For a system with longer period Sd with approach to ug0, Spa is very small.
Response spectrum characteristics
Let u˙˙g 0 , u˙ g 0 , ug 0 be the peak values of ground acceleration, velocity and displacement respectively. Response spectrum values are presented to normalized form in Fig. 17.22. The period range may be separated by period values at a, b, c, d, e and f where Ta = 0.033 s, Tb = 0.125 s, Te = 10 s, Tf = 33 s.
We identify the effects of damping on systems with short period Tn < Ta = 0.033, the peak-pseudo acceleration A = Spa approaches u˙˙g 0 and D = Sd is very small. For a fixed mass, very short period means extremely stiff or essentially rigid. Deformation will be very small and it moves with the ground.
For a system with longer period Sd with approach to ug0, Spa is very small. For a rigid mass the structure is flexible. In that case
For short period system TnTa < Tn < Tc. Spa exceeds u˙˙g 0 with amplification depending on Tn, ρ over a period range Tb to Tc, Spa may be constant = u˙˙g 0 × amplification factor depending on ρ .
For a long period Td < Tn < Tf, Sd generally exceeds ug0 with amplification generally depending on ρ. Over a portion of the period Td to Te(3–10 s) Sd may be idealized as a constant × amplification factor depending on ρ. For intermediate period systems with Tn between Tc < Tn < Td, Spv exceeds u˙g 0 Over the period range Spv may be idealized as a constant value × amplification factor depending on ρ.
Based on the observation of response spectrum, it is logical to divide the spectrum into three ranges:
• Long period range Tn > Td. Displacement-sensitive region because structure response is related mostly to ground displacement.
• Short period range Tn < Tc. Acceleration-sensitive region because structural response is mostly related to ground acceleration.
• Intermediate range Tc < Tn < Td. Velocity-sensitive region because structural response appears to be better related to ground velocity than to other ground motion parameters.
The periods Ta, Tb, Te, Tf on the idealized spectrum are independent of damping but Tc and Td vary with damping.
Idealizing a spectrum by a series of straight lines a, b, c, d, e, f in the four-way logarithmic plot is obviously not a precise process. The period values at a–f and amplification factors are judgemental. The advantages of an idealized spectrum are that we can very easily construct a design spectrum. These values vary from one ground motion with others.
Consider an elastic design spectrum, 84.1% for ground motion u˙˙g 0 = 1 g ; u˙g 0 = 121.92 cm/s; ug0 = 91.44 cm; ρ = 5%. Using the program developed it is possible to construct a design spectrum as shown in Fig. 17.23.
From Fig. 17.23, we can construct a pseudo-acceleration spectrum in terms of g plotted in log scale in Fig. 17.24 for ground acceleration of 1g and damping factor 5%. Similarly for various values of ρ an elastic pseudo-acceleration spectrum can be plotted in log scale as shown in Fig. 17.25 and a design spectrum in Fig. 17.26. If pseudo-acceleration is plotted at a normal scale, the diagram is as shown in Fig. 17.26.
Estimate the maximum sensitive response for the industrial building of Example 17.1 using Newmark–Hall design spectra for an anticipated ground acceleration
of 0.308g and for a damping factor of 0.05. Compare the results with the maximum response obtained from time history analysis.
Damping = 5%
(i) NS direction, T = 0.567 s
From chart (see Fig. 17.23), spectra value Sd = 6.35 cm; Spv = 71.12 cm/s; Spa = 784.35 cm/s2.
Maximum base shear = mSpa
= 131 697.2 × 7.843
(ii) EW direction Sd = 20 mm
Spv = 393.7 mm/s Spa = 7.843 mm/s2 ωn = 20 rad/s
T = 0.313
Maximum base shear = mSpa
= 1032.9 kN
Comparison of the maximum response obtained from time history analysis response spectra and design spectrum analysis is presented in Table 17.4 for NS direction. There is a considerable discrepancy between the results of response spectrum and design spectrum. The former represents the response to a specific earthquake while the latter represents predicted response to any earthquake.