Optimal Performance- Based Seismic Design - Structural Seismic Design Optimization and Earthquake Engineering - page 194

l

i can be calculated based on how it is
obtained. For example, if Eq.(1) is used for its
calculation, this equation should be differentiated
with respect to design variables and Eqs.(31 &
32 should be used. Since we know how to calcu-
late P in terms of design variables, its derivatives
may be easily obtained.
The second term in Eq.(40) is derivative of
the resultant of internal forces in nodes. It can
be calculated by summing up the derivatives of
internal forces in elements. The derivatives of in-
ternal forces in elements, in turn, can be calculated
using Eq.(33) and (34). Note that in Eq.(40), the
second term is the derivative of internal forces in
the previous iteration, and from Eqs.(35 & 36 it
is understood that the internal forces themselves
are determined based of deformations of previ-
ous loadings.
The third term in the right hand side of Eq.(40),
is easily calculated by assembling the derivatives
of the stiffness matrix of all elements with respect
to any design variable. This is because for any
specific amount of deformation, the tangential
stiffness matrix can be expressed explicitly, and
its derivatives can be obtained accordingly.
Summing up the derivatives of displacements
in all load segments, results the derivative of total
displacement in a nonlinear force-displacement
environment.

∂ P ∂ x to Figure 3a, followed a similar procedure for

sensitivity analysis of displacement. On the other
hand, Habibi used the modified Newton-Raphson
method as shown in Figure 3f. In this method,
consideration of unbalanced forces is a must.
Therefore, in the establishment of corresponding
sensitivity analysis, this matter was taken care
of. Another common point in the two sensitivity
methods was the fact that sensitivity assessment
was obtained through an accumulative calcula-
tion process. This is not to say that the sensitivity
calculation is an iterative process, but is to say
that during the analysis process, some derivatives
have to be obtained and summed up for use in
sensitivity computation in upcoming stages.
In Gong's method of analysis, a lumped in-
elasticity was considered, and a plasticity factor
was defined to simulate the partial plasticity with
rigidity degradation in a semi-rigid connection.
This particular model with all its circumstances
was exactly used for sensitivity analysis. Obvi-
ously the outcome of the sensitivity analysis is
consistent with the basic assumptions made.
However, in Habibi's nonlinear analysis, a spread
plasticity model was adopted for consideration
of nonlinear flexibility and nonlinear stiffness
characteristics of a concrete beam column. There
is no doubt that the corresponding sensitivity
analysis procedure is fundamentally different from
Gong's method. Another difference between the
two methods returns to the method of natural
period evaluation. Gong used a Rayleigh method
for determination of natural period of the structure
while Habibi used the analytical method for
evaluation of natural period. Of course, the sen-
sitivity calculations for ∂

Discussion on Sensitivity Analyses

In previous sub-sections, two distinctive sensitiv-
ity analyses were briefly described. Here in this
section some similarities and dissimilarities of the
two methods are discussed to suggest any future
research activities in this field.
A common point in these two methods was the
fact that the formulation of nonlinear sensitivity
analysis in both methods was established based
on the nonlinear procedures that were followed
for nonlinear analyses. For example, Gong who
used an enhanced incremental method similar

T d i are remarkably

different.
As a conclusion, the nonlinear sensitivity
analysis formulation has to be built based on
adopted nonlinear analysis procedure; otherwise,
the results may not be consistent and may result
in difficulty in convergence of the optimization
algorithm. The similarities and dissimilarities of