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Optimal Design of Semi-Rigid Steel Frames using

Practical Nonlinear Inelastic Analysis

Se-Hyu Choi1,* and Seung-Eock Kim2

1Department of Civil Engineering, Kyungpook National University, Daegu 702-701, South Korea2Department of Civil and Environmental Engineering, Sejong University, Seoul 143-747, South Korea

Abstract

Optimal design of semi-rigid steel frames using practical nonlinear inelastic analysis is developed. A practical nonlinearinelastic analysis realistically assesses both strength and behavior of a structural system and its component members in a directmanner. To capture second-order effects associated with P-δ and P-∆ moments, stability functions are used to minimizemodeling and solution time. The Column Research Council (CRC) tangent modulus concept is used to account for gradualyielding due to residual stresses. A softening plastic hinge model is used to represent the degradation from elastic to zerostiffness associated with development of a hinge. Kishi-Chen power model is used to describe the nonlinear behavior of semi-rigid connections. A section increment method is used for minimum weight optimization. Constraint functions are load-carryingcapacities and displacements. A member with the largest unit value evaluated by LRFD interaction equation is replaced oneby one with an adjacent larger member selected in the database. Member sizes determined by the proposed method arecompared with those given by other approaches.

Keywords: nonlinear inelastic analysis, section increment method, optimal design, steel frames.

1. Introduction

Conventional analysis of steel frame structures is usually

carried out under the assumption that the beam-to-column

connections are either fully rigid or ideally pinned.

However, most connections used in current practice are

semi-rigid type whose behavior lies between these two

extreme cases. In the AISC-LRFD Specification (1993),

two types of constructions are designated: Type FR (fully

restrained) construction; and Type PR (partially restrained)

construction. The LRFD Specification permits the evaluation

of the flexibility of connections by rational means when

the flexibility of connections is accounted for in the

analysis and design of frames.

The semi-rigid connections influence the moment

distribution in beams and columns as well as the drift (P-

∆ effect) of the frame. One way to account for all these

effects in semi-rigid frame design is through the use of a

direct nonlinear inelastic analysis. A nonlinear inelastic

analysis indicates a method that can sufficiently capture

the limit state strength and stability of a structural system

and its individual members so that separate member

capacity checks are not required. Since the power of

personal computers and engineering workstations is

rapidly increasing, it is feasible to employ advanced

analysis techniques directly in engineering design office.

During the past 20 years, research efforts have been

devoted to the development and validation of several

nonlinear inelastic analysis methods. The nonlinear inelastic

analysis methods may be classified into two categories:

(1) Plastic-zone method; and (2) Plastic hinge method.

Although the plastic-zone solution is known as the “exact

solution”, it cannot be used for practical design purpose

(Chen and Toma, 1994). This is because the method is too

intensive in computation and costly due to its complexity.

Until now, several practical nonlinear inelastic analyses

for steel frames with rigid and semi-rigid connection

were developed by Lui and Chen (1986), Al-Mashary and

Chen (1991), Kishi and Chen (1990), Liew (1992), Kim

and Chen (1996a; 1996b), Barsan and Chiorean (1997),

Ziemian et al. (1992), Prakash and Powell (1993), Liew

and Tang (1998), Kim et al. (2000), and Choi and Kim

(2002). When these practical analyses are combined with

optimal design procedures leading to minimum weight,

the results must be a considerable contribution to practicing

engineering.

Since the members used for steel frames covered in this

study are mostly ready-made, a discrete optimization

technique should be adopted. Structural optimization

design with discrete design variables is usually very

much complicated. Rajeev and Krishnammorthy (1992),

Lin and Hajela (1992), Pantelides and Tzan (1997), and

*Corresponding authorTel: +82-53-950-7582, Fax: +82-53-950-6564E-mail: [email protected]

142 Se-Hyu Choi and Seung-Eock Kim

Kim and Ma (2003) performed structural optimization

design using Genetic Algorithms (GA).

However, these methods using the elastic analysis of

steel frames have shortcoming that does not satisfy

compatibility between the system of structure and individual

members. To overcome the shortcoming, Pezeshk et al.

(2000) have performed nonlinear inelastic optimal design

using genetic algorithm and resulted in advanced results.

Genetic algorithm as optimum technique, however, has

long computational time because of numerous analyses.

In this paper, optimal design of semi-rigid steel frames

using section increment method is developed to find the

value of optimum design more efficiently. A section

increment method is used for minimum weight optimization.

Constraint functions used are load-carrying capacities and

displacements. If the load-carrying capacities of a

structure are not satisfied, a member with the largest unit

value (calculated by LRFD interaction equations) is

replaced one by one with an adjacent larger member

selected in the database. If the displacements are not

satisfied, a member with the largest displacement ratio is

replaced with a larger member. The database embodies

wide flange sections (W-sections) listed in AISC-LRFD

specification (1993).

2. Practical Nonlinear Inelastic Analysis

2.1. Geometric second-order effects

Stability functions are used to capture the second-order

effects since they can account for the effect of the axial

force on the bending stiffness reduction of a member. The

benefit of using stability functions is that it enables only

one or two elements to predict accurately the second-

order effect of each framed member (Kim and Chen,

1996a; Kim and Chen 1996b).

The force-displacement equation using stability functions

may be written for three-dimensional beam-column element

as

(1)

where P, MyA, MyB, MzA, MzB, and, T are axial force, end

moments with respect to y and z axes and torsion

respectively. δ, θyA, θyB, θzA, θzB, and, φ are the axial

displacement, the joint rotations, and the angle of twist.

S1, S2, S3, and S4 are the stability functions with respect to

y and z axes, respectively.

The stability functions given by Eq. (1) may be written

as

(2a)

(2b)

(2c)

(2d)

where , , and Pis

positive in tension.

2.2. CRC tangent modulus model associated with

residual stresses

The CRC tangent modulus concept is used to account

for gradual yielding (due to residual stresses) along the

length of axially loaded members between plastic hinges.

The elastic modulus E (instead of moment of inertia I) is

reduced to account for the reduction of the elastic portion

of the cross-section since the reduction of the elastic

modulus is easier to implement than a new moment of

inertia for every different section. From Chen and Lui

(1991), the CRC Et is written as

for (3a)

for (3b)

2.3. Parabolic function for gradual yielding due to

flexure

The tangent modulus model is suitable for the member

subjected to axial force, but not adequate for cases of

both axial force and bending moment. A gradual stiffness

P

MyA

MyB

MzA

MzB

T⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

0

EA

L------- 0 0 0 0 0

0 S1

EIy

L-------S

2

EIy

L------- 0 0 0

0 S2

EIy

L-------S

1

EIy

L------- 0 0 0

0 0 0 S3

EIz

L-------S

4

EIz

L------- 0

0 0 0 S4

EIz

L-------S

3

EIz

L------- 0

0 0 0 0 0GJ

L-------

δ

θyA

θyB

θzA

θzB

φ⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=

S1

π ρy π ρy( )sin π2ρ π ρy( )cos–

2 2 π ρy( )cos– π ρy π ρy( )sin–

--------------------------------------------------------------------------- if P 0<

π2ρ h π ρy( )cos π ρy h π ρy( )sin–

2 2 h π ρy( )cos– π ρy h π ρy( )sin+

--------------------------------------------------------------------------------- if P 0>⎩⎪⎪⎨⎪⎪⎧

=

S2

π2ρy π ρy π ρy( )sin–


--------------------------------------------------------------------------- if P 0<

π ρy h π ρy( )sin π2ρy–

2 2 h π ρy( )cos– π ρy h π ρy( )sin+

--------------------------------------------------------------------------------- if P 0>⎩⎪⎪⎨⎪⎪⎧

=

S1

π ρz π ρz( )sin π2ρ π ρz( )cos–

2 2 π ρz( )cos– π ρz π ρz( )sin–

-------------------------------------------------------------------------- if P 0<

π2ρ h π ρz( )cos π ρz h π ρz( )sin–

2 2 h π ρz( )cos– π ρz h π ρz( )sin+

--------------------------------------------------------------------------------- if P 0>⎩⎪⎪⎨⎪⎪⎧

=

S4

π2ρz π ρz π ρz( )sin–


--------------------------------------------------------------------------- if P 0<

π ρz h π ρz( )sin π2ρz–

2 2 h π ρz( )cos– π ρz π ρz( )sin+

------------------------------------------------------------------------------ if P 0>⎩⎪⎪⎨⎪⎪⎧

=

ρy P π2EIy L

2⁄( )⁄= ρz P π2EIz L

2⁄( )⁄=

Et 1.0E= P 0.5Py≤

Et 4P

Py

-----E 1P

Py

-----–⎝ ⎠⎛ ⎞

= P 0.5Py>

Optimal Design of Semi-Rigid Steel Frames using Practical Nonlinear Inelastic Analysis 143

degradation model for a plastic hinge is required to

represent the partial plastification effects associated with

bending. We shall introduce the parabolic function to

represent the transition from elastic to zero stiffness

associated with a developing hinge. When the parabolic

function for a gradual yielding is active at both ends of an

element, the slope-deflection equation may be expressed

as

(4)

where

(5a)

(5b)

(5c)

(5d)

(5e)

. (5f)

The terms ηA and ηB is a scalar parameter that allows

for gradual inelastic stiffness reduction of the element

associated with plastification at end A and B. This term

is equal to 1.0 when the element is elastic, and zero when

a plastic hinge is formed. The parameter η is assumed to

vary according to the parabolic function:

η = 1.0 for α ≤ 0.5 (6a)

η = 4α(1 − α) for α > 0.5 (6b)

where α is a force-state parameter that measures the

magnitude of axial force and bending moment at the

element end. The term (α) may be expressed by AISC-

LRFD and Orbison, respectively:

AISC-LRFD

Based AISC-LRFD bilinear interaction equation

(Kanchanalai, 1977), the cross-section plastic strength of

the beam-column member may be expressed

for (7a)

for (7b)

Orbison

Orbison’s full plastification surface (Orbison, 1982) of

cross-section is given by

(8)

Where, , (strong-axis),

(weak-axis).

2.4. Shear deformation

The stiffness coefficients should be modified to account

for the effect of the additional flexural shear deformation

in a beam-column element. The flexural flexibility matrix

can be obtained by inversing the flexural stiffness matrix

as

(9)

where kii, kij, and kjj are the elements of stiffness matrix

in a planar beam-column. θMA and θMB are the slope of the

neutral axis due to bending moment. The flexibility

matrix corresponding to flexural shear deformation may

be written as

(10)

where GAs and L are shear rigidity and length of the

beam-column, respectively. Total rotation at the A and B

is obtained by combining Eq. (9) and Eq. (10) as

(11)

The force-displacement equation including flexural

shear deformation is obtained by inversing the flexibility

matrix as

P

MyA

MyB

MzA

MzB

T⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

0

EtA

L-------- 0 0 0 0 0

0 kiiykijy 0 0 0

0 kijykjjy 0 0 0

0 0 0 kiizkijz 0

0 0 0 kijzkjjz 0

0 0 0 0 0GJ

L-------

δ

θyA

θyB

θzA

θzB

φ⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=

kiiy ηA S1

S2

2

S1

----- 1 ηB–( )–

⎝ ⎠⎜ ⎟⎛ ⎞EtIy

L--------=

kijy ηAηBS2EtIy

L--------=

kjjy ηB S1

S2

2

S1

----- 1 ηA–( )–

⎝ ⎠⎜ ⎟⎛ ⎞EtIy

L--------=

kiiz ηA S3

S4

2

S3

----- 1 ηB–( )–

⎝ ⎠⎜ ⎟⎛ ⎞EtIz

L--------=

kijz ηAηBS4EtIz

L--------=

kjjz ηB S3

S4

2

S3

----- 1 ηA–( )–

⎝ ⎠⎜ ⎟⎛ ⎞EtIz

L--------=

αP

Py

-----8

9---My

Myp

---------8

9---Mz

Mzp

--------+ +=P

Py

-----2

9---My

Myp

---------2

9---Mz

Mzp

--------+≥

αP

2Py

--------My

Myp

---------Mz

Mzp

--------+ +=P

Py

-----2

9---My

Myp

---------2

9---Mz

Mzp

--------+<

α 1.15p2mz

2my

43.67p

2mz

23.0p

6my

24.65mz

4my

2+ + + + +=

p P Py⁄= mz Mz Mpz⁄= my My Mpy⁄=

θMA

θMB⎩ ⎭⎨ ⎬⎧ ⎫

kjj

kiikjj kij2

–

-------------------kij

kiikjj kij2

–

-------------------–

kij

kiikjj kij2

–

-------------------–kii

kiikjj kij2

–

-------------------

MA

MB⎩ ⎭⎨ ⎬⎧ ⎫

=

θSA

θSB⎩ ⎭⎨ ⎬⎧ ⎫

1

GAsL-------------

1

GAsL-------------

1

GAsL-------------

1

GAsL-------------

MA

MB⎩ ⎭⎨ ⎬⎧ ⎫

=

θA

θB⎩ ⎭⎨ ⎬⎧ ⎫ θMA

θMB⎩ ⎭⎨ ⎬⎧ ⎫ θSA

θSB⎩ ⎭⎨ ⎬⎧ ⎫

–=


(12)

The force-displacement equation may be written for

three-dimensional beam-column element as

(13)

in which

(14a)

(14b)

(14c)

(14d)

(14e)

(14f)

where Asy and Asz are the effective shear areas with

respect to y and z axes, respectively.

2.5. Effect of semi-rigid connection

The connection behavior is represented by its moment-

rotation relationship. Extensive experimental works on

connections have been performed, and a large body of

moment-rotation data has been collected (Goverdhan,

1983; Nethercot, 1985; Chen and Kishi, 1989; Kishi and

Chen, 1990). Using these abundant data base, researchers

have developed several connection models including:

linear; polynomial; B-spline; power; and exponential

models. Herein, the three-parameter power model proposed

by Kishi and Chen (1990) is adopted as shown in Fig. 1.

The Kishi-Chen power model contains three parameters:

initial connection stiffness Rki, ultimate connection

moment capacity Mu, and shape parameter n. The power

model may be written as:

for θ > 0, m > 0 (15)

where m =M/Mu, θ = θr/θ0, θ0 = reference plastic rotation,

Mu/Rki, Mu = ultimate moment capacity of the connection,

Rki = initial connection stiffness, and n = shape parameter.

When the connection is loaded, the connection tangent

stiffness (Rkt) at an arbitrary rotation θr can be derived by

simply differentiating Eq. (15) as:

(16)

When the connection is unloaded, the tangent stiffness

is equal to the initial stiffness as:

(17)

It is observed that a small value of the power index n

makes a smooth transition curve from the initial stiffness

Rkt to the ultimate moment Mu. On the contrary, a large

value of the index n makes the transition more abruptly.

In the extreme case, when n is infinity, the curve becomes

a bilinear line consisting of the initial stiffness Rki and the

ultimate moment capacity Mu.

The connection may be modeled as a rotational spring

in the moment-rotation relationship represented by Eq.

(13). Figure 2 shows a beam-column element with semi-

rigid connections at both ends. If the effect of connection

flexibility is incorporated into the member stiffness, the

incremental element force-displacement relationship of

Eq. (13) is modified as

MA

MB⎩ ⎭⎨ ⎬⎧ ⎫

kiikjj kij2

– kiiAsGL+

kii kjj 2kij AsGL+ + +--------------------------------------------

kiikjj– kij2kijAsGL+ +

kii kjj 2kij AsGL+ + +---------------------------------------------

kiikjj– kij2kijAsGL+ +

kii kjj 2kij AsGL+ + +---------------------------------------------

kiikjj kij2

– kjjAsGL+

kii kjj 2kij AsGL+ + +--------------------------------------------

θA

θB⎩ ⎭⎨ ⎬⎧ ⎫

=

P

MyA

MyB

MzA

MzB

T⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

0

EtA

L-------- 0 0 0 0 0

0 CiiyCijy 0 0 0

0 CijyCjjy 0 0 0

0 0 0 CiizCijz 0

0 0 0 CijzCjjz 0

0 0 0 0 0GJ

L-------

δ

θyA

θyB

θzA

θzB

φ⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=

Ciiy

kiiykjjy kijy2

– kiiyAszGL+

kiiy kjjy 2kijy AszGL+ + +----------------------------------------------------=

Cijy

k– iiykjjy kijy2

kijyAszGL+ +

kiiy kjjy 2kijy AszGL+ + +------------------------------------------------------=

Cjjy

kiiykjjy kijy2

– kjjyAszGL+

kiiy kjjy 2kijy AszGL+ + +----------------------------------------------------=

Ciiz

kiizkjjz kijz2

– kiizAsyGL+

kiiz kjjz 2kijz AsyGL+ + +---------------------------------------------------=

Cijz

k– iizkjjz kijz2

kijzAsyGL+ +

kiiz kjjz 2kijz AsyGL+ + +------------------------------------------------------=

Cjjz

kiizkjjz kijz2

– kjjzAsyGL+

kiiz kjjz 2kijz AsyGL+ + +---------------------------------------------------=

mθ

1 θn+( )

1 n⁄----------------------=

Rkt

dM

dθr

---------Mu

θ01 θn+( )

1 1+ n⁄--------------------------------= =

Rkt

dM

dθr

---------Mu

θ0

------- Rkt= = =

Figure 1. Moment-rotation behavior of three-parametermodel.


(18)

where

(19a)

(19b)

(19c)

(19d)

(19e)

(19f)

(19g)

(19h)

in which RktYA = tangent stiffness of connections A in

the Y-direction, RktYB = tangent stiffness of connections B

in the Y-direction, RktZA = tangent stiffness of connections

A in the Z-direction, and RktZB= tangent stiffness of

connections B in the Z-direction.

2.6. Geometric imperfection modeling

2.6.1. Braced frame

The proposed analysis implicitly accounts for the

effects of both residual stresses and spread of yielded

zones. To this end, proposed analysis may be regarded as

equivalent to the plastic-zone analysis. As a result,

geometric imperfections are necessary only to consider

fabrication error. For braced frames, member out-of-

straightness, rather than frame out-of-plumbness, needs to

be used for geometric imperfections. This is because the

P-∆ effect due to the frame out-of-plumbness is diminished

by braces. The ECCS (1984; 1991), AS (1990), and

CSA (1989; 1994) Specifications recommend an initial

crookedness of column equal to 1/1,000 times the column

length. The AISC Code recommends the same maximum

fabrication tolerance of Lc/1,000 for member out-of-

straightness. In this study, a geometric imperfection of Lc/

1,000 is adopted.

The ECCS, AS, and CSA Specifications recommend

the out-of-straightness varying parabolically with a

maximum in-plane deflection at the mid-height. They do

not, however, describe how the parabolic imperfection

should be modeled in analysis. Ideally, many elements

are needed to model the parabolic out-of-straightness of a

beam-column member, but it is not practical. In this

study, two elements with a maximum initial deflection at

the mid-height of a member are found adequate for

capturing the imperfection. Figure 3 shows the out-of-

straightness modeling for a braced beam-column member.

It may be observed that the out-of-plumbness is equal to

1/500 when the half segment of the member is

considered. This value is identical to that of sway frames

as discussed in recent papers by Kim and Chen (1996a;

1996b). Thus, it may be stated that the imperfection

values are essentially identical for both sway and braced

frames.

2.6.2. Unbraced frame

Referring to the ECCS (1984; 1991), an out-of-

plumbness of a column equal to 1/200 times the column

height is recommended for the elastic plastic-hinge

P

MyA

MyB

MzA

MzB

T⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

0

EtA

L-------- 0 0 0 0 0

0 C∗iiyC∗ijy 0 0 0

0 C∗ijyC∗jjy 0 0 0

0 0 0 C∗iizC∗ijz 0

0 0 0 C∗ijzC∗jjz 0

0 0 0 0 0GJ

L-------

δ

θyA

θyB

θzA

θzB

φ⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=

C∗iiy

Ciiy

CiiyCjjy

RktYB

----------------Cijy

2

RktYB

-----------–+

⎝ ⎠⎜ ⎟⎛ ⎞

R∗Y

--------------------------------------------------=

C∗jjy

Cjjy

CiiyCjjy

RktYA

----------------Cijy

2

RktYA

-----------–+

⎝ ⎠⎜ ⎟⎛ ⎞

R∗Y

--------------------------------------------------=

C∗ijy

Cijy

R∗Y

---------=

C∗iiz

Ciiz

CiizCjjz

RktZB

----------------Cijz

2

RktZB

-----------–+

⎝ ⎠⎜ ⎟⎛ ⎞

R∗Z

-------------------------------------------------=

C∗jjz

Cjjz

CiizCjjz

RktZA

----------------Cijz

2

RktZA

-----------–+

⎝ ⎠⎜ ⎟⎛ ⎞

R∗Z

-------------------------------------------------=

C∗ijz

Cijz

R∗Z

---------=

R∗Y 1Ciiy

RktYA

-----------+⎝ ⎠⎛ ⎞

1Cjjy

RktYB

-----------+⎝ ⎠⎛ ⎞ Cijy

2

RktYARktYB

-----------------------–=

R∗Z 1Ciiz

RktZA

-----------+⎝ ⎠⎛ ⎞

1Cjjz

RktZB

-----------+⎝ ⎠⎛ ⎞ Cijz

2

RktYZRktZB

-----------------------–=

Figure 2. Beam-column element with semi-rigid connections.


analysis. For multi-story and multi-bay frames, the

geometric imperfections may be reduced to 1/200kcks

since all columns in buildings may not lean in the same

direction. The coefficients, kc and k

s account for the

situation where a large number of columns in a story and

stories in a frame would reduce the total magnitude of

geometric imperfections. According to the ECCS (1984;

1991), the member initial out-of-straightness should be

modeled at the same time with the initial out-of-

plumbness if the column parameter is larger

than 1.6. This may be necessary to consider residual

stresses and possible member instability effects for highly

compressed slender columns, however, the magnitude of

the imperfection is not specified in the ECCS (1984;

1991).

Since plastic-zone analysis accounts for both residual

stresses and the spread of yielding, only geometric

imperfections for erection tolerances need be included in

the analysis. The ECCS recommends the out-of-plumbness

of columns equal to 1/300r1r2 times the column height

where r1 and r2 are factors which account for the length

and number of columns, respectively. For the plastic-zone

analysis, the ECCS does not specify the requirement of

the initial out-of-straightness to be modeled in addition to

the out-of-plumbness when the column parameter

is larger than 1.6, since the plastic-zone

analysis already includes residual stresses and spread of

yielding in its formulation.

Since proposed analysis implicitly accounts for both

residual stresses and the spread of yielding, it may be

considered equivalent to the plastic-zone analysis. Thus,

modeling the out-of-plumbness for erection tolerances is

used here without the out-of-straightness for the column,

regardless of the value of the column parameter, so that

the same ultimate strength can be predicted for

mathematically identical braced and unbraced members.

This simplification enables us to use the proposed

methods easily with consistent imperfection modeling.

The CSA (1989; 1994) and the AISC Code of Standard

Practice (1993) set the limit of erection out-of-plumbness

Lc/500. The maximum erection tolerances in the AISC

are limited to 1 in. toward the exterior of buildings and 2

in. toward the interior of buildings less than 20 stories.

Considering the maximum permitted average lean of 1.5

in. in the same direction of a story, the geometric

imperfection of Lc/500 can be used for buildings up to 6-

stories with each story approximately 10 feet high. For

taller buildings, this imperfection value of Lc/500 is

conservative since the accumulated geometric imperfection

calculated by 1/500 times building height is greater than

the maximum permitted erection tolerance.

In this study, we shall use Lc/500 for the out-of-

plumbness without any modification because the system

strength is often governed by a weak story which has an

out-of-plumbness equal to Lc/500 (Maleck et al., 1995)

and a constant imperfection has the benefit of simplicity

in practical design. The explicit geometric imperfection

modeling for an unbraced frame is illustrated in Fig. 4.

3. Optimal Design

3.1. Algorithm

Figure 4 shows the procedure of section increment

method. Initial member sizes are assigned to the lightest

sections. If the results of analysis are not satisfied with

the constraint functions of the structure, the member sizes

are increased one by one. First, nonlinear inelastic

analysis is performed in order to check the load-carrying

capacity of the structure subjected to factored loads. The

value a of each member is calculated by the LRFD

interaction equation written as Eq. (20a) and Eq. (20b). A

member with the largest value a is replaced one by one

with an adjacent larger member selected in the database.

The database used in this paper embodies W shape listed

in the AISC-LRFD Specification. This routine is repeated

until load- carrying capacity of the structural system is

greater than the factored loading.

for (20a)

L Pu EI⁄

L Pu EI⁄

αP

φcPn

----------8

9---My

φbMny

--------------8

9---Mz

φbMnz

--------------+ +=P

φcPn

---------- 0.2≥

Figure 3. Explicit imperfection modeling of braced member.Figure 4. Explicit imperfection modeling of unbraced frame.


for (20b)

where φc and φb are resistance factors for compression

and flexure. P, My, and Mz are axial force, end moments

with respect to y and z axes, respectively. Mny and Mnz are

nominal flexure strength. In this paper, the plastic

moment as nominal flexure strength is used because a

compact section is assumed and lateral torsional buckling

is ignored. Pn is nominal axial compressive strength

determined as

for (21a)

for (21b)

for which λc is

(22)

where Fy is yield stress. A and L are the gross cross

sectional area and the unbraced length. r is the radius of

gyration about the plane of buckling. E is Young's

modulus. K is the effective length factor calculated

approximately by (Dumonteil 1992):

For the braced frame

(23)

For the unbraced frame

(24)

in which GA and GB are the column-to-beam stiffness

ratios at the column ends.

Next, nonlinear inelastic analysis is performed in order

to check the deflection and drift limit of the structure

subjected to service loads. If the beam is not satisfied

with the constraint function of deflection, the beam with

the largest value (b) calculated by Eq. (25) is replaced one

by one with an adjacent larger one selected in the

database. If the column is not satisfied with the constraint

function of story drift, the column with the largest value

(g) calculated by Eq. (26) is replaced one by one with an

adjacent larger one selected in the database. This routine

is repeated until the serviceability conditions are satisfied.

(25)

(26)

where Hi and (∆cv)i are the story height and the

interstory drift of the i-th story. Li and (∆bv)i are the length

and the deflection of the i-th beam.

3.2. Objective function

The objective function taken is the weight of a

structure, which is expressed as

(27)

where ρ is the unit weight. NB and NC are the total

number of beams and columns, respectively. (Vb)i and

(Vc)j are the volume of the i-th beam and the j-th column.

3.3. Constraint function

3.3.1. Load-carrying capacity under load combinations

Nonlinear inelastic analysis follows the format of Load

and Resistance Factor Design. In AISC-LRFD (1993),

the factored load effect does not exceed the factored

nominal resistance of structure. Two kinds of factors are

used: one is applied to loads, the other to resistances. The

LRFD has the format

(28)

where Rn = nominal resistance of the structural member,

Qk = force effect, φ = resistance factor, γk = load factor

corresponding to Qk.

The main difference between current LRFD method

and nonlinear inelastic analysis method is that the right

side of Eq. (28), (φRn) in the LRFD method is the

resistance or strength of the component of a structural

system, but in the nonlinear inelastic analysis method, it

represents the resistance or the load-carrying capacity of

the whole structural system. In the nonlinear inelastic

analysis method, the load-carrying capacity is obtained

from applying incremental loads until a structural system

reaches its strength limit state such as yielding or

buckling. The left-hand side of Eq. (28), ( )

represents the member forces in the LRFD method, but

the applied load on the structural system in the nonlinear

inelastic analysis method.

AISC-LRFD specifies the resistance factors of 0.85 and

0.9 for axial and flexural strength of a member, respectively.

The proposed method uses a system-level resistance

which is different from AISC-LRFD specification using

member level resistance factors. When a structural system

collapses by forming plastic mechanism, the resistance

factor of 0.9 is used since the dominant behavior is

flexure. When a structural system collapses by member

buckling, the resistance factor of 0.85 is used since the

dominant behavior is compression. The constraint function

associated with load-carrying capacity is written as

(29)

αP

2φcPn

--------------My

φbMny

--------------Mz

φbMnz

--------------+ +=P

φcPn

---------- 0.2<

Pn 0.658λc

2

FyA= λc 1.5≤

Pn

0.877FyA

λc

2----------------------= λc 1.5>

λc

KL

πr-------

Fy

E-----=

K3GAGB 1.4 GA GB+( ) 0.64+ +

3GAGB 2.0 GA GB+( ) 1.28+ +------------------------------------------------------------------=

K1.6GAGB 4.0 GA GB+( ) 7.5+ +

GA GB 7.5+ +-------------------------------------------------------------------=

β ∆cv( )i

Hi

300---------–=

γ ∆bv( )i

Li

360---------–=

OBJ ρ Vb( )i

i 1=

NB

∑ Vc( )j

j 1=

NC

∑+⎝ ⎠⎜ ⎟⎛ ⎞

=

γk∑ Qk φRn≤

γk∑ Qk

G 1( ) φR γk∑ Qk– 0≥=


where φR is the load-carrying capacity of a structure,

and is the applied factored load.

3.3.2. Lateral drift and deflection under service load

Based on the studies by the ASCE Ad Hoc Committee

(1986) and by Ellingwood (1989), the interstory drift

limit for story height (H) is selected as H/300 for wind

load. The deflection limit for girder with the span length

(L) is selected as L/360 for service load. The constraint

functions of interstory drift and deflection are written as

(30)

(31)

where Eq. (30) and Eq. (31) are constraint functions on

limiting of interstory drift and deflection, respectively. Hi

and (∆cv)i are the story height and the interstory drift of

the i-th story. Li and (∆bv)i are the length and the

deflection of the i-th beam.

3.3.3. Ductility requirement

Adequate rotation capacity is required for members to

develop their full plastic moment capacity. This is achieved

when members are adequately braced and their cross-

sections are compact. The limits for lateral unbraced

lengths and compact sections are explicitly defined in

AISC-LRFD (1993). Based on AISC-LRFD Table B5.1,

the constraint functions of ductility requirement are

written as

(32)

(33)

(34)

where Eq. (32) is a constraint function on limitation of

unbraced length to avoid lateral instability. Eq. (33) and

Eq. (34) are constraint functions on limitation of width-

thickness ratio for flanges and web to avoid local

buckling. ryi and Lbi are radius of gyration about the weak

axis and unbraced length of i-th member, respectively. bf

and tf are the width and thickness of flange, respectively.

tw is the thickness of web. hc is the depth of web clear of

fillets. Fy is the yield stress of steel.

4. Design Examples

4.1. Planar three-story frame

Figure 6 shows the topology and loading of a two-bay

three-story frame with rigid and semi-rigid connections.

This frame was designed by Pezeshk et al. (2000) using

genetic algorithm. Structural elements are classified into

two groups as shown in Fig. 7. The nine column members

and the six beam members are required to have the same

W-section in the design, respectively. The yield stress

used is 250 MPa (36 ksi) and Young’s modulus is 200,000

γk∑ Qk

G 2( )Hi

300--------- ∆cv( )

i– 0≥=

G 3( )Li

360--------- ∆bv( )

i– 0≥=

G 4( )300ryi

Fy

-------------- Lbi– 0≥=

G 5( )65

Fy

---------bf

2tf-----⎝ ⎠⎛ ⎞

i

– 0≥=

G 6( )640

Fy

---------hc

tw----⎝ ⎠⎛ ⎞

i

– 0≥=

Figure 5. Optimization procedure.

Figure 6. Planar three-story frame.


MPa (29,000 ksi). The beam-column connections are top-

and seat-angles of L6×4×9/16×7 with double web angles

of L4×3.5×5/16×8.5. All fasteners are A325 3/4-in.

diameter bolts. The semi-rigid connections are modeled

by the three-parameter power model proposed by Kishi-

Chen (1990). The values of the three parameters, such as

initial connection stiffness, ultimate connection moment

capacity, and shape parameter, are automatically calculated

during the design process. The effective length factor for

each column member is also directly calculated, while the

out-of-plane effective length factor is specified to be one.

The out of plumbness was assumed to be ψ = 1/500. Each

column section was specified to have a maximum depth

of 10 in. like the study of Pezeshk et al. (2000).

The optimized designs for the frames with rigid and

semi-rigid connections obtained by the proposed section

increment method are compared with those generated by

Pezeshk et al. (2000) as shown in Table 1. The load-

displacement curves of optimized designs of the frames

with rigid and semi-rigid connections are shown in Fig. 8.

The frames with rigid and semi-rigid connections

encountered ultimate state when the applied load ratio

reached 1.29 and 1.20, respectively. Since the frames with

rigid and semi-rigid connections collapse by forming

plastic mechanism, the system resistance factor of 0.9 is

used. The ultimate load ratios λ resulted in 1.16

(= 1.29×0.9) and 1.08 (= 1.20×0.9), which were greater

than 1.0 and the member sizes of the system are adequate.

The maximum deflections of the frames with rigid and

semi-rigid connections were calculated as L/395.5 (27.74

mm) and L/471.0 (23.29 mm), respectively. These meet

the deflection limit of L/360 (30.48 mm). The maximum

inter-story drifts of the frames with rigid and semi-rigid

connections were calculated as H/301.8 (10.10 mm) and

H/311.7 (9.78 mm), respectively. These meet the inter-

story drift limit of H/300 (10.16 mm).

The weights of optimized designs for the frames with

rigid and semi-rigid connections are 13.3% lighter than

and equal to that of Pezeshk’s design, respectively. The

numbers of nonlinear analyses required in optimum

design using the proposed section increment method were

25 and 29 times for the frames with rigid and semi-rigid

connections, respectively. However, the number of

nonlinear analysis required to get optimum value by

Pezeshk et al. was 54,000 since a population size of 60,

a generation size of 30, and 30 runs were used to find

optimum value. Therefore, the proposed optimum design

proves its computational efficiency.

4.2. Space two-story frame

Figure 9 shows the topology and loading of a two-story

frame with rigid and semi-rigid connections. This frame

was designed by Chen (1997) using genetic algorithm.

Figure 7. Design variables (member size group) of planarthree-story frame.

Table 1. Comparison of member sizes of planar three-story frame

Design variablesPezeshk et al. (2000) Proposed

Rigid frame Rigid frame Semi-rigid frame

Group 1 W10×68 W10×68 W24×68

Group 2 W24×62 W21×50 W24×62

Total weight 86.79 kN (19,512 lb) 75.26 kN (16,920 lb) 86.79 kN (19,512 lb)

Applied load ratio - 1.29 1.20

Number of analysis 54,000 25 29

Figure 8. Load-displacement of planar three-story frame.


All columns are oriented such that their strong principal

axes are parallel with the x-axis. The structural members

are classified into eight groups as shown in Fig. 10. The

yield stress and Young’s modulus are taken to be 250

MPa (36 ksi) and 200,000 MPa (29,000 ksi), respectively.

The beam-column connections are top- and seat-angles of

L6×4×9/16×7 with double web angles of L4×3.5×5/

16×8.5. All fasteners are A325 3/4-in. diameter bolts. The

semi-rigid connections are modeled by the three-

parameter power model proposed by Kishi-Chen (1990).

The values of the three parameters, such as initial connection

stiffness, ultimate connection moment capacity, and shape

parameter, are directly calculated during the design

process. The out of plumbness was assumed to be ψ = 1/

500. Each column section was specified to have a

maximum depth of 14 in. like the study of Chen (1997).

The optimized designs for the frames with rigid and

semi-rigid connections obtained by the proposed section

increment method are compared with that generated by

Chen (1997) as shown in Table 2. The load-displacement

curves of optimized designs of the frames with rigid and

semi-rigid connections are shown in Fig. 10. The frames

with rigid and semi-rigid connections encountered ultimate

state when the applied load ratio reached 1.22 and 1.15,

respectively. Since the frames with rigid and semi-rigid

connections collapse by forming plastic mechanism, the

system resistance factor of 0.9 is used. The ultimate load

ratios λ resulted in 1.10 (= 1.22×0.9) and 1.04 (= 1.15×0.9),

which were greater than 1.0 and the member sizes of the

system are adequate. The maximum deflections of the

frames with rigid and semi-rigid connections were

calculated as L/399.2 (27.49 mm) and L/363.3 (30.21

mm), respectively. These met the deflection limit of L/

360 (30.48 mm). The maximum inter-story drifts of the

frames with rigid and semi-rigid connections were

calculated as H/881.7 (5.19 mm) and H/905.3 (5.05 mm),

respectively. These met the inter-story drift limit of H/300

(15.24 mm).

Figure 9. Space two-story frame.

Table 2. Comparison of member sizes of space two-story frame

Design variablesChen (1997) Proposed

Rigid frame Rigid frame Semi-rigid frame

Group 1 W14×109 W14×68 W14×61

Group 2 W12×58 W14×90 W14×74

Group 3 W14×99 W14×61 W14×61

Group 4 W12×87 W14×61 W14×48

Group 5 W27×102 W21×44 W24×55

Group 6 W21×50 W16×26 W16×26

Group 7 W24×62 W24×55 W24×55

Group 8 W21×44 W16×26 W16×31

Total weight 186.5 kN (41,930 lb) 126.1 kN (28,342 lb) 126.0 kN (28,330 lb)

Applied load ratio - 1.22 1.15

Number of analysis 50,000 91 90

Figure 10. Design variables (member size groups) ofspace two-story frame.


The weights of optimized designs for the frames with

rigid and semi-rigid connections are 30.7 and 30.2%

lighter than that of Chen’s design. The numbers of

nonlinear analyses required by the proposed section

increment method were 91 and 90 times in optimum

designs for the frames with rigid and semi-rigid connections,

respectively. However, the number of nonlinear analysis

required to get optimum value by Chen was 50,000 since

a population size of 50, a generation size of 50, and 20

runs were used to find optimum value. Therefore, the

proposed optimum design proves its computational

efficiency.

5. Conclusions

In this paper, an optimal design of semi-rigid steel frames

using practical nonlinear inelastic analysis is developed.

The summaries and conclusions of this study are as

follows.

(1) The practical nonlinear inelastic analysis is used for

the proposed optimal design method. The analysis can

practically account for all key factors influencing behavior

of a space frame: gradual yielding associated with flexure;

residual stresses; geometric nonlinearity; and geometric

imperfections.

(2) The objective function taken is the weight of the

structure and the constraint functions considered are load-

carrying capacity, lateral drift, deflection, and ductility

requirements. The section increment method is used for

optimal design of steel frames with rigid and semi-rigid

connections.

(3) The weights of optimized designs for the planar

three-story frames with rigid and semi-rigid connections

are 13.3% lighter than and equal to that of Pezeshk’s

design, respectively. The weights of optimized designs

for the space two-story frames with rigid and semi-rigid

connections are 32.3 and 32.4% lighter than that of

Pezeshk’s design, respectively.

(4) The proposed optimum design method for the

frames with rigid and semi-rigid connections can save

2,160 and 1,862 analysis times compared with the

method proposed by Pezeshk et al., respectively. The

proposed optimum design method for the frames with

rigid and semi-rigid connections can save 549 and 556

analysis times compared with the method proposed by

Chen, respectively.

(5) The practical nonlinear inelastic analysis and the

optimal design method are combined to provide much

benefit to practical engineering.

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