Optimal Performance- Based Seismic Design - Structural Seismic Design Optimization and Earthquake Engineering - page 180

where, m is the component or element demand
modifier (factor), to account for expected ductility
associated with this action at the selected Structural
Performance Level; i.e., the m-factors modify
the results so as to be similar to the results of a
nonlinear analysis. The m-factors are specified in
Chapters 4 through 8 of FEMA356.
Q CE is the expected strength of the component
or element at the deformation level under consid-
eration for deformation-controlled actions.
Q UD is Deformation-controlled design action
due to gravity loads and earthquake loads.
k is the Knowledge factor (0.75 0r 1.0) de-
pending on the accuracy of collected data and
performance objectives.
For a force-controlled structural component,
in the absence of either a nonlinear or limit state
analysis the following formula applies to the
design force, Q UF .

in question. Otherwise, a lower performance level
is examined.

Nonlinear Analyses

Parallel and simultaneous improvements in com-
putational facilities in commercial software and
the theories supporting nonlinear static seismic
(pushover) analysis, is making the pushover
method more accessible and reliable for engineers,
and from a practical point of view, it is foreseen
to be the most popular analysis procedure in the
future for performance-based seismic design.
The outcome of pushover analysis is the inelastic
capacity curve of the structure. This curve defines
the capacity of the building independent of any
earthquake. To make it useful for evaluation of
performance point for a given earthquake, this
curve has to be converted to spectral ordinates.
This will be discussed later.
On the other hand because of its complexity,
the nonlinear dynamic analysis (the 4 th method)
is hardly used by engineers. However, since it
is a reliable analysis method, it is more often
used for research purposes. When this method
is used the demand data can be directly used for
evaluation of performance of the structure. Design
optimization using nonlinear dynamic analysis is
an extraordinarily difficult subject that is not yet
used for practical design problems.
When nonlinear analysis is used for demand
evaluation, similar to deformations, the internal
forces obtained for components, are used without
substantial modification. One should realize that
unlike to nonlinear dynamic analysis, nonlinear
static (pushover) analysis does not fully reflect
the nonlinear behaviour of a structure. It does not
account for cycle strength and stiffness degrada-
tion. Also, the ductility demand concentration of
individual stories is not tracked in this method.
Two accepted methods i.e., the method of coef-
ficients and method of capacity spectrum have
been proposed to convert the outcome of nonlinear

Q
C C C J

Q = Q ±

E

(4)

UF G

1 2 3

Q UF is the Force-controlled design force due
to gravity plus earthquake loads. J is Force-
delivery reduction factor defined by FEMA356
and is greater than 1. Coefficients C1 , C2 , and
C3 are amplification factors and are defined in
FEMA356. They are used to amplify the design
base shear for achieving displacement target at
the desired performance level. This pre-standard
recommends the following formula for forced
controlled members.

kQ > Q (5)

CL UF

k was previously defined. Q CL is Lower-bound
strength of a component or element at the deforma-
tion level under consideration for force-controlled
actions.
If structural components have capacities more
than the demand, it satisfies the performance level