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Chapter 3
Design Problem Formulation
3.1 Design Performance Levels
To implement performance-based design, one or more building performance levels must
be selected. A performance level is a statement of the desired building behaviour when it
experiences earthquake demands of specified severity. Four building performance levels
are defined in the literature (FEMA-273, 1997), namely, Operational (OP), Immediate
Occupancy (IO), Life Safety (LS) and Collapse Prevention (CP) levels. For each level,
the qualitative description of the building performance is quantified into structural
response parameters, such as target displacement, design base shear, inter-story drifts,
etc. For example, for steel moment frames it is suggested that roof drift ratios of 0.7%,
2.5% and 5% roughly correspond to the target displacements of the IO, LS and CP
performance levels, respectively (FEMA-273, 1997).
The primary parameter used to quantify structural performance is inter-story drift,
which is an excellent parameter for judging the ability of a structure to resist instability
and collapse. The inter-story drift δs of story level s is defined by the following equation:
51
(3.1a)1−−=δ sss vv
where vs and vs-1 are the lateral drifts found through nonlinear pushover analysis at story
level s and (s-1), respectively. Inter-story drift δs is often normalized as the inter-story
drift ratio or angle,
s
ss h
δ=θ (3.1b)
where hs is the height of story level s.
Inter-story drift can be related to the plastic rotation demand imposed on individual
beam-column connection assemblies, and is therefore a good predictor of the
performance of beams, columns and connections (FEMA-350, 2000). Allowable inter-
story drift ratios (so-called inter-story drift capacity) corresponding to the IO and CP
levels are given in FEMA-350. For example, the allowable inter-story drifts of a low-rise
building (1 to 3 stories) at the IO and CP levels are 1.25% and 6.1% of the height of the
storey, respectively.
To complete the specifications of a performance objective, particular earthquake
intensities for which satisfactory performance is to be maintained must be selected. Four
probabilistic hazard levels related to earthquakes having 50%, 20%, 10% and 2%
probability of exceedance in 50 years (mean return periods of 72, 225, 474, and 2475
years) are defined in FEMA-273. The design base shears for the four performance levels
can be evaluated by re-writing Eq. (2.1) as:
52
b
iai
b Wg
SV = (3.2) i=OP, IO, LS, CP
where: superscript i refers to building performance level i; Sai is the spectral response
acceleration; and Wb is the seismic weight of the particular moment frame of the building
under consideration.
The spectral response acceleration Sai for performance level i is calculated by re-
writing Eq. (2.2) as:
( )
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
>
≤<
≤<+
=
ie
e
iiv
ie
iis
ia
ie
ioe
is
ia
ia
TTT
SF
TTTSF
TTTTSF
S
01
00
0
2.0
2.0034.0
i=OP,IO,LS,CP (3.3)
A complete design objective specification is composed of a quantified structural
performance description plus a specified earthquake intensity. A commonly defined
objective called the Basic Safety Objective requires the building to be designed to
achieve both the LS performance level for a 10%/50-year earthquake and the CP
performance level for a 2%/50-year earthquake. Other desired design objectives are
achieving the IO performance level for a 20%/50-year earthquake, and the OP
performance level for a 50%/50-year earthquake. All four of the performance levels and
corresponding earthquake intensities noted in the foregoing are considered by this
research study. An illustrative example determination of design spectra parameters is
presented in Appendix 3.A.
53
3.2 Design Optimization Problem Formulation
Structural optimization seeks optimal values of design variables that achieve the best
outcome of a given objective (or objectives) while satisfying code or designer-specified
criteria. In mathematical notation, the design optimization problem can be cast in the
following general form:
{ }( ) ( )
( )
( )xjj
xUjj
Lj
gUll
Tn
n,j x
n,j xxx
nl gg
)(OBJ),(OBJ),(OBJ o
L
L
L
L
21,
21,
,2,1oSubject t
Minimize 21
=∈
=≤≤
=≤
=
X
x
xxxOBJ (3.4a)
(3.4b)
(3.4c)
(3.4d)
where: OBJ is a vector of no objective functions, comprised of individual objective
functions OBJk, (k=1, 2, …, no); x={x1, x2, …, xnx}T is the vector of nx design variables
that are required to be found in order to minimize the objective function(s); gl(x) is the lth
constraint function bounded by its upper limit, glU; Eqs. (3.4c) and (3.4d) are alternative
side constraints on the design variables. For optimization using continuously varying
design variables, side constraints Eqs. (3.4c) alone are sufficient to limit each design
variable xj to be within its lower bound, xjL, and its upper bound, xj
U. For the design of a
steel building using discrete sizing variables, side constraints Eqs. (3.4d) specify each
design variable xj to be selected from among a predetermined set Xj of discrete sizes.
3.2.1 Objective Functions
An objective function, often known as a cost or performance criterion, is expressed in
terms of the design variables and serves as a decision motivator. The optimal design is
54
the one providing the best value for the objective function while satisfying all the
constraints; thus, the selection of an appropriate objective function is extremely
important.
How to define an optimal or best-performance design in a performance-based
engineering context? What do developers, owners, users, designers, contractors or society
expect from a building designed by the performance-based concepts? Before answering
these questions, it is advantageous to first study the possible attributes of an optimal
design.
The attributes of an optimal performance-based building design could include the
following: 1) good deformability and energy dissipation capacity (structures that meet the
design criteria for strength and ductility can be regarded as having these attributes); 2)
favorable collapse mechanism mode at failure (panel or ‘soft’ story mechanisms should
be avoided); 3) minimum building damage under earthquake loading (the essence of
performance-based design for damage reduction); 4) cost savings (capital construction
cost, or lifetime cost, or both).
When making the selection of objective criteria, one must consider the underlying
analysis method used for the design. For example, if minimum dynamic hysteretic energy
is taken as an objective, the dynamic analysis method must be used. If collapse
mechanisms are taken into account, one must use a nonlinear analysis method that can
trace the progressive deterioration of the structure.
According to the above consideration of attributes, two objective functions
concerning structural cost and building damage under earthquake loading are selected
explicitly for this study. Deformability demand is addressed by the constraints, while
55
concern for ‘weak-story’ collapse mechanism formation at failure is implicitly accounted
for by the damage objective.
Minimum structural cost is a favorable design objective and is universally
adopted in many optimization problems. Some studies have been conducted to develop a
general building cost function (Walker, 1977; Corotis, Jiang & Ellis, 1998; Wen & Kang,
1998). Unfortunately, a complete description of the real cost of a building before its
construction is often nearly impossible because accurate cost data requires information
from many design disciplines, and many factors that influence the cost are unpredictable
and not precisely defined. As the mathematical formulation of a meaningful cost function
in a truly broad context is virtually impossible to achieve for a structural framing system
considered in isolation, as herein, the cost of the members of the structure is alone taken
to define the cost objective function for this study. Assuming that the cost of a member is
proportional to its material weight, that the unit material cost for each member is the
same, and that the member has a prismatic section throughout its length, the least-cost
design can be interpreted as the least-weight design of the structure, and the cost
objective function OBJ1 (also called the weight objective) to be minimized can be
formulated as:
( ) ∑=
ρ=ne
jjj ALOBJ
11 x (3.5)
where: ne is the number of members; ρ is the material mass density; and Lj and Aj are the
fixed length and variable cross-section area of member j, respectively.
56
To facilitate the structural optimization process, the function OBJ1 is normalized
by dividing the frame weight by the maximum possible weight of the frame
Wmax=∑ρLjAjU, (j=1, 2, …, ne), where Aj
U is the upper-bound cross section area for
member j, and the summation extends over all members, i.e.,
( ) ∑=
ρ=ne
jjj AL
WOBJ
1max1
1x (3.6)
In addition to minimizing the structure cost, minimizing the damage to the
building under earthquake loading is another favourable design objective (perhaps more
desirable), especially after the 1994 Northridge earthquake that caused tremendous
economic loss (and thereby directly ignited performance-based engineering). Here, it is
required to quantitatively formulate structural damage in terms of structural response.
One way to quantify the degree of damage to a structure is by a damage index
(Park and Ang, 1985; Cosenza, 1993; Rodriguez, 1994; Biddah, Heidebrecht, and
Naumoski, 1995; Teran-Gilmore, 1997). A damage index is expressed as a combination
of the damage caused by excessive deformation and that caused by repeated cyclic
loading. Several different damage index expressions are available, but none of them are
widely applicable. Since the damage index concept is still an issue under development
and, furthermore, since its evaluation involves the calculation of dynamic hysteretic
energy, which is beyond the capability of static pushover analysis, this research study
does not use damage indices to quantify building damage.
Another way to quantify the degree of damage to a building framework is to
establish the relationship between damage and inter-story drift. Inter-story drift is the
primary parameter in evaluating structural performance (Bhatti and Pister, 1981; FEMA-
57
274, 1997), and is widely regarded as a major parameter characterizing the extent of
plastic deformation of a building (FEMA-350, 2000; Gupta and Krawinkler, 2000). To
this end, it is necessary to only consider the plastic inter-story drift distribution at extreme
performance levels, such as the CP level, since damage in the elastic range of structure
response is of minor consequence. It was observed in many collapsed structures that
deformation concentration took place at a soft (or weak) story under severe earthquake
loading, which directly led to building collapse. It is therefore reasonable to assume that a
structure will undergo less damage if such deformation concentration is avoided; i.e., that
less damage will occur when the structure exhibits a more uniform inter-story drift
distribution when undergoing significant plasticity. Thus, the damage-mitigating
objective can be stated as pursuing a uniform inter-story drift distribution since this is
equivalent to achieving a uniform ductility demand over all stories (which avoids the
‘soft-story’ phenomenon (Chopra, 1995)). For this study, the building damage function
(so-called uniform ductility objective function) is defined in terms of the inter-story drifts
at the CP performance level as (choosing the CP level instead of the LS level at which to
formulate the damage function is prompted by the fact that the greatest ductility demand
occurs at the CP level),
( ) ( ) ( )2/1
1
2
2 ⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛ ∆−
δ= ∑
=
ns
s
CP
s
CPs
HhOBJ
xxx (3.7)
where: ∆CP and δsCP are the roof drift and the inter-story drift of story s at the CP
performance level; hs is the height of story s; H is the height of the building; and ns is the
number of building stories. By definition, the value of OBJ2(x) is not less than zero, and
58
only under the extreme case of a perfectly uniform inter-story drift distribution is
OBJ2(x)=0. Therefore, in addition to cost, the second objective of the design optimization
is to minimize the value of the damage function Eq. (3.7) for a building.
To facilitate the structural optimization process, the uniform ductility objective
function Eq. (3.7) is normalized by the number of stories ns. Furthermore, since a linear
story drift distribution is equivalent to a uniform inter-story drift distribution, the
normalized form of Equation (3.7) can be written as:
( ) ( )( )
2/11
1
2
2 11⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛−
∆= ∑
−
=
ns
sCP
sCPs
HHv
nsOBJ
xxx (3.8)
where vsCP is the drift of story s at the CP performance level and Hs is the vertical
distance from the base of the building to story level s. In effect, Eq. (3.8) defines the
coefficient of variation of the story drift distribution, since vsCP/Hs and ∆CP/H represent
the story-drift ratio and the mean-drift ratio, respectively.
3.2.2 Design Variables
The aim of structural design is to find the values of design variables which, for this study,
are taken to be the cross section sizes of the members. The design sizing variables may be
taken as being continuous or discrete. A continuous design variable may take any value
in a pre-set range of variation, while a discrete design variable can only take a value from
among a finite set of permissible values. A practical structural steel design usually
requires the use of discrete sections that are commercially available from steel
manufacturers. A widely adopted technique for optimizing a structure using discrete
59
variables (also adopted by this study) is to first conduct the design using continuous
variables and then achieve the final design by employing a certain discrete section
selection strategy.
For planar frameworks designed for (equivalent) static loading, there are four
basic cross-sectional properties for each member, i.e., the area A, moment of inertia I,
elastic modulus S (associated with first yield) and plastic modulus Z (associated with full
plasticity). These four properties for commercially available standard steel sections are
related together through functional relationships which, for this study, are expressed as:
I = C1⋅ A2 + C2 ⋅A + C3
S = C4 ⋅A + C5
Z = fs ⋅S
Z = C6 ⋅ (1/A)C7
(3.9a)
(3.9b)
(3.9c)
(3.9d)
where C1 to C7 are constants determined by regression analysis. These relations have
been formulated by this study for W14, W24, W27, W30, W33, and W36 wide-flange
hot-rolled steel sections from the AISC LRFD design manual (1997). The corresponding
constants C1 to C7 are tabulated in Table 3.1, and the curves of some sections are drawn
in Figures 3.1 to 3.3. For a specified type and nominal depth of section, instantaneous
updating of the section properties I, S, and Z is achieved through Equations (3.9) for a
given section area A. That is, having such relationships, the cross section area A can be
taken as the only design variable, thereby reducing the number of design variables
significantly.
60
Since performance-based design of a framework involves deformation well into
the inelastic response stage, all member sizes must be selected from compact sections that
allow for the development of plastic hinges.
3.2.3 Design Constraints
In structural engineering design, the variable values cannot be chosen arbitrarily; rather
they have to satisfy certain specified requirements called design constraints. Constraints
that represent limitations to the behaviour or performance of the system are termed
behavioural constraints, such as those constraints imposing strength, displacement and
stiffness restrictions for safety and performance reasons. Constraints that depend on the
availability, fabricability or other physical limitations are called side constraints; for
example, the availability of commercial steel shapes for designing members of steel
frameworks. According to the roles they play in determining the design variables, the
constraints can be further classified as being primary or secondary. Those that play major
roles in the design selection are referred to as primary constraints, while those that are not
expected to significantly participate in the design selection are labelled as secondary
constraints. The following constraints are considered by this study:
1) Drift constraints, which are primary constraints that serve to control building drift
(including roof drift and inter-story drift).
2) Strength constraints, which are secondary constraints that require the seismic demand
to not exceed the member strength.
3) Side constraints. The lower and upper limits imposed on the values of the design
variables.
61
3.2.3.1 Drift Constraints
The inertia loads due to an earthquake are generally applied to a structure as lateral
forces, which cause the structure to sway laterally. Moderate lateral deflection of a
building may cause human discomfort, and minor damage of nonstructural components,
such as cracking of partitions or cladding and the leakage of pipes. Extreme inelastic
lateral deflection due to a severe earthquake can cause the failure of mechanical,
electrical and plumbing systems, or cause suspended ceilings and equipment to fall,
thereby posing threats to human life; such loading can also increase the possibility of
building instability, thereby compromising structural safety. Thus, it is essential to
control the lateral drift of building frameworks under seismic loading.
There are two kinds of lateral deflections to be considered in design practice. One
is the overall building drift (roof drift). The other is the inter-story drift, defined as the
drift difference between two adjacent floors (see Eq. (3.1)). The overall building drift
represents the average lateral translation. Since a structural performance level is usually
defined as a state corresponding to a target displacement, it is common to adopt the target
displacement as the maximum allowable roof drift for a specified performance level. For
example, roof drifts of 0.7%, 2.5% and 5% of the height of the building are taken as the
allowable roof drifts for the IO, LS and CP performance levels in the design optimization
process developed by this study (FEMA-273, 1997).
In FEMA-273, nonstructural components are classified as acceleration-sensitive
and deformation-sensitive components. Most architectural components, such as exterior
wall elements, interior partitions, veneers, and ceilings, are regarded as deformation-
62
sensitive. The main way to alleviate the damage of deformation-sensitive components is
to limit the inter-story drifts of the building.
The allowable inter-story drift is determined by four factors: 1) desirable
performance level; 2) architectural components; 3) global inter-story drift capacity; 4)
local connection drift angle capacity (FEMA-350, 2000). For example, according to
FEMA the global inter-story drift capacities of a low-rise building at the IO and CP
performance levels are 1.25% and 6.1% of the height of the storey, respectively (FEMA-
350, 2000). The limiting inter-story drift ratios for adhered veneer exterior wall
components are 0.01 and 0.03 for the IO and LS performance levels, respectively
(FEMA-273, 1997). For practical design, the adopted allowable inter-story drift ratio
should be the one that is the most stringent among structural and architectural
requirements.
One may ask why the inter-story drift constraints are still required if, as herein,
there exists a uniform inter-story drift distribution objective imposed at the CP
performance level (see Eq. (3.8)). The primary reason is that provision of a uniform inter-
story drift distribution at the CP level usually does not result in a uniform inter-story drift
distribution at less critical performance levels, such as the IO and LS levels (see Chapter
6).
The inter-story and roof drift constraints are expressed as,
) ) δ≤δ s
∆≤∆
63
(3.10
(s = 1, 2, …, ns(3.11)
where: δs is inter-story drift of story s, ∆ is the roof drift, and ⎯δ and⎯∆ are the allowable
inter-story and roof drifts, respectively.
3.2.3.2 Strength Constraints
Although the drift constraints usually govern the member selection process for moment-
resistance frames under earthquake loading, the member strength requirements still must
be checked in accordance with the provisions of the applicable design code or standard.
One way to account for member strength requirements for a seismic design is to
express them explicitly in terms of the design variables in the same manner as are the
lateral drift constraints, such as in the works of Xu (1994). Though this approach is
direct, the major obstacle is excessive computational expense due to a large number of
strength constraints. Alternatively, member strength requirements can be treated as
secondary constraints because most of them are usually inactive. This approach, first used
by Chan (1993), is adopted in this study and described in the following. Immediately
after the pushover analysis of a framework is conducted, a strength design process is
applied to determine the minimum required size of each member (from a given set of
standard sections) to satisfy the strength criteria in accordance with the governing design
standard. These strength sizes are then taken as the lower sizing bounds on design
variables for the current design cycle involving drift constraints. In this way, the design
optimization of a moment frameworks under seismic loading explicitly accounts for the
lateral drift constraints while implicitly accounting for the member strength requirements.
At this point, it is necessary to specify a particular design standard in order to
further describe how strength requirements are implemented for a design, since they are
different from one standard to another. In this research, the Load and Resistance Factor
64
Design Specification for Structural Steel Buildings (AISC LRFD, 1999) is used. Since we
are considering seismic loads, the Seismic Provisions for Structural Steel Buildings
(AISC, 2000) are followed at the same time. The main design provisions of concern to
this study are described in the following.
1) Load Combination
The following load combination is considered by this study:
(3.12) QE + QG = QE + 1.0D + 0.25L + 0.2S
where: QE is the earthquake load; and QG is the associated combination of gravity loads,
in which D, L and S are dead load, live load and snow load, respectively.
There are many other possible loading combinations that are not covered by Eq.
(3.12). However, since lateral earthquake loading mainly controls the design of moment
frameworks, the load combination Eq. (3.12) is felt sufficient for the purposes of this
study. Note that additional loading combinations can be readily included if required.
2) Member Strength Design
For implementation of the member strength design process, all the girders are treated as
beam members (i.e., negligible axial force), while all the columns are treated as beam-
column members (i.e., subject to both axial force and bending moment). Beam member
design is governed by LRFD Clause F1 provisions, while beam-column member design
is governed by LRFD Clause H1 provisions (AISC LRFD, 1999). All beams are assumed
to be braced by the floor slab such that lateral-torsional buckling is prevented. All
columns are designed explicitly accounting for lateral-torsional buckling. The effective
65
length factors of columns are determined as for braced frames since second-order effects
have already been accounted for in the pushover analysis. The degradation of the stiffness
of members is ignored when calculating the column effective length.
Two amplification factors B1 and B2 are used in LRFD Clause C1 to calculate the
bending moment demand, where B1 accounts for member P-δ instability effects assuming
there is no lateral translation of the framework, and B2 accounts for frame P-∆ instability
effects resulting from lateral translation of the framework. For this study, since member
and frame instability effects are both taken into account by the pushover analysis, the
factors B1 and B2 are each taken equal to unity (1) in LRFD Equation H1-1.
3) Strong-Column Weak-Beam
In earthquake engineering, one of the design goals is to provide the structure with good
energy dissipation capacity. In this regard, the 'Strong-Column Weak-Beam' (SC/WB)
concept is advocated in the Seismic Provisions (2000) as a means to help meet this
objective for moment frames. Specifically, the benefit of the SC/WB concept is that the
columns are designed strong enough such that flexural yielding generally occurs in beams
alone at multiple levels of the framework, thereby achieving a higher level of energy
dissipation (Schneider et al., 1991; Roeder, 1987). Weak-column frames, on the other
hand, are likely to exhibit undesirable response involving weak- or soft-story collapse
mechanisms. The SC/WB concept is implemented in the Seismic Provisions (2000)
through the following constraint applied at beam-to-column connections:
0.1*
*
>=∑∑
pb
pccb
M
Mr (3.13)
66
where: rcb is a so-called column-beam moment ratio; ∑Mpc* is the sum of the moments in
the column above and below the joint, calculated as ∑Mpc*= ∑ [(Z)(σy−N/A)] (where N is
the axial force for the column and σy is the design yield strength); ∑Mpb* is the sum of
moments in the beams at the joint, calculated as ∑Mpb*= ∑[1.1(Z)(σye)] (where σye is the
expected yield strength and the factor of 1.1 is introduced to recognize the potential over-
strength of beams due to other considerations); and subscripts b and c refer to beams and
columns at the connection under consideration. It is noted that the SC/WB provision is
mandatory only for special moment frameworks in the Seismic Provisions (2000).
Through applying the relationship established in Eq. (3.9d), the constraint Equation
(3.13) can be expressed explicitly in terms of design variables (cross section areas).
4) Plastic Design Provisions
In performance-based seismic design, structures are expected to deform far into the
inelastic stage, and plastic hinges are anticipated to occur in flexural beams and columns.
Therefore, it is desirable that frame members are selected from among compact sections
that allow for plastic rotation. This study adopts the classification of sections in Table
B5.1 of AISC LRFD (1999) and Table I-9-1 of the Seismic Provisions (2000).
Specific provisions C2.2 and E1.2 in AISC LRFD (1999) are followed for the
selection of column sections. Provision C2.2 states that the axial force in the columns
caused by gravity load plus horizontal loads shall not exceed 0.75φc times AgFy. Provision
E1.2 restricts the column slenderness parameter l/r to be not greater than 1.5π yFE / ,
where l is the laterally unbraced column length.
67
Beams and columns are generally designed as double-symmetric sections (e.g.,
W-shape). The selection of structural members to satisfy code-stipulated strength
requirements usually entails an iterative process. In each design cycle, assuming there is
no internal member force redistribution, the set of standard sections specified for each
member is searched to identify the minimum required section size. Once the strength-
based member sizes are determined, they are taken as lower-bound sizes for the stiffness-
based design optimization.
3.2.3.3 Side Constraints
The lower and upper bound areas for member cross-sections are respectively taken as the
smallest and largest commercially available sectional areas found from the AISC design
manual (1994). When strength constraints are considered in the design process, the lower
bound cross section areas are updated if it is found that larger areas are required to meet
the member strength demands.
3.2.4 Design Formulation and Solution Strategy
From the foregoing, the performance-based design optimization model can be formulated
as (note that the damage objective function OBJ2 applies only for the CP performance
level, while the inter-story and roof drift constraints are considered for all four building
performance levels OP, IO, LS and CP):
( ) ( )( )
HHv
nsBJO and:
ALW
OBJ :
ns
sCP
sCPs
ne
jjj
2/11
1
2
2
1max1
11
1)(Minimize
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛−
∆=
ρ=
∑
∑
−
=
=
xx
x
x (cost)
(damage)
(3.14a)
(3.14b)
68
:
( )( )
),1( Level) (OPOPOP
OPOPs nss
∆≤∆
=δ≤δ
x
x L
( )( )
),1( Level) (IOIOIO
IOIOs nss
∆≤∆
=δ≤δ
x
x L
( )( )
),1( Level) (LSLSLS
LSLSs nss
∆≤∆
=δ≤δ
x
x L
(3.14i)
(3.14j)
(3.14k)
( )( )
),,2,1(
),1( Level) (CPCPCP
nejA
nss
jj
CPCPs
L
L
=∈
∆≤∆
=δ≤δ
a
x
x
where
derive
the or
accord
(i)
(ii)
(iii)
and optional SC/WB constraints at specified connections:
( ) 01.1 ≤σ⋅⋅+⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −σ⋅−∑ ∑
c bbye
cy Z
ANZ
aj is the set of discrete section areas possible for design variable j,
s from Eq. (3.13).
The general philosophy of the numerical procedure for optimizati
iginal problem Eqs. (3.14) with a sequence of explicit approx
ing to the following procedure:
Set the design cycle index b=1;
Start from an initial trial design point x1;
Conduct the structural analysis, and formulate the explicit appr
optimization problem at the current design point xb;
69
(3.14c)
(3.14d)
(3.14e)
i
(3.14f)
(3.14g)
(3.14h)
and discrete sizing restrictions:
Subject to drift constraints
(3.14l)
and Eq. (3.14l)
on is to replace
mate problems
oximate design
(iv) Apply an optimization algorithm to search for an improved design point xb+1;
(v) Test whether xb+1 is the optimum. If so, stop the procedure; otherwise, set b = b +
1 and repeat steps (iii) to (v).
The foregoing general procedure is carried out in two phases. Phase I treats the design as
a continuous variable design problem. Then, a discrete design strategy is employed to
start phase II based on the phase I results. After phase II, a feasible optimal set of discrete
section sizes is found.
In the above procedure, step (iii) is critical in solving the design optimization
problem successfully, since structural design usually involves large numbers of variables
and constraints, which requires tremendous computational effort. Schmit and Farshi
(1974) introduced approximation concepts into the structural design process, including
design variable linking, temporary constraint deletion, and construction of high quality
explicit approximations of retained constraints using reciprocal variables and first-order
Taylor series. These approximations led to the emergence of computationally efficient
mathematical programming techniques for structural design optimization.
In Equations (3.14), the cost objective OBJ1(x) is alone an explicit function of the
design variables, while the damage objective OBJ2(x) and all drift constraints
(Eqs.(3.14c) to (3.14j)) are implicit functions of the design variables. In such a case, it is
necessary to use an approximation technique to formulate the objective function OBJ2
and all drift constraints explicitly in terms of design variables so as to facilitate the
computer solution of the design optimization problem. In fact, as herein, reciprocal
variables are often adopted so as to achieve high quality approximations. The use of
70
reciprocal variables is inspired by the fact that displacement response is a strictly linear
function of the reciprocal-areas of elements for statically determinate structures. As well,
the use of reciprocal-area variables generally improves the quality for linear
approximations of displacement response of statically indeterminate structures.
Upon introducing the reciprocal sizing variable (i.e., reciprocal-area) for each
member j,
jj A
x 1= (3.15)
and then employing first-order Taylor series expansions, Eqs. (3.14) can be reformulated
as the explicit design problem:
( ) ( ) ( ) ( )
xxdx
dOBJOBJBJO and:
x
LW
OBJ :
ne
jjj
j
ne
j jj
∑
∑
=
=
−⎥⎥⎦
⎤
⎢⎢⎣
⎡+=
ρ=
1
0
0
2022
1max1
11)(Minimize
xxx
x
( )[ ] ( ) ( )
( )[ ] ( ) ( )
),,2,1(
),,,(
,,,,,1(
toSubject
1
0
00
1
0
00
nejx
CPLSIOOPixxdx
d
LSIOOPinssxxdx
d
:
jj
ine
jjj
j
ii
ine
jjj
j
isi
s
L
L
=∈
=∆≤−⎥⎥⎦
⎤
⎢⎢⎣
⎡ ∆+∆
==δ≤−⎥⎥⎦
⎤
⎢⎢⎣
⎡ δ+δ
∑
∑
=
=
X
xx
xx
(3.16a)
(3.16b)
)CP (3.16c)
(3.16d)
(3.16e)
with optional SC/WB constraints at specified beam-column connections:
0≤−+∑∑b
Ucbbbc
cc gxcxc (3.16f)
71
where: superscript ' 0 ' represents values for the current design; dδ/dxj, d∆/dxj, and
dOBJ2/dxj are inter-story drift, roof drift, and ductility objective function derivatives with
respect to the design variable xj, respectively; and, as derived below, the coefficients cc,
cb, gcbU define linear SC/WB constraints at the connections under consideration.
The SC/WB constraint Eq. (3.16f) is derived as described in the following.
Firstly, Z is approximated by a first-order Taylor series formulated at the current design
point x0 by applying Equation (3.9d), i.e.,
( )0
00 xx
dxdZZZ −⋅⎟
⎠⎞
⎜⎝⎛+=
176
1
767
71 where −−
=⎟⎠⎞
⎜⎝⎛= C
C
xCCA
CCdxdZ (3.17b)
Then, Eq. (3.17a) is substituted into Equation (3.14l) to get,
( ) ( ) (1.1 00
000
0 σ⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟
⎠⎞
⎜⎝⎛++⎟
⎠⎞
⎜⎝⎛ −σ
⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟
⎠⎞
⎜⎝⎛+− ∑∑
b bcy
c c
xxdxdZZ
ANxx
dxdZZ
Finally, Eq. (3.16f) is obtained by defining the following terms in Eq. (3.17c):
( )bye
b
b
cy
c
c
dxdZc
AN
dxdZc
σ⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛=
⎟⎠⎞
⎜⎝⎛ −σ
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−=
0
0
1.1
72
(3.18a)
(3.18b)
(3.17a)
) 0≤bye (3.17c)
( )∑∑ σ
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−−⎟
⎠⎞
⎜⎝⎛ −σ
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−=
bbye
bc cy
c
Ucb x
dxdZZ
ANx
dxdZZg 0
000
00 1.1 (3.18c)
To complete the approximate formulation of the design optimization problem
Eqs. (3.16), it remains to determine the three derivatives dOBJ2/dxj, dδ/dxj and d∆/dxj ,
also known as sensitivity coefficients, as is done in the next chapter.
73
( in.2 )
( in.4 )
W33 W30
W14 W24
I
A
250 2001501000 50
32000
24000
16000
8000
0
Figure 3.1 I-A Relationship for Commercial W-Shape Sections
0
500
1000
1500
2000
0 50 100 150 200 250
A
( in.2 )
W33
W30
W24 W14
S ( in.3 )
Figure 3.2 S-A Relationship for Commercial W-Shape Sections
74
Z (in.3) 2000 1600 W24 W33
W30 W14 1200
800
400
Figure 3.3 Z-1/A Relationship for Commercial W-Shape Sections
0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
1/A (1/in.2)
75
TABLE 3.1 SECTION PROPERTIES RELATIONSHIPS
I = C1 A2 + C2 A + C3 S = C4 A + C5 Z = fs S Z = C6 (1/A)C7Section
Type C1 C2 C3 C4 C5 fs C6 C7W14 0.140 35.53 -61.94 5.94 -25.63 1.18 3.95 -1.117W18 0.185 60.38 -133.96 6.96 -13.32 1.15 5.16 -1.103W24 0.201 105.91 -376.30 9.14 -27.95 1.15 6.33 -1.109W27 0.182 136.41 -626.86 10.17 -33.45 1.15 7.32 -1.097W30 0.119 178.83 -1288.26 11.42 -55.77 1.14 7.44 -1.113W33 0.027 227.99 -2070.32 12.69 -76.56 1.14 7.88 -1.120W36 0.180 232.00 -1610.41 13.06 -65.12 1.15 9.37 -1.090
TABLE 3.2 SIZE BOUNDS ON AISC W-SECTIONS
Section Type
Lower/Upper Bound Section Designations
Cross-Section Area (in.2)
W14×22 6.49 W14 W14×808 237.0
W18×35 10.3 W18 W18×311 91.5 W24×55 16.2 W24 W24×492 144.0 W27×84 24.8 W27 W27×539 158.0 W30×90 26.4 W30 W30×477 140.0 W33×118 34.7 W33 W33×354 104.0 W36×135 39.7 W36 W36×848 249.0
1 in.2 = 645.16 mm2
76
Appendix 3.A
Design Spectra Parameters
3.A.1. Site Parameters for 2%/50-year and 10%/50-year Earthquakes
For the purposes of illustration, we adopt design spectra parameters from FEMA-273
(1997) maps for a site located at Latitude 36.9° N and Longitude 120° W. It is assumed
that the site is Class D or stiff soil. The following maps provide the site parameters for
10%/50-year and 2%/50-year earthquakes:
1) Map 29: Probabilistic Earthquake Ground Motion for California/Nevada of 0.2
sec. spectral Response Acceleration (5% of critical Damping), 10% Probability of
Exceedance in 50 years.
2) Map 30: Probabilistic Earthquake Ground Motion for California/Nevada of 1.0
sec. spectral Response Acceleration (5% of critical Damping), 10% Probability of
Exceedance in 50 years.
3) Map 31: Probabilistic Earthquake Ground Motion for California/Nevada of 0.2
sec. spectral Response Acceleration (5% of critical Damping), 2% Probability of
Exceedance in 50 years.
77
4) Map 32: Probabilistic Earthquake Ground Motion for California/Nevada of 1.0
sec. spectral Response Acceleration (5% of critical Damping), 2% Probability of
Exceedance in 50 years.
The above maps give accelerations for site Class B only. They need to be adjusted
for the other site classes. The adjustment coefficients are found in Tables 2-13 and 2-14
in FEMA-273. The design spectra parameters Ss, S1, Fa and Fv obtained for site class D
from the above noted maps and tables are listed in Table 3.A. The period T0 (see Figure
2.1) is available from the site parameters as,
sa
v
SFSF
T 10 = )
3.A.2 Site Parameters for 20%/50-year and 50%/50-year Earthquakes
FEMA-273 does not provide maps for 20%/50-year and 50%/50-year e
does provide equations for calculating corresponding site parameters
earthquake hazards considered in Section 3.A.1. To this end, parameter
20%/50-year and 50%/50-year earthquakes are calculated through Eq
FEMA-273 as below,
nR
iiPSS ⎟
⎠⎞
⎜⎝⎛=
47450/10
where: subscript i (= s, 1) represents an acceleration response of short per
long period (1 sec.); n is a zone factor (n=0.44 at the site of Latitud
Longitude 120° W); Si10/50 are parameters of the 10%/50-year earthquake
mean return period given by,
78
(3.A.1
arthquakes, but
from the two
s Ss and S1 for
uation (2-3) of
(3.A.2)
iod (0.2 sec.) or
e 36.9° N and
; and PR is the
)1ln(02.0 5011
EPRe
P −−= (3.A.3)
where PE50 is the probability of exceedance in 50 years of the earthquake under
consideration. For example, for the 20%/50-year earthquake located at the site having
Latitude 36.9° N and Longitude 120° W,
2251
1)2.01ln(02.0 =
−= −e
PR
From Eq. (3.A.2), the parameters for the 20%/50-year earthquake are then calculated as,
50/10
44.0
50/1050/20 72047.0474225
iii SSS ⋅=⎟⎠⎞
⎜⎝⎛=
That is: Ss, 20/50=(0.72047)(0.29) = 0.209 (g), and S1, 20/50=(0.72047)(0.14) = 0.10 (g)
where Ss,10/50 =0.29 and S1,10/50=0.14 are taken from Table 3.A for the given site.
The parameters for the 50%/50-year earthquake at the same site are similarly
found as,
72e1
10.5)0.02ln(1 =
− = −RP
50/10
44.0
50/1050/50 4364.047472
iii SSS ⋅=⎟⎠⎞
⎜⎝⎛=
Ss, 50/50=(0.4364)(0.29) = 0.126 (g), and S1, 50/50=(0.4364)(0.14) = 0.061 (g)
The site parameters for all four earthquake intensities are summarized in Table 3.A.
The same procedure was applied to find the parameters for another site at Latitude
41° N and Longitude 115.2° W, and the results are also given in Table 3.A.
79
TABLE 3.A PERFORMANCE LEVEL SITE PARAMETERS
Site
Location Site
Class Performance
Level Earthquake
Level Ss(g)
S1(g)
Fa Fv
OP 50%/50∗ 0.126 0.061 1.60 2.40 IO 20%/50 0.209 0.100 1.60 2.40 LS 10%/50 0.290 0.140 1.57 2.24
Latitude 36.9°N,
Longitude 120°W
D
CP 2%/50 0.500 0.230 1.40 1.94 OP 50%/50 0.109 0.035 1.60 2.40 IO 20%/50 0.180 0.058 1.60 2.40 LS 10%/50 0.250 0.080 1.60 2.40
Latitude 41°N,
Longitude 115.2°W
D
CP 2%/50 1.100 0.410 1.06 1.59 ∗Sa Exceedance probability/years
80
Chapter 4
Sensitivity Analysis
4.1 Introduction
For the numerical implementation of structural design optimization, the sensitivities of
structural responses are often required in order to explicitly formulate the constraints and,
sometimes, objective functions. A sensitivity is the rate of change of a structural response,
such as displacement or internal force, with respect to change of the design variables. The
evaluation of sensitivity coefficients is called sensitivity analysis, and constitutes a major
portion of the total calculation in a structural design process. The sensitivity coefficients
are also important in their own right as they identify trends for the performance of
structural systems, and can serve as a guide for the redesign stages of a manual iterative
design procedure since they can be used to predict the structural response with respect to
small variation of the design variables.
In general, there are three classes of methods to evaluate design sensitivities: the
finite difference method (FDM); the direct differentiation method (DDM) (also called the
pseudo-load method); and the adjoint variable method (AVM) (also called the dummy-
81
load or virtual-load method) (Arora, 1979; Haug, Choi and Komkov, 1986). Though
FDM is the most straightforward method, it is normally only used to verify the results of
the other two methods since it is computationally expensive and even deficient in terms
of accuracy and reliability (Adelmand and Haftka 1986).
The design sensitivity analysis of linear elastic structural systems is well-
documented. However, the sensitivity analysis of nonlinear inelastic structural systems,
such as steel moment frames loaded into the plastic response range, is far more
complicated and computationally intensive because the state of internal forces at any
given load level depends on the prior loading history.
There has been considerable research since the mid-1980’s on history-dependent
design sensitivity analysis. Ryu, Haririan, Wu, and Arora (1985) studied design
sensitivity in the realm of non-linear analysis. They compared the difference between
linear and nonlinear sensitivity analyses and pointed out that the DDM and AVM were
suitable for nonlinear sensitivity calculation. Wu and Arora (1987) recognized that the
response sensitivity at a given time required the calculation of partial derivatives of
internal forces with respect to the design variables. However, since analytical expressions
were not available, the finite difference method was used in their study to compute the
partial derivatives of internal forces to account for inelastic material behaviour (called the
semi-analytical approach). Haftka and Mroz (1986), Choi and Santos (1987), Cardoso
and Arora (1988) developed the variational formulations for nonlinear design sensitivity
analysis. In these expressions, constraints were formulated as functional while
perturbations of the design variables were treated as pseudo-initial strains. The variational
approaches were suitable only for simple continuum structures. Vidal, Lee and Haber
82
(1991) presented an incremental direct differentiation method for the design sensitivity
analysis of history-dependent materials. Their study showed it was critical to utilize the
consistent tangent operator in sensitivity formulations in order to obtain reliable results.
Haftka (1993) concluded that the semi-analytical approach by Wu and Arora could be
viewed as an overall finite difference approach based on a single Newton iteration.
Polaneff and co-workers (1993) employed a central difference scheme to evaluate the
derivatives of internal forces. They argued that numerical stability and accuracy were
dramatically improved when compared with the forward difference method. Ohsaki and
Arora (1994) obtained sensitivity coefficients from incremental equilibrium equations.
The sensitivities of the incremental displacements were calculated and accumulated to
obtain the sensitivities of the total displacements at the current loading level. In this
procedure, the yielding load (so-called 'yielding time') of each member is considered to
be a function of the design variables. These 'yielding times' were recorded and
differentiated with respect to the design variables in order to solve the sensitivity
discontinuity. This procedure is path-dependent and extremely difficult for problems
where 'yielding times' and their sensitivities are hard to find. Lee and Arora (1995)
investigated the discontinuity of design sensitivity due to a piecewise linear constitutive
law. They pointed out that sensitivity coefficients along the loading path could be
obtained by differentiating the total equilibrium equations. This is an important discovery
because 'yielding times' and their sensitivities are not necessary for this method.
Yamazaki (1998) suggested that incremental equilibrium equations could be
differentiated to find incremental sensitivities that accumulate to give total sensitivities.
83
The sensitivity discontinuity at material transition points is overcome by using the
differentiated constitutive law at the yielding points.
While significant advances have been made, the literature reveals that applications
of nonlinear design sensitivity analysis have been limited to bar trusses and simple
continuum structures such as plates, shells and single beams, under static loading alone.
This chapter presents a performance-based design sensitivity analysis procedure for
inelastic steel moment frames under seismically induced inertial loading. This analysis
problem, which has received little or no attention in the literature, is complicated by the
fact that the inertial loads vary whenever the design of a structure is modified because the
vibration behavior of the structure is also modified. Analytical formulations defining the
sensitivity of displacements to modifications in member sizes are based on a load-control
pushover analysis procedure.
4.2 Sensitivity of Damage Objective
The derivative of the damage objective OBJ2 with respect to design variable xj [see Eq.
(3.16b)] is found by first finding its partial derivatives with respect to lateral inter-story
drift vsCP and roof drift ∆CP, i.e.,
⎟⎟⎠
⎞⎜⎜⎝
⎛−
∆⋅
∆⋅⋅=
∂∂
1/1
2
2
HHv
HOBJnsH
vOBJ
CPs
CPs
CPs
CPs
( )∑−
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∆−⎟
⎟⎠
⎞⎜⎜⎝
⎛−
∆⋅
⋅=
∆∂∂ 1
12
2
2
/
/11 ns
sCP
sCPs
CPs
CPs
CP H
HvHHv
OBJnsOBJ
84
and then finding,
j
CP
CP
ns
s j
CPs
CPsj dx
dOBJdx
dvv
OBJdx
dOBJ ∆∆∂
∂+⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂∂
= ∑−
=
21
1
22 (4.1)
where dvsCP/dxj and d∆CP/dxj are the derivatives of the CP-level story drift vs and roof
drift ∆ with respect to xj, respectively, which are derived in the next section.
4.3 Sensitivity of Drift Displacements
The equilibrium equations for the nonlinear analysis can be written as,
( ) ( )xPxuF =, (4.2a)
where: F(u, x) is the overall internal nodal force vector; P(x) is the overall externally
applied nodal load vector; u is the vector of nodal displacements; and x is the vector of
design variables.
For this study, Eq. (4.2a) is formulated at the specified four performance levels as,
(4.2b)
( ) ( )xPxuF ii =, i = OP,IO,LS,CP
Differentiate Eq. (4.2b) with respect to design variable xj , to get,
j
i
j
i
j
i
i
i
dxd
xdxd PFu
uF
=∂∂
+∂
∂i = OP,IO,LS,CP
which can be re-written as,
j
i
j
i
j
ii
xdxd
dxd
∂∂
−=FPuK (4.3) i = OP,IO,LS,CP
85
where Ki=∂F/∂u is the global tangential stiffness matrix (Wu and Arora, 1987) at the ith
loading (or performance) level (available from the pushover analysis process). The partial
derivative ∂F/∂xj in Eq. (4.3) means that the displacement field u is set to a constant
while differentiating the equations with respect to xj. The total derivatives du/dxj and
dP/dxj are computed while the remaining design variables xk (k≠j) are held as constants
(Note that du/dxj and dP/dxj are frequently expressed as ∂u/∂xj and ∂P/∂xj in elastic
sensitivity analysis).
Earthquake loading is an inertia force that depends on the seismic mass and lateral
stiffness of the structure. The derivative dPi/dxj in Eq. (4.3) is the sensitivity of the nodal
(inertia) load vector with respect to changes in the variable xj, which reflects the influence
of structural modifications on (equivalent static) earthquake loading (its detailed
derivation is addressed in Section 4.4).
Consider a particular displacement uli at performance level i for a given framework,
which is related to the overall vector of nodal displacements u as,
i=OP,IO,LS,CP (4.4)iT
lilu ub=
where bl is a Boolean vector (consisting of 0's and 1's) that depends on the nature of
displacement ul . For instance, if ul = v1-v2 is the inter-story drift of a story, the vector bl is
obtained by setting the component entries corresponding to degrees of freedom v1 and v2
to unity while setting the rest of the entries to zero; i.e., blT={1, -1, 0, …, 0} assuming
that v1 and v2 correspond to degrees of freedom 1 and 2, respectively.
The sensitivity of displacement uli with respect to changes in design variable xj is
then found as, from Equations (4.3) and (4.4),
86
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
−⋅=⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
−⋅==−
j
i
j
iTi
lj
i
j
iiT
lj
iTl
j
il
xdxd
xdxd
dxd
dxdu FP
UFPKbub 1 (4.5) i=OP,IO,LS,CP
where UliT
=bl T Ki -1 is called the adjoint displacement field under the adjoint load bl
(Haug, Choi and Komkov, 1986). Equation (4.5) clearly shows that the sensitivity dul/dxj
consists of two parts, where the term UliT⋅dPi/dxj is the contribution from varying inertia
loads while the term -UliT⋅∂Fi/∂xj is that due to static loads (This decomposition of
displacement variation is visually shown in Figure 4.1).
The major difficulty in nonlinear sensitivity analysis is to evaluate the partial
derivatives of internal nodal forces, i.e., the term ∂Fi/∂xj in Eq. (4.5). Since the pushover
analysis uses the incremental-load method (see Section 2.1), the vector of internal nodal
forces at performance level i is equal to the summation of incremental internal nodal
forces along the loading history, i.e.,
( )∑=
∆=i
m
mi
1FF (4.6) i=OP,IO,LS,CP
where m is the load-step index and ∆F(m) represents the increment of internal nodal forces
at load step m. Differentiating Eq. (4.6) with respect to design variable xj, we obtain,
( )( ) ( )( ) ( )
( )∑ ∑ ∑∑
∑= = =
=
= ⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∆⋅
∂∂
=∂
⎥⎦
⎤⎢⎣
⎡∆⋅∂
=∂
∆∂=
∂
∂ i
m
i
m
ne
k
mk
j
mkT
kj
ne
k
mk
mk
Tki
m j
m
j
i
xxxx 1 11 1
1 uTK
TuTKT
FF
(4.7) i = OP,IO,LS,CP
87
where: Tk is the direction-cosine matrix for member k; ∆u is the vector of incremental
nodal displacements; and Kk is the stiffness matrix of element k (having the same
dimension as the global stiffness matrix K).
From Equation (2.9) for index j=k, and letting Gk=Nk⋅Gk' where Nk is the axial
force of element k and Gk' is a constant matrix, we have at load step m,
( )
( )( ) ( )
( ) ( )( )
j
mkgm
km
gkkj
mk
j
msk
km
skj
k
j
mk
xdxNd
xxx ∂
∂++
∂∂
+∂∂
=∂∂ − C
GCGCSCSK '1
(4.8)
where: ∂Csk/∂xj and ∂Cgk/∂xj are the derivatives of correction matrices Csk and Cgk,
respectively; ∂Sk/∂xj is the derivative of elastic stiffness matrix Sk, and dNk(m-1)/dxj is the
sensitivity of the element axial force at the beginning of load step m (i.e., the axial force
from the previous loading step is used to establish the current geometric stiffness matrix).
Equation (4.8) requires finding the sensitivity of axial forces N when second-order
effect are accounted for in the analysis, which results in a computationally intensive
process since it must be carried out over the full loading history. Since it is known that
the variation of axial forces in moment frames due to small perturbations of the design is
negligable, it is assumed here that the axial force of each element remains constant when
the structure undergoes a small design variation at each load step, i.e.,
0=j
k
dxNd (4.9)
88
Furthermore, since the displacement field ∆u is held constant in Eq. (4.7), the variation of
a local design variable only causes variation in the internal forces for that particular
element, such that,
( )jkif
x j
mk ≠=
∂∂
0K
(4.10)
For k=j, the derivative of the element stiffness matrix can be expressed as, from Eqs. (4.8)
and (4.9),
( )( )
( )
( )
( )( )
( )
( )
( )
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
∂
∂+
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
∂
∂+
∂∂
=∂
∂∑∑=
−
−=
−
−
2
1
1
1
2
1
1
1n j
mjn
mjn
mjgm
jn j
mjn
mjn
mjS
jmjS
j
j
j
mj
xp
pxp
pxxC
GC
SCSK
(4.11)
where: pjn
(m-1) (n=1,2) are plasticity-factors for element j determined in the previous
loading step (see Appendix 2.B); and partial derivatives ∂Cs/∂p and ∂Cg/∂p are found in
Appendix 4.A.
Substituting Eq. (4.10) into Eq. (4.7), we have,
( )( )∑
= ⎥⎥⎦
⎤
⎢⎢⎣
⎡∆⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂
∂=
∂
∂ i
m
mj
j
mjT
j
From Eq.(4.12a), the partial derivative ∂F/∂xj at load step m can be written as (see
Appendix 2.C),
( ) ( ) ( ) ( ) ( )( )m
jj
mjT
jj
m
j
m
j
m
j
m
xxxxxuT
KTFFFF
∆⋅⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
=∂∆∂
+∂
∂=
∂∂ −− 11
j
i
xx 1uT
KT
F(4.12a) i=OP,IO,LS,CP
(4.12b)
89
where ∂F(m-1)/∂xj is partial derivative ∂F/∂xj found at load step (m-1) and ∂∆F(m)/∂xj is
found from Eqs. (4.6) and (4.12a). Obviously, ∂F(m)/∂xj is ∂Fi/∂xj when the base shear at
load step m equals Vbi (i.e., the design base shear of performance level i). In other words,
∂F/∂xj needs to be calculated at each load step through Eq. (4.12b) in order to obtain
∂Fi/∂xj for the four specified performance levels.
To find the derivative of the elastic stiffness matrix ∂Sj /∂xj in Eq. (4.11), it is
necessary to establish the relationship between member section area and moment of
inertia. In previous studies by Lee (1983) and Xu (1994), a linearized relationship was
formulated as,
(4.13) jjj AAI 0η=
where: η is a constant that depends upon the cross-sectional shape of the element; Ij is the
cross-section moment of inertia; Aj0 is the cross-section area for the current design cycle,
and Aj is that which needs to be found for the next design cycle. The section-properties
relationship defined by Eq. (4.13) is adopted by this study.
Upon adopting reciprocal design variables xj=1/Aj to improve the quality of the
approximation (see Section 3.2.4) we have, from Eq. (4.13),
( )j
j
j
jj
j
jj
j
j
jj
j
jj x
IxA
Ax
AAxI
xxA
xA −=
∂
∂η=
∂
η∂=
∂
∂−=
∂
∂= 0
02 ;1;1 (4.14)
and, therefore,
j
j
j
j
xxSS
−=∂∂
(4.15)
90
From Eq. (2.13), the derivative of the plasticity-factor p in Eq. (4.11) is found as,
( ) ( ) ( )
( )
( )
)2,1(1
1
111
=∂
∂
∂
∂+
∂
∂⋅
∂
∂=
∂
∂ −
−
−−−
nx
R
R
pxI
Ip
xp
j
mpjn
mpjn
mjn
j
j
j
mjn
j
mjn (4.16)
where:
( )( )[ ] ( )
j
mjnm
jnj
j
j
mjn
xp
pxI
Ip 1
11
1−
−−
−=∂
∂∂ (4.17)(n=1,2) ∂
( )
( )
( )[ ]( )2,1
3
121
1
1
=−
=∂
∂ −
−
−
nEI
Lp
R
p
j
jmjn
mpjn
mjn (4.18)
From Eqs. (4.14) to (4.18), Eq. (4.11) becomes,
( )( )
( )
( )( )
( )
( )( )( )
( ) ( )( ) ( )
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂
∂⋅
−+−⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂+−=
∂
∂ −−−−
=−−∑
j
mpjn
j
jmjn
j
mjnm
jnn
mjn
mjgm
jmjn
msj
jm
sjjjj
mj
xR
EILp
xp
pppxx
12111
2
111 3
111 C
GC
SCSK
(4.19)
where ∂Rjnp/∂xj is the derivative of the rotational stiffness of plastic-hinge section n,
which is found by differentiating Eq. (2.11b) with respect to the design variable xj , (note
that M in Eq. (2.11b) is replaced by Meq from Eq. (2.16) for the combined stress case), to
get,
91
( ) ( ) ( )
( )
( )
j
mjneq
mjneq
mpjn
j
jny
jny
mpjn
j
mpjn
xM
MR
xM
MR
xR
∂
∂
∂
∂+
∂
∂
∂
∂=
∂
∂ −
−
−−− 1,
1,
111 (4.20) (n =1, 2)
where Mjny is the reduced yielding moment at end-section n of element j for combined
stress states. Superscript r (used in Section 2.4.4 to represent 'reduced') is omitted in Mjny,
since Eq. (4.20) is applicable to both the single and combined stress states by defining
Mjny=My and Meq,jn=M under the pure bending state (see Appendix 2.C). The partial
derivatives of Rjnp with respect to Mjny and Meq,jn are found from Eq. (2.11b) as,
( ) ( ) ( ) ( )
( ) ( )[ ] 31,
21
1,
411
1
1
−φ
−+=
∂
∂−−
−−−
jnym
jneqpmp
jn
mjneqs
jny
mpjn
jny
mpjn
MMR
MfM
RM
R
( )
( )( )
( ) ( )[ ] 31,
21
4
1,
1
1
1
−φ
−−=
∂
∂−−−
−
jnym
jneqpmp
jn
jnysm
jneq
mpjn
MMR
MfMR
If only moment (i.e., a single stress state) is considered for the yielding
then,
j
j
s
ye
s
yej
jj
jny
xZ
ffZ
xxM
∂
∂σ=⎟⎟
⎠
⎞⎜⎜⎝
⎛ σ
∂∂
=∂
∂
Otherwise, from Eq. (2.15) for a=1, we have,
jp
yjn
jjpjp
yjn
sj
jye
j
jny
NN
AMNN
fxZ
xM 01
−⎟⎟⎠
⎞⎜⎜⎝
⎛−
∂
∂σ=
∂
∂
92
(4.21)
(4.22)
criteria,
(4.23)
(4.24)
where Njny is the axial force of element j at first-yield of end-section n, and dNjn
y/dxj=0
from Eq. (4.9).
The partial derivative ∂Meq,jn (m-1)/∂xj in Eq. (4.20) is found by differentiating Eq.
(2.16) with respect to xj, to get,
( )
jp
yjn
jpsjjp
mj
jjpj
mjn
jp
mj
j
jp
jp
yjn
sj
mjneq
NN
MfAN
NAM
xM
NN
xM
NN
fx
M 0)1(
01)1(0)1(
, 1 ζ−⎟⎟⎠
⎞⎜⎜⎝
⎛+
∂
∂+
∂
∂⎟⎟⎠
⎞⎜⎜⎝
⎛−=
∂
∂ −−−−
(4.25)
where:
j
jye
j
jp
xZ
xM
∂∂
=∂∂
σ0
(4.26)
The partial derivarive ∂Mjn (m-1)/∂xj in Eq. (4.25) is the derivative of end moments in
the local co-ordinate system, and is obtained from the derivative of the element force
vector as,
( ) ( ) ( )( )1
121−
−−−
∆∂
∂+
∂
∂=
∂
∂ mj
j
mj
j
mj
j
mj
xxxu
Kff (4.27)
where Mjn (m-1) is one entry of the end force vector fj
(m-1) at load step m-1 (m ≥ 2).
Recognizing the fact that plasticity-factors pjn are all equal to unity and axial forces
Nj are all equal to zero at the first loading step (m=1), from Eqs. (2.C.3d), (4.19) and
(4.27) we obtain for that step,
( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )111111
111 111jjj
jjj
jjsjj
jj
j
j
j
j
j
j Axxxxxx
ffuSuCSuKff
∆−=∆−=∆−=∆−=∆∂
∂=
∂
∆∂=
∂
∂
(4.28a)
93
where Csj(1)=1 since plasticity-factors pjn=1, and from Eq. (4.12a),
( ) ( ) ( )( )1
111
uTK
TFF∆⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂
∂=
∂∆∂
=∂∂
jj
jTj
jj xxx (4.28b)
where Kj(1) is the element elastic stiffness matrix at the first load step.
If no plasticity is detected for any load step m, after substituting Eq. (4.19) into Eq.
(4.27) we find for that step,
( ) ( ) ( )
( )( )
( )( )
( )mjj
j
mjm
jjjj
mjm
jj
mj
j
mj
j
mj A
xxxxxxf
fuS
fu
Kff∆−
∂
∂=∆−
∂
∂=∆
∂
∂+
∂
∂=
∂
∂ −−− 111 1(4.29)
However, if yielding is detected for any section, Eq. (4.25) is applied to find the partial
derivative ∂Meq,jn (m)/∂xj which, in turn, is substituted in Eq. (4.20) to find the partial
derivative ∂Rjnp(m)/∂xj . Then partial derivative ∂Kj
(m)/∂xj is found from Eq. (4.19), then
Eq. (4.12b) is evaluated to find the partial derivative ∂F(m)/∂xj. In this way, ∂Fi/∂xj is
found for the four performance levels successively.
Finally, upon substituting Eq. (4.19) in Eq. (4.5), the sensitivity of displacement uli
with respect to reciprocal-area variables xj is found as,
( )( )
( )( )
( )
( )( )( )
( ) ( )( ) ( )( )∑ ∑
=
−−−−
=−−
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∆⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂−+−⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂−
+=
i
m
mj
j
mpjn
j
jmjn
j
mjnm
jnn
mjn
mjgm
jmjn
msj
jm
sjjj
Tj
Til
j
iTi
lj
il
x
REI
Lpx
pp
ppx
dxd
dxdu
1
12111
2
111 3
111 uT
CG
CSCSTU
PU
94
i=OP,IO,LS,CP (4.30)
Equation (4.30) reduces to the well-known static elastic displacement sensitivity
formulation when pjn = 1, m = i = 1, and dP/dxj = 0; i.e.,
[ ]uTKTU jjTj
Tl
jj
l
xdxdu 1
= (4.31)
Finally, displacement sensitivity with respect to direct cross-section area variables
Aj can be found from the following equation,
j
il
jj
j
j
il
4.4 Sensitivity of Nodal Loads
A major difference between optimizing for static loads and earthquake loads is the fact
that inertia-related seismic loads are dependent upon the natural period of the structure.
Since the natural period of a structure is a function of the structural stiffness and building
seismic mass, modifying the structure causes the earthquake loading to change.
The total lateral load applied to a framework at a specified performance level is
equal to the corresponding design base shear. The overall nodal load vector P is related to
the inertial load vector Pl through Eq. (2.6), such that the sensitivity of nodal load vector
Pi at performance level i is found as,
j
il
j
g
j
il
j
i
dxd
dxd
dxd
dxd PD
PPDP⋅=+⋅= (4.33) ( i=OP,IO,LS,CP )
j
il
dxdu
AdAdx
dxdu
dAdu
2
1−== (4.32)
95
where dPg/dxj=0 since the Pg load vector is a constant, and dPli/dxj is found by
differentiating Eq. (2.3) with respect to the design variable xj, to get,
vj
ib
j
il
dxdV
dxd
CP
= (4.34) ( i=OP,IO,LS,CP )
where dVbi/dxj is the sensitivity of the design base shear which, from Eq. (3.3), is found
as,
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
>−
≤<
≤≤
=⋅∂∂
=
ie
j
e
e
iivb
ie
i
ie
j
ei
0
is
iab
j
e
e
iab
j
ib
TTdxdT
TSF
gW
TTT
TTdxdT
TSF
gW
dxdT
TS
gW
dx02
1
00
0
2.00
2.003
dV (4.35) ( i=OP,IO,LS,CP )
in which, from Eq. (2.A.10),
( ) j
sns
s
svessns
kkkve
ns
s j
s
s
e
j
e
dxdvCVT
vmvCVTdx
dvvT
dx ∑∑
∑=
=
= ⎟⎟⎠
⎞⎜⎜⎝
⎛
π
⋅⋅−
⋅⋅⋅
π=
∂∂
=1
2,1
2
1,1
2
1 422dT (4.36)
In Eq. (4.36): V1 is an assigned base shear set small enough so that it does not push the
structure into the inelastic range; Cv,s and Cv,k are the inertia load distribution factors at
story levels s and k, respectively (see Eq. (2.4)); vs and vk are lateral drifts under the
action of base shear V1 at story levels s and k, respectively; and dvs /dxj is the static elastic
sensitivity of story drift vs , since V1 is a constant, and is found from Eq. (4.31).
96
4.5 Sensitivity Analysis Procedure
The load-control pushover sensitivity analysis is carried out by the following procedure:
1) Apply a base shear V1, which is distributed heightwise over the structure in
accordance with Eqs. (2.3) and (2.4), and conduct an elastic analysis to find story
drifts vs. Compute the elastic period of the structure Te by Eq. (2.A.10), the spectral
acceleration responses Sai at performance levels i=OP, IO, LS and CP by Eq. (3.3),
and the corresponding design base shears Vbi by Eq. (3.2). Calculate elastic
sensitivities dvs /dxj by Eq. (4.31); then substitute Te and dvs /dxj into Eq. (4.36) to find
the sensitivity dTe /dxj. Substitute Te and dTe/dxj in Eq. (4.35) to find the sensitivity of
design base shears dVbi/dxj for each performance level. Find the load sensitivity
dPi/dxj through Eqs. (4.33) and (4.34).
2) Set load-step index m=1 and conduct the initial pushover analysis to find u(1), P(1),
pj1(1), pj2
(1), etc (see Appendix 2.C).
3) Compute the derivatives of local element force vectors ∂f (1)/∂xj by Eq. (4.28a), and
the derivative of the global internal force vector ∂F(1)/∂xj by Eq. (4.28b).
4) Set m=m+1 and conduct the next step of the incremental pushover analysis.
5) Compute the derivatives of local element force vectors ∂f (m)/∂xj by Eq. (4.27), where
∂Meq,jn(m)/∂xj is found through Eq. (4.25) if yield is detected at any section. Compute
the derivative of the global internal force vector ∂F(m)/∂xj by Eq. (4.11a). Store ∂Fi/∂xj
to disk when Vb(m)=Vb
i, where Vb(m) is the base shear load at load step m.
6) If the base shear reaches the maximum design base shear for the most critical
performance level, stop the pushover analysis; otherwise, go back to step 4.
97
7) Create adjoint load vectors bl (see Eq. (4.4)).
8) Set i=1 (i.e., the OP performance level).
9) Retrieve ∂Fi/∂xj.
10) Recreate global tangential stiffness matrix Ki (see Eq. (4.3)).
11) Solve for virtual displacement vector Uli (see Eq. (4.5)).
12) Calculate duli/dxj by Eq. (4.5).
13) Set i=i+1 (i.e., to 2, 3 or 4, defining the IO, LS and CP performance levels,
respectively); if i > 4, stop sensitivity analysis; otherwise, go to step 9.
In the foregoing procedure, step 1 is specially devised to find the sensitivities of
nodal load vectors Pi at the i=1, 2, 3, 4 specified performance levels (note that step 1 is
not counted as one of the loading steps for pushover analysis). It is critical to accurately
find the vectors dPi/dxj to conduct the seismic design. For this purpose, the following
point is important in the numerical realization: frame girders are modeled as beam
members, i.e., corresponding axial deformations are neglected (this is usually the case
since it is common practice in seismic design to assume floors to be rigid diaphragms)
and, hence, the nodes on the same floor level have identical lateral displacement and
identical sensitivity coefficients for story drifts.
Equation (4.30) is not used directly in the foregoing procedure because it is the
expanded form of Eq. (4.5). Attention needs to be paid to the sense of the internal forces
in the computer coding. That is, Mp and My must have the same sense as the
corresponding moment M; e.g., if M at a section is positive (negative), then its
corresponding Mp and My capacities are both positive (negative).
98
The computer time required for design sensitivity analysis is mainly due to the
work of solving for the adjoint displacement vectors Uli in Eq. (4.5). Generally, this
additional computer time is only a small fraction of that required for a complete pushover
analysis.
4.6 Modal Pushover Sensitivity Analysis
The previous sections in this chapter have concentrated on conventional single-mode
pushover sensitivity analysis. To conduct multi-mode (modal) pushover sensitivity
analysis, it is necessary to differentiate Eq. (2.19) with respect to the design variables
according to the Chain rule (Kaplan, 1973), to get,
j
imnm
im c
imnm
im j
im
im
cnm
imim
jj
c
dxdu
uu
dxdu
dudu
udxd
dxdu ∑∑∑
===
==⎟⎠
⎞⎜⎝
⎛=
11
2/1
1
2 (4.37)
where the derivative duc/duim is obtained from Eq. (2.19) by differentiating uc with
respect to uim (note that the subscript i used to identify vibration modes in Eq. (2.19) is
here replaced by im in order to distinguish it from the subscript i used in this chapter to
identify performance levels). Equation (4.37) indicates that the sensitivity of the
combined response uc to changes in the design is equal to the weighted combination of
the sensitivities of the individual responses uim , where the weighting factors are equal to
the corresponding displacement ratios uim/uc.
Having sensitivities duc/dxj through Eq.(4.37), the load-control modal pushover
sensitivity analysis procedure is carried out as follows:
1) Set mode index im=1 (i.e., the first vibration mode).
99
2) Perform pushover analysis for the single mode im as described in Section 2.5.
3) Conduct sensitivity analysis for the single mode im in accordance with the
procedure described in Section 4.5.
4) Set im=im+1; if im>nm (where nm is the number of vibration modes under
consideration), go to step 5, otherwise go to step 2.
5) Perform combination of modal responses according to Eq. (4.37).
100
P
u
O
Current design New design
dus
dP
du du = dup+ duswhere: du = total displacement variation dus = variation due to static load dup = variation due to varying load
dup
Ki
Figure 4.1 Decomposition of Displacement Variation
101
Appendix 4.A
Derivatives of Correction Matrices Cs and Cg
( )
( ) ( )( ) ( )
( ) ( )( ) ( ) ⎥
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−−+
−−
−−−+
−=
∂∂
26460000
422820000000000
000212460
00018240000000
41
2222
22222
222
2222
2211
pppL
pLppp
ppL
pLpp
pppsC
( )
( ) ( )( ) ( )
( ) ( )( ) ( )⎥
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−−+
−−
−−+
−=
∂∂
121
1121
1121
11211
2212
212460000
18240000000000
00026460
000422820000000
41
ppL
pLpp
pppL
pLppp
pppsC
102
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
=∂∂
nnnn
nnnn
pg
pg
pg
pg
pg
pg
pg
pg
p
66656362
36353332
00
010000
000000
00
000010
000000
n
gC n = 1, 2
where:
( )221211
32
2221
323
211
32 2514025613526416)4(5
4 ppppppppppppLp
g−−+++−
−=
∂∂
( )221122
31
2121
313
212
32 251406483210464)4(5
4 ppppppppppppLp
g−−+−−+
−=
∂∂
( )21221
32121
223
211
33 2825643419216)4(5
4 pppppppppppp
g−+++−
−=
∂∂
( )1312
212
3121
213
212
33 286443243296)4(5
4 pppppppppppp
g−++−+−
−=
∂∂
( )232
321
22
221213
211
36 1121693668128)4(5
4 pppppppppppp
g−+++−
−=
∂∂
( )1212
3122
21213
212
36 1126416646872)4(5
4 pppppppppppp
g−+++−
−=
∂∂
103
( )221211
32
2221
323
211
62 251406483210464)4(5
4 ppppppppppppLp
g−−+−−+
−=
∂∂
( )212122
31
2121
313
212
62 2514025613526416)4(5
4 ppppppppppppLp
g−−+++−
−=
∂∂
( )131
312
21
212213
212
63 1121693668128)4(5
4 pppppppppppp
g−+++−
−=
∂∂
( )2221
321
221213
211
63 1126416646872)4(5
4 pppppppppppp
g−+++−
−=
∂∂
( )12212
31221
213
212
66 2825643419216)4(5
4 pppppppppppp
g−+++−
−=
∂∂
( )232
2211
3221
223
211
66 286443243296)4(5
4 pppppppppppp
g−++−+−
−=
∂∂
nn pg
pg
∂∂
−=∂∂ 6265
nn pg
pg
∂∂
−=∂∂ 3235
104
Appendix 4.B
Model Building Frameworks
Two building frameworks used repeatedly throughout this study as model frameworks for
numerical experiments and design examples are described in the following. Both
frameworks are found in the literature (Gupta and Krawinkler, 1999) and are slightly
modified for this study. Another study (Hasan, Xu and Grierson; 2002) also used the
frameworks to illustrate the pushover analysis technique adopted by this study.
4.B.1 Three-Story Building Framework
This is a perimeter moment frame of a building, which was designed according to the
Uniform Building Code (UBC 1994). All four bays are each 30 feet (9.14 m) wide
(centerline dimensions) and all three stories are each 13 feet (3.96 m) high. The frame has
rigid moment connections, with all the column bases fixed at the ground level. All the
columns use 50 ksi (345 MPa) steel (expected yield stress = 57.6 ksi or 397 MPa) wide-
flange sections, while all the beams use 36 ksi (248 MPa) steel (expected yield stress =
49.2 ksi or 339 MPa) wide-flange sections. The exterior columns all have the same
W14×257 section, while the interior columns all have the same W14×311 section. The
first, second, and roof story beams have W33×118, W30×116, and W24×68 sections,
respectively. Constant gravity loads of 2.2 kips/ft. (32 kN/m) are applied to the first and
105
second story beams, while gravity loads of 1.97 kips/ft. (28.7 kN/m) are applied to the
roof beams. The seismic weight is 1054 kips (4688 kN) for each of the first and second
stories, and 1140 kips (5071 kN) for the roof.
4.B.2 Nine-Story Building Framework
This is also a perimeter moment frame of a building. All five bays span 30 feet (9.14 m)
(centerline dimensions) and stories are 13 feet (3.96 m) high, except that the first story is
18 feet (5.49 m) high. The frame has rigid moment connections, with all the column
bases fixed at the ground level. All the columns use 50 ksi (345 MPa) steel (expected
yield stress = 57.6 ksi or 397 MPa) wide-flange sections, while all the beams use 36 ksi
(248 MPa) steel (expected yield stress = 49.2 ksi or 339 MPa) wide-flange sections. All
member sections are shown in Figure 4.B.2, where all beams on the same floor level are
noted to have the same section. The interior columns at grid lines b, c, d and e have the
same section over the height of the building. Constant gravity loads of 2.2 kips/ft. (32
kN/m) are applied to the beams in the first to the eighth story, while 1.97 kips/ft. (28.7
kN/m) are applied to the roof beams. The seismic weights are 1111 kips (4942 kN) for
the first story, 1092 kips (4857 kN) for each of the second to eighth stories, and 1176 kips
(5231 kN) for the roof.
106
Column Sections: W14×257W14×311 W14×311 W14×311W14×257
1.97 kip/ft.
4 @ 30'
3 @
13'
W33×118
W30×116
W24×68
2.2 kip/ft
2.2 kip/ft.
edcba
Roof
2nd Story
1st Story
(Structure and gravity loads are symmetric about centerline) 1 ft. = 0.3048 m; 1 kip/ft. = 15.59 kN/m
Figure 4.B.1 Three-Story Model Framework
107
2.21 kip/ft.
W30×99
W27×87
W36×135
W36×135
W36×135
W36×160
W36×135
W36×160
W24×68 1.97 kip/ft.
5 @ 30' 18
' 8
@ 1
3'
14×2
57
14×2
83
14×2
83
W14×5
00
14×4
55
14×4
55
14×3
70
14×3
70
14×2
57
14×2
57
14×2
57
W14×3
70
14×3
70
14×3
70
14×2
83
14×2
83
14×2
33
14×2
33
f e d c b a
Roof 8th
7th
6th
5th
4th
3rd
2nd
1st
(Structure and gravity loads are symmetric about centerline) 1 ft. = 0.3048 m; 1 kip/ft. = 15.59 kN/m
Figure 4.B.2 Nine-Story Model Framework
108
Appendix 4.C Numerical Realization and Examples for Sensitivity Analysis Even though the moment-rotation (M-φ) relation expressed by Eq. (2.10) has continuous
first derivatives at φ=0 and φ=φp, a flaw exists in this equation from the viewpoint of
design sensitivity analysis since the post-elastic rotational stiffness of the member end-
section suddenly changes from an infinite to a finite value when first-yield occurs, which
leads to an instantaneous reduction of the plasticity factor p value at first yield. This
phenomenon is equivalent to a stiffness discontinuity at first yield, which results in a
sensitivity discontinuity. A straightforward remedy for this difficulty is to use a smaller
loading increment for the pushover and design sensitivity analyses, but this is usually not
desirable since it increases the computational effort. Fortunately, this sensitivity
discontinuity usually is not a problem for the design synthesis process (more details are
discussed on this matter in the following examples).
Another issue concerning the M-φ relation is the numerical difficulty resulting from
the zero-valued post-elastic rotational stiffness of the plastic-hinge section when φ>φp
(see Figure 2.4), since the derivative of the post-elastic rotational stiffness ∂Rp/∂xj is not
available when φ>φp. To circumvent this problem, a value of φp great enough to avoid the
occurrence of zero post-elastic rotational stiffness for plastic-hinge sections is used,
which has little influence on the results of pushover analysis.
109
The linear relationship between cross section area and moment of inertia (see Eq.
(4.13)) must be paid special attention in order to make calculated sensitivity coefficients
accurate enough for practical usage. Furthermore, the derivative of the plastic section
modulus, ∂Z/∂xj, in Eqs. (4.23), (4.24) and (4.26) must be carefully calculated since its
value has a significant impact on the quality of sensitivity coefficients in the plastic
response range. To this end, it is noted that there are approximate linear relationships
between A and I, and A and S for a specified type and nominal depth of section, as shown
in Figures 3.1 and 3.2. Plastic modulus Z can be expressed as a linear function of section
area A as Z = fs [c4⋅A + c5] through Eq. (3.9b) and (3.9c). Therefore, ∂Z/∂x = fs⋅c4⋅(-A2). In
this way, the sensitivity coefficients ∂Z/∂x are readily suitable for practical steel structural
design using discrete commercially available sections.
4.C.1 Example One: Three-Story Moment Frame
The three-story by four-bay steel moment frame shown in Figure 4.B.1 (hereafter referred
to as a three-story model framework) is used to illustrate the proposed sensitivity analysis
procedure. Only material nonlinearity is considered in this example.
The frame consists of 27 members and the cross-section area of each member is
taken as a design variable. The member sections, gravitational loads and seismic weights
are the same as those for the 3-story model frame in Appendix 4.B.
In pushover analysis, each girder is modeled by two beam elements by considering
the mid-span as a point for a potential plastic hinge, while each column is represented by
a single beam-column element. Post-elastic rotation φp is taken to be 0.09. The heightwise
distribution of lateral loading is calculated by Eq. (2.4) for µ=2. The first-step base shear
110
is taken to be 16 kips (71 kN). Thereafter, the base shear load increment is taken to be 4
kips (17.8 kN).
The base shear V1 is also taken to be 16 kips (71 kN), which is applied to the
structure to find the elastic period of the frame to be Te=0.88 seconds. The sensitivity
coefficients dTe/dAj for the elastic period are found through Eq. (4.36) to be as shown in
column 2 of Table 4.C.1; as expected, the sensitivity coefficients are symmetric about the
structural centerline. The forward finite-difference method was employed to verify the
sensitivity coefficients in Table 4.C.1 for a 1% perturbation of all member sizes. The
ratios of the analytical dTe/dAj value to the finite difference value found are shown in the
last column of Table 4.C.1, where it can be observed that the results of the two methods
agree well with each other.
All four of the performance levels OP, IO, LS and CP are considered for the
pushover and design sensitivity analyses. The design base shears corresponding to the
four performance levels are 532, 939, 1251 and 1454 kips (1 kip = 4.448 kN),
respectively. The corresponding pushover curve is plotted in Figure 4.C.1. The structure
is fully elastic at the OP level, slightly inelastic at the IO level, somewhat more inelastic
at the LS level, and significantly inelastic at the CP level.
The sensitivity coefficients for the roof drift ∆ at the four performance levels are
listed in Table 4.C.2, the forward finite-difference method was used to verify these
sensitivity coefficients for a 1% perturbation of member areas (except as otherwise noted).
The ratio of the calculated sensitivity coefficient to the finite difference value is listed in
parenthesis in Table 4.C.2, and it is observed that the calculated sensitivity coefficients
match the finite difference values very well except for some members at the LS level.
111
Upon examining the occurrence of the plasticity-factors at the LS level shown in Figure
4.C.2, it is found that the lower end of column e/0-1 just yields but that its plasticity-
factor value abruptly drops from 1.0 to 0.75. As noted in the previous section, this sudden
stiffness degradation leads to a discontinuity in the sensitivity coefficient. On the other
hand, all of the evaluated sensitivity coefficients at the CP level agree very well with the
finite difference values since the structure is significantly plastic, as indicated in Figure
4.C.3, such that no sudden stiffness degradation occurs.
Sensitivity coefficients may be used to predict structural responses. For example,
suppose that beam 1/a-b in Figure 4.C.2 (i.e., the beam on the first floor between grid a
and b) is modified from a W33×118 (A=34.7 in.2 / 22387 mm2) section to a W33×130
(A=38.3 in.2 / 24710 mm2) section. The roof drifts ∆ at the four performance levels for the
modified frame are predicted using a first-order Talyor series and the sensitivity
coefficients in Table 4.2, i.e.,
( ) )( 0
0
0jj
j
AAdAd
−⎟⎟⎠
⎞⎜⎜⎝
⎛ ∆+∆=∆ (4.C.1)
to find,
( ) level. CP at the in. 5950.247.343.381059.34548387.25 4 =−×−=∆ −
( )
( )
( )
level, LS at the in. 9910.67.343.381097.3581203.7
level, IO at the in. 7012.37.343.381022.357138.3
level, OP at the in. 0873.27.343.381099.1409267.2
4
4
4
=−×−=∆
=−×−=∆
=−×−=∆
−
−
−
Formal reanalysis of the modified structure finds that the roof drifts at the four
performance levels are 2.0859, 3.7014, 6.9556 and 24.3259 inches (1 in. = 25.4 mm),
112
respectively, which, considering that the variation in member size is as much as 10.4%,
agrees fairly well with the values found using calculated sensitivity coefficients.
4.C.2 Example Two: Nine-Story Moment Frame
The nine-story model framework in Appendix 4.B is used to illustrate the sensitivity
analysis procedure taking into account both second-order effects and the combined stress
yielding condition. The member sections, gravitational loads, seismic weights, and frame
dimensions are the same as those for the model frame in Figure 4.B.2. The heightwise
distribution of lateral earthquake inertial loads is calculated by Eq. (2.4) for µ=1. Only
one performance level is considered for this example, and its design base shear is taken to
be 2100 kips (9341 kN).
The base shear load increment is taken to be 30 kips (133 kN) until first-yield is
detected. Thereafter, the base shear load increment is reduced to 6 kips (26.7 kN). The
corresponding pushover curve is plotted in Figure 4.C.4, from which it can be noted that
the structure undergoes significant plastification at the design base shear performance
level (The reason that this example uses the relatively small 6 kips base shear increment
after first yield is to capture the reduced yielding moments Myr more accurately and
therefore reduce the errors in calculating sensitivity coefficients).
The base shear V1 is taken to be 30 kips (133 kN), and the elastic period of the
structure is found to be Te=2.076 seconds. Some representative sensitivity coefficients for
the roof drift ∆ are listed in column 2 of Table 4.C.3 for the columns in the first and
second story having the greatest axial force. The forward FDM was used to verify the
sensitivity coefficients for a 1% perturbation of member sizes.
113
4.C.3 Discussion
Sensitivity analysis forms the basis for design optimization in this study. In order to
provide high quality sensitivity coefficients, it is very important to develop sensitivity
formulations that are based upon, and completely consistent with, the pushover analysis
techniques presented in Chapter 2.
Since the performance-based design only concerns a limited number of damage
states, it is not necessary to find the design sensitivity coefficients over the full loading
history. In fact, as roof and inter-story drift displacements are alone of concern, the
adjoint variable method (AVM) is very economical in evaluating the design sensitivity
coefficients. For example, the design sensitivity analysis of the three-story frame example
only involves solving the linear Eqs. (4.5) seven times [three times to determine adjoint
displacement vectors Uli in order to find dvs/dxj (s =1, 2, 3) in Eq. (4.36); and one time for
each of the 4 performance levels to determine the adjoint displacement vector in order to
find d∆/dxj]. As a result, the additional computer time required for sensitivity analysis is
only a small fraction of that required for a complete pushover analysis.
An important feature of seismic design is that the inertial loading is itself a function
of the design variables. The accurate evaluation of earthquake loading sensitivity is
essential for the overall design sensitivity analysis. The method proposed in Section 4.4 to
evaluate the sensitivity of nodal loads, including seismic effects, has proven to be
successful. In fact, since structural displacements and their sensitivities are so readily
available from the pushover analysis, the implementation of the method in Section 4.4 is
quite straightforward.
114
An important fact about plastic sensitivity coefficients needs to be mentioned here.
From Table 4.C.2, the displacement sensitivity coefficients at the CP level are two orders
of magnitude greater than the elastic sensitivity coefficients at the OP and IO levels. It
can be concluded that the plastic structural response is much more sensitive to the
variation of member sizes than the elastic response is.
The numerical results of the two examples illustrate the applicability of the derived
sensitivity analysis formulations. Some sources of error and other key issues in numerical
realization are identified in the examples presented in Chapters 6 and 7.
115
0
1600
10 15 20 25 30 ∆ (in.)
0 5
400
800
1200
Vb (kips)
OPIO
LS
CP
Figure 4.C.1 Pushover Curve for Three-Story Framework
0.31
0.070.05 0.05
0.05 0.06 0.320.06 0.31 0.060.16
0.06 0.070.05
0.04 0.080.050.080.050.080.05 0.06
1.00 0.21
a b c d e
0.75 0.210.21
0.04 0.07
Figure 4.C.2 Plasticity-Factors at the LS Level
116
a b c d e
0.010.01 0.01
0.01 0.01 0.020.01 0.02 0.020.010.02
0.01 0.010.01
0.01 0.01 0.010.35
0.01 0.010.35
0.01 0.010.35
0.01
0.03 0.030.030.030.03
0.01 0.01
Figure 4.C.3 Plasticity-Factors at the CP Level
∆ (in.) 0
500
1000
1500
2000
2500
0 10 20 30 40 50 60
Specified Performance Level Vb (kips)
Figure 4.C.4 Pushover Curve for Nine-Story Framework
117
TABLE 4.C.1 THREE-STORY FRAMEWORK: SENSITIVITY COEFFICIENTS
FOR THE ELASTIC PERIOD
Member Aj(at grid # & level #)
dTe/dAj (10-4)
Ratio of Analytical Value to Finite Difference Value
a/0-1 -2.10 0.99 b/0-1 -2.64 1.01 c/0-1 -2.59 0.99 d/0-1 -2.64 0.99 e/0-1 -2.10 0.99 a/1-2 -0.90 1.01 b/1-2 -1.93 1.00 c/1-2 -1.85 1.02 d/1-2 -1.93 0.99 e/1-2 -0.90 0.99 a/2-3 -0.31 0.97 b/2-3 -0.77 1.00 c/2-3 -0.76 1.04 d/2-3 -0.77 1.02 e/2-3 -0.31 0.97 1/a-b -8.45 1.01 1/b-c -7.01 1.01 1/c-d -7.01 0.99 1/d-e -8.45 0.99 2/a-b -7.62 0.99 2/b-c -6.73 1.01 2/c-d -6.73 1.00 2/d-e -7.62 0.99 3/a-b -4.96 1.01 3/b-c -4.58 1.01 3/c-d -4.58 0.99 3/d-e -4.96 0.99
0 Level
1st Level
2nd Level
3rd Level
edcba
118
TABLE 4.C.2
THREE-STORY FRAMEWORK: SENSITIVITY COEFFICIENTS FOR THE ROOF DRIFTS AT VARIOUS PERFORMANCE LEVELS
d∆ / dAj (10-4) Member Aj
(at grid # & level #) OP IO LS CP a/0-1 -2.12 (1.01) -3.59 (1.01) -40.08 (0.95) -904.04 (1.01)b/0-1 -3.20 (1.02) -4.59 (1.02) -11.65 (0.85) -750.33 (0.99)c/0-1 -3.10 (1.01) -5.30 (1.02) -13.34 (0.83) -766.80 (0.99)d/0-1 -3.09 (0.99) -4.63 (0.99) -11.70 (0.82) -750.30 (1.01)e/0-1 -3.10 (0.99) -2.73 (0.96) -39.92 (0.95) -900.68 (1.01)a/1-2∗ -1.02 (1.01) -1.99 (1.01) 26.72 (0.96) 294.36 (1.01)b/1-2 -3.82 (1.01) -5.35 (1.03) 40.25 (1.01) 603.03 (0.98)c/1-2 -3.60 (1.02) -8.95 (1.02) 37.38 (1.01) 575.12 (0.97)d/1-2 -3.71 (1.02) -6.25 (1.02) 40.31 (1.01) 603.03 (0.98)e/1-2∗ -2.40 (1.01) 1.57 (1.01) 27.03 (1.01) 295.22 (1.02)a/2-3∗ -0.97 (0.99) -1.80 (0.99) 4.93 (0.96) 103.72 (1.03)b/2-3∗ -4.14 (1.01) -5.48 (1.02) 13.66 (0.97) 255.82 (1.02)c/2-3∗ -3.98 (0.99) -8.41 (0.99) 12.78 (0.93) 249.12 (1.03)d/2-3∗ -4.03 (1.01) -7.46 (1.01) 13.35 (0.97) 255.83 (1.03)e/2-3∗ -2.75 (1.01) -2.93 (1.01) 7.11 (0.96) 105.09 (1.01)1/a-b -14.99 (1.01) -35.22 (1.05) -358.97 (0.99) -3454.59 (0.99)1/b-c -11.53 (1.01) -25.45 (1.01) -355.23 (0.99) -3869.26 (0.99)1/c-d -11.72 (1.01) -26.15 (1.01) -355.34 (0.99) -3869.49 (0.99)1/d-e -12.61 (1.01) -41.57 (1.01) -361.15 (0.99) -3456.43 (0.99)2/a-b -24.28 (0.99) -65.73 (1.01) -501.03 (0.99) -3466.59 (0.99)2/b-c -20.75 (1.01) -50.06 (1.02) -542.32 (1.01) -3757.28 (1.02)2/c-d -20.72 (1.01) -47.44 (0.99) -542.56 (0.99) -3757.09 (0.98)2/d-e -22.89 (0.99) -70.38 (0.99) -499.74 (1.01) -3466.78 (1.01)3/a-b -24.63 (0.99) -59.02 (0.99) -423.09 (1.01) -3085.55 (1.01)3/b-c -20.51 (1.01) -44.91 (1.01) -441.43 (0.99) -3217.54 (0.99)3/c-d -20.75 (0.99) -42.43 (0.99) -440.62 (1.01) -3217.59 (0.99)3/d-e -20.64 (0.98) -40.68 (0.98) -454.15 (1.01) -3090.89 (0.99)∆ (in.) 2.0927 3.7138 7.1203 25.8387
Note: 1. * Central finite-difference method was here used. 2. Inside the brackets are the ratios of analytical value to finite difference value. 3. 1 in. = 25.4 mm
119
TABLE 4.C.3 NINE-STORY FRAMEWORK: SENSITIVITY COEFFICIENTS
FOR THE ROOF DRIFT
Member Aj(at grid # & level #)
d∆ / dAj Ratio of Analytical Value to Finite Difference Value
a/0-1 -0.489 0.94 b/0-1 -0.461 0.97 c/0-1 -0.463 1.06 d/0-1 -0.463 1.06 e/0-1 -0.461 0.97 f/0-1 -0.505 0.96 a/1-2 0.081 0.93 b/1-2 0.130 1.04 c/1-2 0.127 0.97 d/1-2 0.127 0.97 e/1-2 0.130 1.05 f/1-2 0.081 0.94 1/a-b -2.257 1.01 1/b-c -2.344 1.01 ∆ (in.) 56.814
1 in. = 25.4 mm
a b c d e
3rd Level
2nd Level
1st Level
0 Level
9st Level
f
120