken as proportional to peak ground

acceleration:

• the horizontal peak acceleration ga , for the horizontal components

• the vertical peak acceleration vga , for the vertical component

Eurocode 8 uses the same spectral shape for different performance levels or limit states

and the difference between hazard levels is taken into account through the peak ground

acceleration to which the spectrum is anchored.

7.2. Elastic spectra of the horizontal components

The elastic response spectral accelerations for the two horizontal components is described

by the expressions below and presented in figure 11

( ) ( )0 : 1 2.5 1B a gB

TT T S T a ST

η⎡ ⎤

≤ ≤ = + −⎢ ⎥⎣ ⎦ (9)

Constant pseudo – acceleration range:

( ): 2.5B C a gT T T S T a S η≤ ≤ = ⋅ (10)

Constant pseudo – velocity range:

( ): 2.5 CC C a g

TT T T S T a ST

η ⎡ ⎤≤ ≤ = ⋅ ⎢ ⎥⎣ ⎦ (11)

Constant spectral – displacement range:

( ) 24 sec : 2.5 C DD a g

T TT T S T a ST

η ⎡ ⎤≤ ≤ = ⋅ ⎢ ⎥⎣ ⎦ (12)

where

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ga - design ground acceleration on rock;

S – soil factor

( )10 / 5 0.55η ζ= + ≥ is a correction factor for viscous damping ratio ζ

The values of BT , CT , DT and the soil factor S are taken in accordance with the ground type.

Eurocode recognizes five standard ground types and two special ones (Table 2).

Table 2. Ground types in Eurocode 8 for the definition of the seismic action

Description vs,30 (m/s) NSPT cu (kPa) A Rock outcrop, with less than 5 m cover of weaker >800 – – material B Very dense sand or gravel, or very stiff clay, several 360–800 >50 >250 tens of meters deep; mechanical properties gradually increase with depth C Dense to medium-dense sand or gravel, or stiff clay, 180–360 15–50 70–250 several tens to many hundreds meters deep D Loose-to-medium sand or gravel, or soft-to-firm clay <180 <15 <70 E 5–20 m surface alluvium layer with vs < 360 m/s underlain by rock (with vs > 800 m/s ) S1 ≥10 m thick soft clay or silt with plasticity index <100 – 10–20 > 40 and high water content S2 Liquefiable soils; sensitive clays; any soil not of type A to E or S1

The spectra’s shown in figure 11 can be categorized as follows:

• Type 1 – for moderate to large magnitude earthquakes

• Type 2 – for low magnitudes ones (e.g. with surface magnitude less than 5.5)

Figure 11. Elastic response spectra of Type 1 (left) and 2 (right) recommended in EC8, for PGA onrock equal to 1 g and for 5% damping

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The values of TB, TC, TD and S recommended in Eurocode 8 for the five standard ground types

A to E are given in Table 3. They are based on Rey et al. (2002) and European strong motion

data.

Table 3. Recommended parameter values for the standard horizontal elastic

response spectra

Ground type Spectrum type 1 Spectrum type 2

S TB (s) TC (s) TD (s) S TB (s) TC (s) TD (s)

A 1.00 0.15 0.4 2.0 1.0 0.05 0.25 1.2

B 1.20 0.15 0.5 2.0 1.35 0.05 0.25 1.2

C 1.15 0.20 0.6 2.0 1.50 0.10 0.25 1.2

D 1.35 0.20 0.8 2.0 1.80 0.10 0.30 1.2

E 1.40 0.15 0.5 2.0 1.60 0.05 0.25 1.2

7.3. Some points to consider when using RSA in E – TABS

7.3.1. Minimum number of modes

The minimum number of Eigen modes to take into consideration during RSA

according to EC8 is given by the formula 3 stk n= and at least one natural period below

Tk = 0.2 sec.

where

stn - is the number of storeys above the foundation or the top of a rigid basement;

7.3.2. Lower bound factor β

Factor β gives a lower bound for the horizontal design spectrum acting as a

safeguard against excessive reduction of the design forces due to the flexibility of the

system (real of presumed in the design). Its recommended value in Eurocode 8 is β = 0.2.

7.3.3. Fundamental period 1T

• For buildings up to 40 m EC8 recommends the following expression

3/41 tT C H= (13)

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where

Ct is 0,085 for moment resistant space steel frames, 0,075 for moment

resistant space concrete frames and for eccentrically braced steel

frames and 0,050 for all other structures;

H is the height of the building, in m, from the foundation or from the top

of a rigid basement.

• for structures with concrete or masonry shear walls the value Ct in expression

(13) may be taken as being

1 0.075 cT A= (14)

where

( )( )20, 2c i wiA A l H⎡ ⎤= ⋅ +⎣ ⎦∑ (15)

Ac is the total effective area of the shear walls in the first storey of the

building, in m2;

Ai is the effective cross-sectional area of the shear wall i in the first storey

of the building, in m2;

lwi is the length of the shear wall i in the first storey in the direction

parallel to the applied forces, in m, with the restriction that lwi/H should

not exceed 0,9.

H is the height of the building, in m, from the foundation or from the top

of a rigid basement.

• An alternative expression to estimate 1T is

1 2T d= (15)

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d is the lateral elastic displacement of the top of the building, in m, due to

the gravity loads applied in the horizontal direction.

7.3.4. Behavior factor q

For building structures designed for energy dissipation and ductility, the value of

the behavior factor q, by which the elastic spectrum used in linear analysis is divided,

depends:

• on the ductility class selected for the design

• on the type of lateral-force-resisting-system, and

• (in Eurocode 8) on the regularity of the structural system in elevation

The q – factor value is linked to the local ductility demands in members through

the ductility classification as in EC8 shown in the table below

Table 3. Recommended parameter values for the standard horizontal elastic response spectra

Lateral-load resisting structural system DC M DC H

Inverted pendulum system 1.5 2

Torsionally flexible structural system 2 3

Uncoupled wall system, not belonging in one of the two categories above 3 4αu/α1

Any structural system other than those above 3αu/α1 4.5αu/α1

αu/α1 is the ratio of the seismic action that causes development of a full

plastic mechanism, to the seismic action at formation of the first plastic

hinge in the system – both in the presence of the gravity loads

considered concurrent with the design seismic action.

For buildings regular in plan, the recommended values are:

• αu/α1 = 1.0 for wall systems with just two uncoupled walls per horizontal

direction;

• αu/α1 = 1.1 for:

one-storey systems and frame-equivalent dual (i.e., frame-wall) ones, and

wall systems with more than two uncoupled walls in the horizontal

direction considered.

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• αu/α1 = 1.2 for:

one-bay multi-storey frame systems and frame-equivalent dual ones,

wall-equivalent dual systems, and

coupled wall systems.

• αu/α1 = 1.3 for multi-storey multi-bay frames or frame-equivalent dual systems.

In buildings which are irregular in plan according to the classification criteria of

Eurocode 8 presented in Sections 2.1.5 and 2.1.6, the default value of αu/α1 is the

average of:

• 1.0 and

• the default values given above for the buildings regular in plan

8. CONCLUSIONS

8.1. Most seismic design codes present zonation maps and response spectra derived

probabilistically, even though these design loads are often associated with a totally

arbitrary selection of the return period.

8.2. The main advantage that a seismic code offers in terms of earthquake loading is

allowing the engineer to bypass considerable effort required for site – specific hazard

assessment, but every engineer should be aware of the considered assumptions and

limitations.

8.3. Although the purpose of Eurocode 8 is the harmonization of seismic design across

Europe, each country have to produce its own seismic hazard map showing PGA

values for the 475 – year return period.

8.4. Now days a highly specialized discipline such as "Seismic hazard analysis" is

evolving and advancing. The art of this method lies in assessing the uncertainties

associated with the data and the applicability of the models to the specific region and

site under consideration.

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REFERENCES

[1] Anil K. Chopra (1995) Dynamics of Structures. Prentice – Hall, Inc., USA.

[2] Steven L. Kramer (1996) Geotechnical Earthquake Engineering. Prentice – Hall, Inc.,

USA.

[3] Tushar K. Datta (2010) Seismic Analysis of Structures. John Wiley & Sons (Asia), Pte

Ltd., Singapore.

[4] Ahmed Y. Elghazouli (2009) Seismic Design of Buildings to Eurocode 8. Spon Press,

United Kingdom.

[5] Michael N. Fardis (2009) Seismic Design, Assessment and Retrofitting of Concrete

Buildings based on Eurocode 8. Springer Science + Business Media. USA

[6] CEN (2004a) European Standard EN 1998-1:2004 Eurocode 8: Design of structures for earthquake resistance, Part 1: General rules, seismic actions and rules for buildings. Comite Europeen de Normalisation, Brusells