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