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.
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
Ground acceleration, percent of gravity
figure 1 below.
Time, sec
Figure 1. Typical earthquake accelerogram 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 Tn of the system and its damping
2
ratio ζ . 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.
u + 2ζωnu + ωn2u = −ug ( t )
(1)
Figure 2. Deformation response of single degree of freedom systems 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 Tn of the system, or a related parameter such as circular frequency ωn , is called the response spectrum for that quantity.
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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.
Figure 3. Acceleration response spectrum 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 Tn ) is changed. Let us focus on the first SDOF with Tn = 0.5 sec , the computed peak displacement is
u0 = 2.67 inches. The second system has a period of Tn = 1.0 sec and the peak displacement is u0 = 5.97 inches, in continuity the third one shows Tn = 2.0 sec and peak displacement of u0 = 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 Tn while keeping fixed the value of damping ratio ζ (for this case ζ = 2% ). The deformation response spectrum provided by this procedure is shown below in Figure 4.
4
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
V = ωn D =
2π D Tn
(2) 2
⎛ 2π ⎞ A=ω D =⎜ ⎟ D ⎝ Tn ⎠ 2 n
(3)
Figure 4. Response spectra (a) deformation response spectrum (b) pseudovelocity response spectrum (c) pseudo-acceleration response spectrum 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
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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 Tn the values of spectral displacement D and spectral acceleration A can be read from the diagonal axes.
Figure 5. Combined D-V-A response spectrum, damping ration ζ = 2% 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 Tn = 0.05...20 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 6
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 ground motion, damping ration ζ = 0, 2,5,10 and 20%
Figure 6. Response spectrum for El Centro ground motion with normalized scales D / u g , V / u g , A / ug 7
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 Ta , Tb , Te and T f 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 +1 σ spectra with probability distributions for V at Tn = 0.25, 1 and 4 sec; ζ = 5% Dashed lines show an idealized design spectrum. (After Chopra "Dynamics of Structures") 8
Figure 8. Construction of elastic design spectrum (Chopra "Dynamics of Structures") Table 1. Amplification factors; Elastic design spectra
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 α A , αV and α D substantially change with the soil conditions. 9
Figure 9. Design pseudo – acceleration spectrum 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). rno = rnst An
(4)
where rno - n-th contribution to any response quantity rnst - modal static response An - pseudo acceleration corresponding to natural period Tn and damping ration ζ n
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6.2. Modal combination rules In order to determine the peak value ro ≡ max t r ( t ) one needs to combine the peak modal responses rno ( n = 1, 2,..., N ) . Actually this is impossible, because as we can see from figure 10, the modal responses rnt reach their peaks at different time instants and the combined response r ( t ) attains its peak at yet different time.
Figure 10. Base shear and fifth-story shear; modal contributions Vbn (t ) and V5 n (t ) and total responses, Vb (t ) and V5 (t ) (After Chopra)
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 N
ro ≤ ∑ rno
(5)
n =1
Actually this rule is too conservative and it is not popular in design of structures.
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•
SRSS – Square root of sum of squares 1/2
⎛ N ⎞ ro ⎜ ∑ rno2 ⎟ ⎝ n =1 ⎠
(6)
This modal combination rule applies well to structures with well separated natural periods. •
CQC – Complete quadratic combination 1/2
⎛ N N ⎞ ro ⎜ ∑∑ ρin rio rno ⎟ ⎝ i =1 n =1 ⎠
(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 ρin can be calculated according to Der Kiureghian by the formula below
ρin =
8ζ 2 (1 + βin ) βin3/2
(1 − β )
2 2 in
Where βin =
+ 4ζ 2 βin (1 + βin )
(8)
2
ωi ωn
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 rnst that is multiplied by the spectral ordinate An to retrieve the peak modal response rno 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 ωn and natural modes φn 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 Tn and damping ration ζ n , read Dn and An , 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 f yn and torques fθ n
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 f yn and fθ n .
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 S a (T ) is normally used but if spectral displacements are of interest (e.g. for displacement – based assessment or design) they can be derived from S a (T ) as explained in the paragraphs above. The EC8 spectra includes ranges of : 13
•
constant spectral pseudo – accelerations for natural periods between TB and TC
•
constant pseudo – velocity for natural periods between TC and TD
•
constant pseudo – displacements for periods longer than TD
Also in EC8 the elastic response spectrum is taken as proportional to peak ground acceleration: •
the horizontal peak acceleration ag , for the horizontal components
•
the vertical peak acceleration avg , 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 ⎡ T ⎤ 0 ≤ T ≤ TB : S a (T ) = ag S ⎢1 + ( 2.5η − 1) ⎥ ⎣ TB ⎦
(9)
Constant pseudo – acceleration range: TB ≤ T ≤ TC : S a (T ) = ag S ⋅ 2.5η
(10)
Constant pseudo – velocity range:
⎡T ⎤ TC ≤ T ≤ TC : S a (T ) = ag S ⋅ 2.5η ⎢ C ⎥ ⎣T ⎦
(11)
Constant spectral – displacement range: ⎡T T ⎤ TD ≤ T ≤ 4 sec : S a (T ) = ag S ⋅ 2.5η ⎢ C 2D ⎥ ⎣ T ⎦
(12)
where
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ag - design ground acceleration on rock; S – soil factor
η = 10 / ( 5 + ζ ) ≥ 0.55 is a correction factor for viscous damping ratio ζ
Figure 11. Elastic response spectra of Type 1 (left) and 2 (right) recommended in EC8, for PGA on rock equal to 1 g and for 5% damping The values of TB , TC , TD 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 A B C D E S1 S2
Description vs,30 (m/s) NSPT Rock outcrop, with less than 5 m cover of weaker >800 – material Very dense sand or gravel, or very stiff clay, several 360–800 >50 tens of meters deep; mechanical properties gradually increase with depth Dense to medium-dense sand or gravel, or stiff clay, 180–360 15–50 several tens to many hundreds meters deep Loose-to-medium sand or gravel, or soft-to-firm clay <180 <15 5–20 m surface alluvium layer with vs < 360 m/s underlain by rock (with vs > 800 m/s ) ≥10 m thick soft clay or silt with plasticity index <100 – > 40 and high water content Liquefiable soils; sensitive clays; any soil not of type A to E or S1
cu (kPa) – >250 70–250 <70 10–20
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) 15
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
S
Spectrum type 2
TB (s) TC (s) TD (s)
TB (s) TC (s) TD (s)
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 k = 3 nst and at least one natural period below
Tk = 0.2 sec. where
nst - 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 T1
•
For buildings up to 40 m EC8 recommends the following expression
T1 = Ct H 3/4
(13) 16
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
T1 = 0.075 Ac
(14)
where 2 Ac = ∑ ⎡ Ai ⋅ ( 0, 2 + ( lwi H ) ) ⎤ ⎣ ⎦
Ac
(15)
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 T1 is
T1 = 2 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
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 Any structural system other than those above
αu/α1
DC H
4αu/α1
3αu/α1 4.5αu/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:
¾ one-storey systems and frame-equivalent dual (i.e., frame-wall) ones, and ¾ wall systems with more than two uncoupled walls in the horizontal direction considered. 18
•
αu/α1 = 1.2 for:
¾ one-bay multi-storey frame systems and frame-equivalent dual ones, ¾ wall-equivalent dual systems, and ¾ 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
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