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= -MIx g (3)
Pre-multiplied by ® r , equation (3) becomes
mq + cq + kq = -£, (4)
Where
r=$> J MI (5)
(6)
(7)
(8)
If it is a proportional or classical damping system, according to Duhamel integration, the modal
coordinates can be calculated on the basis of equation (4). Then, the displacement can be calculated on
the basis of equation (2). Hence, the structural members' force and deformations can also be obtained.
Maximum displacement
At the initial stage of structural design, a structural engineer should consider reasonable deformations
under various earthquake intensities. Nevertheless, the prediction of important characteristic of input
ground motion and the basic structural parameters are uncertain. Based on displacement spectrum, the
maximum value of displacement is given. Because the inelastic displacement spectrum is not presented
by Code for seismic design of buildings [2001], the displacement spectrum is transform from the
acceleration spectrum, as followed:
Oi^ (9)
So, the maximum value of thej-f/z modal coordinates can be expressed by:
W=aK>f>« 0°)
When the periods and modes of a structure at previous step is known, according to modal analysis,
both the relative-displacement of the structural members and the inter-storey drifts can be calculated
using SRSS (Square Root of the Sum of the Square). And then force corresponding to them can also be
obtained.