SEISMIC RESPONSE SPECTRUM
By Dr. Jagadish. G. Kori Professor & Head Civil Engineering Department Govt. Engineering College, Haveri-581110 INTRODUCTION • In order to perform the seismic analysis and design of a structure to be built at a particular location, the actual time history record is required. • However, it is not possible to have such records at each and every location. • Further, the seismic analysis of structures cannot be carried out simply based on the peak value of the ground acceleration as the response of the structure depend upon the frequency content of ground motion and its own dynamic properties. • To overcome the above difficulties, earthquake response spectrum is the most popular tool in the seismic analysis of structures. INTRODUCTION • Response spectrum is an important tool in the seismic analysis and design of structures. It describes the maximum response of damped single degree of freedom system to a particular input motion at different natural periods. • Response spectrum method of analysis is advantageous as it considers the frequency effects and provides a single suitable horizontal force for the design of structure. Methods of Seismic Analysis • Two basic methods are widely used for dynamic seismic analysis, namely, Response Spectrum and Time History methods
1. Response Spectrum methods allows determination of maximum modal response of a singly supported structural system or a multiple supported system where all supports receive the same excitation.
2. Time History method of analysis permits the simultaneous application of different excitations at each support point of uncoupled model of the system of Time History of recorded ground acceleration at interest . Capitola, California in the 1989 Loma Prieta earthquake 1989 ORIGIN OF THE RESPONSE SPECTRUM METHOD
• In 1971, with the occurrence of the San Fernando, California, earthquake, the modern era of RSM was launched. • This earthquake was recorded by 241 accelerographs, and by combining these data with all previous strong-motion records it became possible to perform the first comprehensive empirical scaling analyses of response spectral amplitudes. TIME HISTORY DATA
THE MOST DIRECT DESCRIPTION OF AN EARTHQUAKE MOTION IN TIME DOMAIN IS PROVIDED BY ACCELEROGRAMS THAT ARE RECORDED BY INSTRUMENTS CALLED STRONG MOTION ACCELEROGRAPHS .
THE ACCELEROGRAPH RECORDS THREE ORTHOGONAL COMPONENTS OF GROUND ACCELERATION AT A CERTAIN LOCATION.
THE PEAK GROUND ACCELERATION DURATION, AND FREQUENCY CONTENT OF EARTHQUAKE CAN BE OBTAINED FROM AN ACCELEROGRAMS. AN ACCELEROGRAM CAN BE INTEGRATED TO OBTAIN THE TIME VARIATIONS OF THE GROUND VELOCITY AND GROUND DISPLACEMENT. TIME HISTORY DATA
Time, El Centro ground motion sec Acceleration, g (N-S Component) 0.00 0.00630 0.02 0.00364 May 18, 1940 0.04 0.00099 0.06 0.00428 0.08 0.00758 0.10 0.01087 0.12 0.00682 0.14 0.00277 0.16 -0.00128 0.18 0.00368 0.20 0.00864 0.22 0.01360 0.24 0.00727 0.26 0.00094 http://peer.berkeley.edu/smcat/ 0.28 0.00420 0.30 0.00221 http://db.cosmos-eq.org/scripts/default.plx TIME HISTORY DATA 0.4
0.2
0 ug,g 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 -0.2 Time, sec
-0.4 FOR EARTHQUAKE EXCITATION – i. ANALYTICAL SOLUTION IS NOT POSSIBLE; ii. NUMERICAL METHODS ARE EMPLOYED TO FIND OTHER QUANTITIES LIKE iii. a. VELOCITY; b. DISPLACEMENT ETC. DIFFERENT NUMERICAL METHODS ARE: CENTRAL DIFFERENCE METHOD AVERAGE ACCELERATION METHOD NEWMARK’S METHOD ETC. TIME HISTORY DATA ANALYSIS DEFORMATION RESPONSE SPECTRUM Deformation Response Spectrum
For a given EQ excitation calculate |umax | from SDOF response with a certain ξ and within a range of natural periods or frequencies.
|umax | for each frequency will be found from the computed u(t) history at this frequency.
A plot of |umax | vs. natural period is constructed representing the deformation (or displacement) response spectrum (Sd).
From this figure, one can directly read the maximum relative displacement of any structure of natural period T (and a particular value of ξ as damping)
VELOCITY RESPONSE SPECTRUM
Plot of V vs. T N ACCELERATION RESPONSE SPECTRUM
Plot of A vs. T N COMBINED D-V-A SPECTRUM
A =V = ω Dn ωn
T 2π n A =V = D 2π Tn
RESPONSE SPECTRUM CHARACTERISTICS
Response spectrum ( ζ= 0,2,5, and 10%) and peak values of ground acceleration, ground velocity, and ground displacement for El Centro ground motion.
Response spectrum for El Centro ground motion plotted
with normalized scale A/ϋgo , V/ůgo , and D/ugo ; ζ = 0, 2 , 5 and 10%. RESPONSE SPECTRUM CHARACTERISTICS
Response spectrum for El Centro ground motion shown by a solid line together with an idealized version shown by a dashed line ; ζ = 5% Response Spectrum Characteristics Mid Frequency Low Frequency Transition from High Frequency Out-of-Phase Out-of-Phase to In-Phase Rigid Static Response In-Phase Response Response
F F F 1 2 ZPA Frequency
F1 = frequency at which peak spectral acceleration is observed F2 = frequency above which the SDOF (modal) oscillators are in-phase with the transient acceleration input used to generate the spectrum and in phase with each other
FZPA = frequency at which the spectral acceleration returns to the zero period acceleration; maximum base acceleration of transient acceleration input used to generate the spectru m ACCELERATION RESPONSE SPECTRUM EL CENTRO EARTHQUAKE 5% DAMPING
IT IS NOT PRACTICALLY POSSIBLE TO CALCULATE EXACT STRUCTURAL PERIOD .
SPECTRAL ACCELERATION FOR SHORT PERIOD IS VERY IRREGULAR.
FOR PRACTICAL USE IT HAS TO MADE ‘SMOOTH’ Elastic Design Spectrum
Use recorded ground motions (available) Use ground motions recorded at similar sites: Magnitude of earthquake Distance of site form earthquake fault Fault mechanism Local Soil Conditions Geology/travel path of seismic waves
Motions recorded at the same location. For design, we need an envelope. One way is to take the average (mean) of these values DESIGN RESPONSE SPECTRUM
(Design Spectrum may include more than one earthquake scenario) Factor Influencing Response Spectra The response spectral values depends upon the following parameters, •I) Energy release mechanism •II) Epicentral distance •III) Focal depth •IV) Soil condition •V) Richter magnitude •VI) Damping in the system •VII) Time period of the system RESPONSE SPECTRUM METHOD OF ANALYSIS Introduction Response spectrum method is favoured by earthquake engineering community because of:
It provides a technique for performing an equivalent static lateral load analysis.
It allows a clear understanding of the contributions of different modes of vibration .
It offers a simplified method for finding the design forces for structural members for earthquake.
It is also useful for approximate evaluation of seismic reliability of structures. Contd… The concept of equivalent lateral forces for earth- quake is a unique concept because it converts a dynamic analysis partly to dynamic & partly to static analysis for finding maximum stresses.
For seismic design, these maximum stresses are of interest, not the time history of stress.
Equivalent lateral force for an earthquake is defined as a set of lateral force which will produce the same peak response as that obtained by dynamic analysis of structures .
The equivalence is restricted to a single mode of vibration. Contd… The response spectrum method of analysis is developed using the following steps. A modal analysis of the structure is carried out to obtain mode shapes, frequencies & modal participation factors.
Using the acceleration response spectrum, an equivalent static load is derived which will provide the same maximum response as that obtained in each mode of vibration.
Maximum modal responses are combined to find total maximum response of the structure. Contd… The first step is the dynamic analysis while , the second step is a static analysis. The first two steps do not have approximations, while the third step has some approximations. As a result, response spectrum analysis is called an approximate analysis; but applications show that it provides mostly a good estimate of peak responses. Method is developed for single point, single component excitation for classically damped linear systems. However, with additional approximations it has been extended for multi point-multi component excitations & for non- classically damped systems. Seismic code provisions All countries have their own seismic codes.
For seismic analysis, codes prescribe all three methods i.e. RSA & seismic coefficient method.
Codes specify the following important factors for seismic analysis: • Approximate calculation of time period for seismic coefficient method.
•Ch Vs T plot. • Effect of soil condition on A S a or& C h g g Contd… • Seismicity of the region by specifying PGA.
• Reduction factor for obtaining design forces to include ductility in the design.
• Importance factor for structure.
Provisions of a response spectrum in some country code. The codes include: • IBC – 2000 • NBCC – 1995 • EURO CODE – 1995 • NZS 4203 – 1992 • IS 1893 – 2002 Contd…
IS CODE (1893-2002) • Time period is calculated by empirical formula and distribution of force is given by:
2 Whj j Fj = V b N (5.65) 2 ∑ Whj j j=1
Sa • Ce vsT& vsTare the same; they are given by: g 1+15T 0≤ T≤0.1s S a = 2.5 0.1 ≤ T≤0.4s for hard soil (5.62) g 1 0.4≤ T≤ 4.0s T Contd…
1+15T 0≤T≤0.1s S a = 2.5 0.1 ≤T≤0.55s formediumsoil (5.63) g 1.36 0.55 ≤ T ≤ 4.0s T 1+15T 0≤T≤0.1s S a = 2.5 0.1 ≤T≤0.67s for softsoil (5.64) g 1.67 0.67 ≤ T ≤ 4.0s T For the three types of soil S a/g are shown in Fig 5.13 Seismic zone coefficients decide about the PGA values. Contd… 6/6
3
Hard Soil
2.5 Medium Soil /g) a
Soft Soil 2
1.5
1
0.5 Spectral acceleration coefficientcoefficient SpectralSpectral acceleration acceleration (S (S
0 0 0.5 1 1.5 2 2.5 3 3.5 4
Time period (sec) Variations of ( Sa /g) with time period T Fig 5.13 Seismic force evaluation
• During base excitation – Structure is subjected to acceleration • From Newton’s second law – Force = mass x acceleration • Hence, seismic force acting on structure = Mass x acceleration Seismic force evaluation
• For design, we need maximum seismic force • Hence, maximum acceleration is required – This refers to maximum acceleration of structure – This is different from maximum acceleration of ground – Maximum ground acceleration is termed as peak ground acceleration, PGA – Maximum acceleration of rigid structure is same as PGA.
. . . Seismic force evaluation
• Seismic force = mass x maximum acceleration – Can be written as: • Force = (maximum acceleration/g) x (mass x g) = (maximum acceleration/g) x W – W is weight of the structure – g is acceleration due to gravity • Typically, codes express design seismic force as:
V = (Ah) x (W) – V is design seismic force, also called design base shear – Ah is base shear coefficient Seismic force evaluation
• Maximum acceleration of structure depends on – Severity of ground motion – Soil conditions – Structural characteristics • These include time period and damping • More about time period, later
• Obviously, base shear coefficient, Ah, will also depend on these parameters Seismic force evaluation
• Seismic design philosophy is such that, design seismic forces are much lower than actual seismic forces acting on the structure during severe ground shaking – Base shear coefficient has to ensure this reduction in forces • Hence, base shear coefficient would also have a parameter associated with design philosophy Seismic force evaluation
• Thus, base shear coefficient depends on: – Severity of ground motion – Soil condition – Structural characteristics – Design philosophy
IS 1893 (Part 1):2002
• Ah = (Z/2). (I/R). (S a/g) – Z is zone factor – I is importance factor – R is response reduction factor
– Sa/g is spectral acceleration coefficient IS 1893 (Part 1):2002
• Zone factor, Z – Depends on severity of ground motion – India is divided into four seismic zones (II to V) – Refer Table 2 of IS 1893(part1):2002 – Z = 0.1 for zone II and Z = 0.36 for zone V IS 1893 (Part 1):2002
• Importance factor, I – Ensures higher design seismic force for more important structures – Values for buildings are given in Table 6 of IS :1893 • Values for other structures will be given in respective parts • For tanks, values will be given in Part 2 IS 1893 (Part 1):2002
• Response reduction factor, R – Earthquake resistant structures are designed for much smaller seismic forces than actual seismic forces that may act on them. This depends on • Ductility • Redundancy • Overstrength – See next slide IS 1893 (Part 1):2002 Δ
Total Horizontal Maximum force F Load if structure remains elastic el Due to Linear Elastic Ductility Response Non linear Maximum Response F Load Capacity y First Due to Significant Redundancy Load at F Yield First Yield s Due to Overstrength F Design force des Total Horizontal Load Load HorizontalHorizontal Total Total
0 Δ Δ Δ w y max Roof Displacement (Δ)
Maximum Elastic Force (Fel) Response Reduction Factor = Design Force (Fdes) IS 1893 (Part 1):2002
• Response reduction factor (contd..) – A structure with good ductility, redundancy and over strength is designed for smaller seismic force and has higher value of R • For example, building with SMRF has good ductility and has R = 5.0 as against R = 1.5 for unreinforced masonry building which does not have good ductility – Table 7 gives R values for buildings • values will be given in IS:1893 (Part 2) IS 1893 (Part 1):2002
• Spectral acceleration coefficient, S a/g – Depends on structural characteristics and soil condition • Structural characteristics include time period and damping – Refer Fig. 2 and Table 3 of IS:1893 – See next slide IS 1893 (Part 1):2002
For 5% damping IS 1893 (Part 1):2002
• For other damping, S a/g values are to be multiplied by a factor given in Table 3 of IS:1893 – Table 3 is reproduced below
% 0 2 5 7 10 15 20 25 30 damping Factor 3.20 1.40 1.00 0.90 0.80 0.70 0.60 0.55 0.50
For higher damping, multiplying factor is less Hence, for higher damping, S a/g is less