1 March 22, 2002Singapore, Elgamal Response Spectrum Ahmed Elgamal

2 March 22, 2002Singapore, Elgamal Dynamics of a Simple Structure The Single-Degree-Of-Freedom (SDOF) Equation References Elements Of Earthquake Engineering And Structural Dynamics, André Filiatrault, Polytechnic International Press, Montréal, Canada, ISBN (Section 4.2.3). Dynamics of Structures, Anil K. Chopra, Prentice Hall, New Jersey, ISBN (Chapter 3).

3 March 22, 2002Singapore, Elgamal Equation of motion (external force) k 1 u fsfs fDfD c 1 ů f s = k uf D = c ů fSfS fIfI p(t) fDfD fIfI fSfS fDfD FBD p(t) u (t) m c k Mass-spring damper system f I + f D + f s = p(t) Free-body diagram

4 March 22, 2002Singapore, Elgamal u g (t) Earthquake Ground Motion f I + f D + f s = 0 You may note that Earthquake Excitation Fixed base (t) External force = u g (t)

5 March 22, 2002Singapore, Elgamal Undamped natural frequency Property of structure when allowed to vibrate freely without external excitation m k  Undamped natural circular frequency of vibration (radians/second)    2 f natural cyclic frequency of vibration (cycles/second or 1/second or Hz) f 1 T  natural period of vibration (second) T is the time required for one cycle of free vibration If damping is present, replace  by  D where 2 D 1  natural frequency,and   m2 c fractionof critical damping coefficient c c c  (dimensionless measure of damping) called damping ratio km2 c  2m2c c 

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9 March 22, 2002Singapore, Elgamal (0.05 for example) if but w = u + u g or

10 March 22, 2002Singapore, Elgamal Deformation Response Spectrum For a given EQ excitation calculate |u max | from SDOF response with a certain  and within a range of natural periods or frequencies. |u max | for each frequency will be found from the computed u(t) history at this frequency. A plot of |u max | vs. natural period is constructed representing the deformation (or displacement) response spectrum (S d ). From this figure, one can directly read the maximum relative displacement of any structure of natural period T (and a particular value of  as damping) From: Chopra, Dynamics of Structures, A Primer

11 March 22, 2002Singapore, Elgamal Concept of Equivalent lateral force f s If f s is applied as a static force, it would cause the deformation u. Thus at any instant of time: f s = ku(t), or in terms of the mass f s (t) = m  2 u(t) The maximum force will be S d = deformation or displacement response spectrum = pseudo-acceleration response spectrum The maximum strain energy E max stored in the structure during shaking is: = pseudo-velocity response spectrumwhere From: Chopra, Dynamics of Structures, A Primer

12 March 22, 2002Singapore, Elgamal From: Chopra, Dynamics of Structures, A Primer

13 March 22, 2002Singapore, Elgamal From: Chopra, Dynamics of Structures

14 March 22, 2002Singapore, Elgamal From: Chopra, Dynamics of Structures, A Primer

15 March 22, 2002Singapore, Elgamal From: Chopra, Dynamics of Structures

16 March 22, 2002Singapore, Elgamal From: Chopra, Dynamics of Structures

17 March 22, 2002Singapore, Elgamal From: Chopra, Dynamics of Structures Motions recorded at the same location. For design, we need an envelope. One way is to take the average (mean) of these values (use statistics to define curves for mean and standard deviation, see next)

18 March 22, 2002Singapore, Elgamal From: Chopra, Dynamics of Structures Mean response spectrum is smooth relative to any of the original contributing spectra

19 March 22, 2002Singapore, Elgamal From: Chopra, Dynamics of Structures

20 March 22, 2002Singapore, Elgamal From: Chopra, Dynamics of Structures (Design Spectrum may include more than one earthquake scenario)