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Response Spectrum Ahmed Elgamal

Response Spectrum Ahmed 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 2-553-00629-4 (Section 4.2.3).

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Response Spectrum Ahmed Elgamal

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  1. Singapore, Elgamal Response Spectrum Ahmed Elgamal

  2. Singapore, 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 2-553-00629-4 (Section 4.2.3). Dynamics of Structures, Anil K. Chopra, Prentice Hall, New Jersey, ISBN 0-13-855214-2 (Chapter 3).

  3. Singapore, Elgamal fI p(t) fs fD k c 1 1 u ů fS fD fD fI p(t) fS FBD u(t) c p(t) m k Mass-spring damper system Equation of motion (external force) Free-body diagram  fI + fD + fs = p(t) fs = k u fD = c ů

  4. Singapore, Elgamal External force (t) = Earthquake Excitation Fixed base ug(t) ug(t) Earthquake Ground Motion fI + fD + fs = 0 You may note that

  5. Singapore, Elgamal k w = m w = f p 2 1 = T f w = w - x 2 1 D c x = w 2 m c c = = c 2 km c = w = c 2 m 2 km c Undamped natural frequency Property of structure when allowed to vibrate freely without external excitation Undamped natural circular frequency of vibration (radi an s/second) natural cyclic frequency of vibration (cycles/second or 1/second or Hz) natural period of vibration (second) T is the time required for one cycle of free vibration w w If damping is present, replace by D where natural frequency , and fraction of critical damping coefficient (dimensionless measure o f damping) called damping ratio

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  9. Singapore, Elgamal if (0.05 for example) but w = u + ug or

  10. Singapore, Elgamal Deformation Response Spectrum For a given EQ excitation calculate |umax| from SDOF response with a certain  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  as damping) From: Chopra, Dynamics of Structures, A Primer

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

  12. Singapore, Elgamal From: Chopra, Dynamics of Structures, A Primer

  13. Singapore, Elgamal From: Chopra, Dynamics of Structures

  14. Singapore, Elgamal From: Chopra, Dynamics of Structures, A Primer

  15. Singapore, Elgamal From: Chopra, Dynamics of Structures

  16. Singapore, Elgamal From: Chopra, Dynamics of Structures

  17. Singapore, 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. Singapore, Elgamal From: Chopra, Dynamics of Structures Mean response spectrum is smooth relative to any of the original contributing spectra

  19. Singapore, Elgamal From: Chopra, Dynamics of Structures

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

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