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} , u_{g}
_{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 T_{a}
= 0.033 s, T_{b} = 0.125 s, T_{e} = 10 s, T_{f}
= 33 s.
We identify the effects of
damping on systems with short period T_{n} < T_{a}
= 0.033, the peak-pseudo acceleration A = S_{pa}
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 S_{d} with
approach to u_{g}_{0}, S_{pa} is very
small. For a rigid mass the structure is flexible. In that case
For short period system T_{n}T_{a}
< T_{n} < T_{c}. S_{pa}
exceeds u˙˙_{g} _{0} with amplification
depending on T_{n}, ρ over a period range T_{b} to T_{c},
S_{pa} may be constant = u˙˙_{g} _{0}
× amplification factor depending on ρ .
For a long period T_{d}
< T_{n} < T_{f}, S_{d}
generally exceeds u_{g}_{0} with amplification generally
depending on ρ. Over a
portion of the period T_{d} to T_{e}(3–10 s) S_{d}
may be idealized as a constant ×
amplification factor depending on ρ. For intermediate period systems with T_{n}
between T_{c} < T_{n} < T_{d},
S_{pv} exceeds u˙_{g} _{0}
Over the period range S_{pv} 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 T_{n} > T_{d}.
Displacement-sensitive region because structure response is related
mostly to ground displacement.
• Short
period range T_{n} < T_{c}.
Acceleration-sensitive region because structural response is mostly
related to ground acceleration.
• Intermediate
range T_{c} < T_{n}
< T_{d}. Velocity-sensitive region because structural
response appears to be better related to ground velocity than to other ground
motion parameters.
The periods T_{a},
T_{b}, T_{e}, T_{f} on the
idealized spectrum are independent of damping but T_{c} and T_{d}
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.
Example 17.7
Consider an elastic design
spectrum, 84.1% for ground motion u˙˙_{g} _{0}
= 1 g ; u˙_{g}
_{0} = 121.92 cm/s; u_{g}_{0} = 91.44 cm; ρ = 5%. Using the program
developed it is possible to construct a design spectrum as shown in Fig. 17.23.
Solution
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.
Example 17.8
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.
Solution
Damping = 5%
(i) NS
direction, T = 0.567 s
From chart (see Fig. 17.23), spectra value S_{d}
= 6.35 cm; S_{pv} = 71.12 cm/s; S_{pa} = 784.35
cm/s^{2}.
Maximum
base shear = mS_{pa}
= 131 697.2
× 7.843
1032.9 kN
(ii)
EW direction S_{d} = 20 mm
S_{pv} = 393.7 mm/s S_{pa}
= 7.843 mm/s^{2} ω_{n} = 20 rad/s
T = 0.313
Maximum base shear = mS_{pa}
= 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.