Optimal Performance- Based Seismic Design - Structural Seismic Design Optimization and Earthquake Engineering - page 198

Figure 8. The ten story reinforced concrete frame,
Example 1

Upon establishing the explicit formulation of
the inelastic displacement, Eq. (49), Zou and Chan
could explicitly write the optimization problem
of minimizing the steel construction cost of a
multistory RC building with drift constraints in
terms of the design variables, ρ i .
In Zou and Chan's study (2005), since the
dimensions of frame elements were fixed, the
objective function of the OPBSD problem W s was
the volume or weight of reinforcements. The drift
constraints were written in terms of design vari-
ables, ρ , in the form of quadratic functions at the
target displacement. The design optimization
problem became in Box 2. Where d j U is the upper
limit on story drift and ψ1i and ψ2i are first and
second derivatives of differential displacement
δ ∆ j with respect to ρ. They are expressed in terms
of two end moments and first and second deriva-
tives of plastic hinge rotations, θ, with respect to
ρ . (See Zou and Chan 2005).
The above optimization problem was solved
utilizing Optimality Criteria method for a ten story
2D frame of Figure (8), in which, the concrete
cylinder strength and steel reinforcement yield
strength, fy , for all members were assumed to be
20 and 335 MPa, respectively.
The problem was solved for two different
gravity loads, to show the effect of axial force on
the pushover performance and capacity of frame-
work. The followings have been reported as the
outcome of design optimization process.
In the load Case A, the gravity is 30 kN/m while
that of Case B is 10 kN/m. The results show that,
as expected, more reinforcements are required for
the beams in Case A relative to Case B. This is
because the gravity forces have more influence on
bending moment of beams compared to columns.
For columns, the situation is somehow differ-
ent. Most of columns in Case B required more
reinforcement in comparison to Case A and there
were more plastic hinges in columns of Case B
compared to A. This might be because the gravity
forces have a reverse effect compared to lateral

loads on tensile reinforcement of columns. After
design optimization, there were 45 plastic hinges
in Case A while there were 55 plastic hinges in
Case B. The distribution of plastic hinges was
albeit different in the two Cases. Overall, the
reinforcement in Case B was more than Case A.

Example 2

As the second example consider the three story
2D frame in Figure (9) that was studied by Xu
et al. (2006).