Civil Engineering Reference
In-Depth Information
where
M
i
is the seismic mass of the
i
th level, g the acceleration of gravity.
(e) Perform a static pushover of the structure subjected to the storey forces computed in step (d) and
corresponding to each mode independently.
(f) Estimate element (or local) and structure (or global) forces and displacements by means of SRSS
combinations of each modal quantity for the
k
th step of analysis. Add the above quantities, i.e.
forces and displacements, to the relevant quantity of the (
k
- 1)th step.
(g) Compare the values established in step (f) to the limiting values for the specifi ed performance
goals at both local and global levels, as provided, for example, in Section 4.7. Return to step (b)
until the target performance is achieved.
A variant on the above is empirically using the input motion spectrum to scale the modal contribu-
tions to the applied force or displacement vector, referred to as adaptive pushover with spectrum scaling.
This procedure is not mathematically rigorous, but resembles the method of base shear calculation
adopted in codes, and also used in modal spectral analysis. The lack of rigour results from the applica-
tion of superposition to an inherently inelastic problem. Spectrum scaling provides an interesting angle,
whereby the static capacity curve is no longer unique to a structure, but is a function of the input
motion.
The steps of the adaptive pushover analysis utilizing the scaling of acceleration spectrum are as
follows:
(a) Apply the gravity loads in a single step.
(b) Perform an eigenvalue analysis of the structure at the current stiffness state. The elastic stiffness
can be used for the initial step. Eigenvalues and eigenvectors are computed.
(c) Determine the modal participation factors Γ
j
for the
j
th mode using equation (4.17.5) in Section
4.6.1.1 .
(d) Compute the modal storey forces at each fl oor level for the
N
modes deemed to satisfy mass
participation of at least 85-90% of the total mass. These forces
F
i,j
are estimated at the
i
th level
for the
j
th mode (being 1 ≤
j
≤
N
) as given below:
()
F
=
ΓΦ
a
WS
T
g
(4.30)
ij
,
j
i
ij
,
,
j
j
where
S
a,
j
(
T
j
) is the spectral acceleration relative to the
j
th mode with period of vibration equal
to
T
j
.
(e) Compute the modal base shears
V
j
as follows:
N
=
∑
V
=
F
(4.31)
j
i j
,
i
1
where
N
is the number of stories.
(f) Combine the force determined in step (e). Use, for example, the SRSS combination rule as shown
below:
N
=
∑
V
2
(4.32)
=
V
j
i
1
(g) The storey modal base shears
V
j
computed in step (e) are uniformly scaled:
V
j
=
SV
(4.33.1)
n
j
