RESEARCH ARTICLE

Wai Fah CHEN

Advanced analysis for structural steel building design

E Higher Education Press and Springer-Verlag 2008

Abstract The 2005 AISC LRFD Specifications for

Structural Steel Buildings are making it possible for

designers to recognize explicitly the structural resistance

provided within the elastic and inelastic ranges of beha-

vior and up to the maximum load limit state. There is an

increasing awareness of the need for practical second-

order analysis approaches for a direct determination of

overall structural system response. This paper attempts

to present a simple, concise and reasonably comprehens-

ive introduction to some of the theoretical and practical

approaches which have been used in the traditional and

modern processes of design of steel building structures.

Keywords advanced analysis, structural steel building

design, introduction

1 Introduction

The purpose of structural design is to produce a physical

structure capable of withstanding the environmental con-

ditions to which it may be subjected. Many factors affect

the design process, from loading to foundation to dimen-

sion lay out to risk and cost, but basically the ultimate

design is a reflection of the properties of the structural

material and the geometrical imperfection of its structural

members, and in particular its mechanical properties and

the residual stresses induced in the structural members

during manufacturing and fabrication, which define the

characteristic response of the material and the member to

the environmental forces.

At present, in engineering design practice, there is a

fundamental two-stage process in the design operation:

firstly, the forces acting on each structural member in

the structure must be calculated; secondly, the load car-

rying capacity of each of these structural members to

those forces acting on it must be determined. The first

stage involves an analysis of the distribution of forces

and moments acting on each of these structural members;

the second stage involves knowledge of the load carrying

capacity of these members to resist these forces and

moments acting on them. The more comprehensive this

knowledge, the more exact will be the design and the more

reliable will be the structure.

Since the load carrying capacity of structural membersdepends on the type of loads acting on the member, geo-

metrical imperfections, properties of material and residual

stresses, the knowledge of load-carrying capacity of these

structural members has been determined mostly on the

basis of full scale tests in the form of pin-ended column

strength curves for axially loaded members, simply sup-

ported beam strength curves for bending dominatedmem-

bers, and beam-column interaction curves for membersunder combined axial force and bending moment. These

member strength curves are formally coded as the member

strength curves or equations for design practice.

Having divided structural members in a framed struc-

ture into three classes, namely columns, beams, and beam-

columns, and determined their respective strengths by full

scale tests with ideal end or boundary conditions, the next

stage must be to drastically simplify the material behavior

under stress in such a way as to readily assist the engineerin analyzing the stress distribution in the structure in order

to size up the structural members in a framed structure. At

present, the practicing engineer bases design primarily on

the simple model of linear elasticity of the material for

those early designs based on the allowable stress method.

In this process, time-dependent effects of material are

assumed insignificant. This is indeed a drastic simplifica-

tion of material properties over long term behavior. Sowith this time independent simplification, the design pro-

cess is now focused and concentrated on reducing the

stress level or the structure is loaded only in the working

load level. The large safety factor is therefore used to

adjust the design to take into account inelastic features

in the material to avoid failure. Most structural analyses

in engineering practice have been based on linear elastic

analysis. First-order linear elastic analysis has been thehallmark of structural engineering in early years, while

the second-order linear elastic analysis for a structural

Received December 13, 2007; accepted March 10, 2008

Wai Fah CHEN (*)Department of Civil Engineering, University of Hawaii at Manoa,Honolulu, HI 96822, USAE-mail: chenwf@eng.hawaii.edu

Front. Archit. Civ. Eng. China 2008, 2(3): 189–196DOI 10.1007/s11709-008-0024-8

system has been developed and is increasingly being uti-

lized in recent years.

2 First-order elastic structural analysis withK-factors

The boundary conditions of a framed member in a frame

structure are quite different from that of an isolated mem-

ber that is used as the basis for the development of column

strength curves (pin-pin end conditions) or beam strength

curves (simply supported end conditions). In order to size

up the framed members, the framed member boundary

conditions must be adjusted to the equivalent pin-pin

end conditions for the case of column design, for example,

so that the column strength curves can be properly used in

determining the required size of the framed member under

consideration (see Fig. 1).

To achieve this equivalency, the effective length fac-

tor or the K-factor has been widely used in the past to

relate the pin-ended column strength curves to the

framed member design in a structural system. The effec-

tive length method generally provides a good method

for the design of framed structures. This method has

been widely used for the development of modern steel

design codes including the allowable stress design and

the plastic design in early years; and the load and res-

istance factor design in more recent years [1]. However,

despite its popularity, the approach has several major

limitations and shortcomings.

The first of these is that it does not give an accurate

indication of the factor against failure, because it does not

consider the interaction of strength and stability between

the member and the structural system in a direct manner.

It is a well recognized fact that the actual failure mode of

the structural system often does not have any resemblance

whatsoever to the elastic buckling mode of the structural

system that is the basis for the determination of the effec-

tive length factor K.

The second, and perhaps the most serious limitation, is

probably the rationale of the current two-stage process in

design: elastic analysis is used for the determination of

distribution of forces acting on each member of a struc-

tural system, whereas the member’s ultimate strength

curves are developed for design either on the basis of full

scale tests or by inelastic analysis with each member

treated as an isolated component [2,3]. There is no veri-

fication of the compatibility between the isolated member

and the member as part of a frame. The individual mem-

ber strength equations as specified in specifications are

not concerned with system compatibility. As a result,

there is no explicit guarantee that all members will sustain

their design loads under the geometric configuration

imposed by the framework.

The other limitations of the effective length method

include the difficulty of computing a K factor, which is

not user-friendly for computer-based design, and the

inability of the method to predict the actual strength of

a framed member, among others. To this end, there is an

increasing tendency of the need for practical analysis/

design methods that can account for the compatibility

between the member and system. With the rapid develop-

ment of computing power and the availability of desk-top

computers and user-friendly software, the development of

an alternative method to a direct design of structural sys-

tem without the use of K-factors becomes more attractive

and realistic.

3 Second-order elastic structural analysiswith K-factors

The adoption of elastic structural analysis with K-fac-

tors for steel design may be divided into two stages of

progress. The simplest first stage of progress with K-

factors in the design process is the first-order elastic

analysis with amplification factors to include the sec-

ond-order effects as generally provided by the specifi-

cations [4]. This is described in the preceding section.

Logically, the next stage of progress is a direct second-

order elastic analysis without the use of amplification

factors for second-order effects [4]. Both methods are

based on the formation of first plastic hinge defined as

the failure of the system (see Fig. 2).

As mentioned previously, the effective length factor will

generally yield good designs for framed structures, but it

does have the following drawbacks:

Fig. 1 Interaction between structural system and its com-ponent members

190 Wai Fah CHEN

1) It cannot capture accurately the interaction between

the structural members and the structural system behavior

and strength;

2) It cannot reflect the proper inelastic redistributions

of internal forces in a structural system;

3) It cannot predict the failure modes of a structural

system;

4) It is not easy to implement in an integrated computer

design application with the use of alignment charts in the

K-factor calculation process;

5) It is generally a time-consuming process requiring

separate member capacity checks with different K-factors

for different framed members.

Even for the most recent AISC-LRFD procedures, sim-

ilar difficulties are encountered when performing a seismic

design because the same strength interaction equation

checks must be performed [4]. Some of the difficulties

are even more so on the seismic designs since additional

questions are raised such as:

1) How is the structure going to behave during an earth-

quake?

2) Which part of the structure is the most critical area?

3) What will happen if part of the structure yields or

fails?

4) What might happen if forces greater than those spe-

cified in the code occur?

None of these questions can be answered by the con-

ventional LRFD method with K- factors. In contrast, the

advanced analysis to be described in the following can

provide this information (see Fig. 3).

4 Various advanced analysis methods

Advanced analysis refers to any method that captures the

strength and stability of a structural system and its indi-

vidual members in such a way that separate member capa-city checks are not required. Thus, there is no need for K-

factor calculation. Usually these analyses are also referred

more formally as the second-order inelastic analysis for

frame design (see Fig. 2). With the present computing

power, it is a rather straight forward process to combine

the theory of stability [5] with the theory of plasticity [6]

for structural system analysis [7]. The real challenge is

therefore to make this type of new approach to designwork and competitive with the current methods in engin-

eering practice [8].

4.1 Elastic-plastic hinge method

The elastic-plastic hinge method is the simplest approxi-

mation of inelastic behavior of material by assuming all itsinelastic effect is concentrated at the plastic hinge loca-

tions. In this idealization, it assumes that the element

remains elastic except at the ends where zero-length plas-

tic hinges can form. This method accounts for inelasticity

but not the spread of yielding or plasticity at sections, or

the influence of residual stresses.

Here, as in previous sections for elastic analysis,

depending on the geometry used to form the equilibriumequations, the elastic-plastic hinge method may be divided

into first-order and second-order plastic analysis. For the

Fig. 2 General analysis types for steel building structure design

Advanced analysis for structural steel building design 191

first-order elastic-plastic hinge analysis, the undeformed

geometry is used, and nonlinear geometric effects are

neglected. As a result, the predicted ultimate load is the

same as conventional rigid-plastic analysis. In the second-

order elastic plastic analysis, the deformed shape is con-

sidered and geometric nonlinearities can be included using

stability functions which enable use of only one beam-

column element per member to capture the second-order

effects. A comprehensive presentation of the plastic design

and second-order analysis method can be found in the

book by Chen and Sohal [9].

The second-order elastic-plastic hinge analysis is only

an approximate method [10,11]. For slender members

whose dominant failure mode is elastic instability, the

method provides a good approximation; but for stocky

members as well as for beam-column elements subjected

to combined axial load and bending moment, this method

overestimates the actual strength and stiffness in the inel-

astic range due to spread of yielding effects.

This method is therefore a good first approximation of

the second-order inelastic analysis for frame design within

the applicable range described above. It requires further

refinement before it can be recommended for analysis of a

wide range of framed structures [12].

4.2 Refined plastic-hinge method

The refined plastic-hinge methods are based on some sim-

ple modifications of the elastic-plastic hinge method

described above. The notional-load concept is first intro-

duced to the conventional elastic-plastic hinge method by

applying additional fictitious equivalent lateral loads to

account for the influence of residual stresses, member

imperfections, and distributed plasticity that are not

included in the conventional procedures. With certain

modifications, this refined approach is accepted in the

European Convention for Construction Steelwork [13],

the Canadian Standard, and the Australian Standard.

However, Liew’s research [10] shows that this method

under-predicts the strength in the various leaning column

frames by more than 20% and over-predicts the strength

by up to 10% in the isolated beam-columns subjected to

axial forces and bending moments.

In the following, we shall make further modifications

and simplifications of the elastic-plastic hinge method to

improve its performance and at the same time to make it

practical and work in engineering practice. These modifi-

cations are grouped into three categories: geometry,

material, and connection. Details of these modifications

can be found in the two doctoral theses [10,14,15] and

their subsequent papers (for example, Refs. [16,17]).

This refined plastic-hinge method will now be called the

practical advanced method for frame design in what fol-

lows.

4.3 Plastic-zone method

The plastic-zone method is considered to be ‘‘exact’’ since

it is based on the most refined finite element analysis for a

structural system. This plastic-zone analysis has been well

developed and documented by a research team at Cornell

University over the last two decades under the leadership

of Professor McGuire [18]. The team members include

Ziemian [19], White [11], Attala and Deierlein [20], among

others. As a comparison, the elastic-plastic hinge model is

considered to be the simplest; while the elastic-plastic-

zone model exhibits the greatest refinement.

In the plastic-zone method, each member is discretized

into many sections along the length and each segment is

subdivided into many finite elements. The material

Fig. 3 Analysis and design methods: conventional vs advanced analysis/design

192 Wai Fah CHEN

properties of each finite element are specified for specific

properties and residual stresses. The behavior of the mem-

ber is obtained by numerical integration of these finite ele-

ments. The second-order geometric effects are captured by

updating the geometry at each incremental load step. The

exact solutions were verified with some benchmark beam-

column tests. This method is ideal for verification of various

simplified methods developed for engineering practice.

5 Practical advanced method

The practical advanced method was developed at Purdue

University over the last 15 years beginning with the work

of Liew [10], verified with the well-documented cal-

ibration frames [7,21], and completed with the adoption

of the advanced analysis method in the 2005 AISC Load

and Resistance Factor Design Specification for Structural

Steel Buildings. The details of this development were sum-

marized in the 1997 book ‘‘LRFD Steel Design Using

Advanced Analysis’’ by Chen and Kim [22]. Herein, as

in our previous experiences we learned from engineering

practice, the practical advanced analysis method must be

reduced to a sequence of equivalent linear elastic analysis

as was the case for the widely popular allowable stress

design in early years as well as the elastic-plastic hinge

design in later years in order to gain acceptance in engin-

eering practice for general use.

In the following, I shall briefly summarize the highlights

of this development in terms of idealizations and simpli-

fications of geometry, material and connection to achieve

this goal while meeting the current LRFD requirements.

In this process of development, it initially sets out to use

the familiar but simplified stability functions to capture

the second-order effects, going on to show how the spread

of plasticity due to residual stresses can be accounted for

with a reduction of material tangent modulus, and finally

outlining the influence of geometric imperfections that

may be simulated by a further reduction of tangent modu-

lus in the ultimate processes of design of structural system

with an equivalency of simple elastic analysis.

5.1 Second-order effects

To capture the second-order effects, the simplified

stability functions reported by Chen and Lui are adopted

[23].

5.2 Cross-section plastic strength

The LRFD cross-section plastic strength curves are

adopted for both strong and weak-axis bending (see

Fig. 4) [5]. The same reduction factors used in the

LRFD specification are selected as 0.85 for axial strength

and 0.9 for flexural strength.

5.3 Residual stresses

The CRC tangent modulus is employed to account for the

gradual yielding effect due to residual stresses along the

length of members under axial loads. In this approach, the

elastic modulus E instead of the moment of inertia I is

reduced to account for the reduction of the elastic portion

of the cross-section, because the reduction of elastic

modulus is easier to implement than that of the moment

of inertia for different sections. The reduction rate in stiff-

ness for both strong and weak axis is taken to be the same

as reflected by the CRC tangent modulus curve Et (see

Fig. 5).

5.4 Geometric imperfections

The degradation of member stiffness due to geometric

imperfections may be simulated by a further reduction

of member stiffness. This may be achieved by a further

Fig. 4 Smooth stiffness degradation for work-hardeningplastic hinge based on LRFD sectional strength curve

Fig. 5 CRC and reduced tangent modulus, Et, for memberswith residual stresses and geometric imperfections

Advanced analysis for structural steel building design 193

reduction factor of the tangent stiffness E0t~0:85Et. This

further reduction modulus is applicable for both braced

and unbraced members and frames (see Fig. 5).

5.5 Semi-rigid connections

The actual behavior of a semi-rigid connection in a struc-

tural steel building falls between the two extremes of

pinned-joint and rigid-joint. These two extreme idealiza-

tions have been widely used in the past in conventional

structural analysis and design. In recent years, attention

has been focused toward a more accurate modeling of

such connections (see Fig. 6). To this end, the 1994

AISC-LRFD Specifications designated two types of con-

struction in their provisions: Type FR (fully restrained)

and Type PR (partially restrained). The Type FR is the

traditional rigid connection; while the PR Type is known

as the semi-rigid connections in the conventional termino-

logy of structural steel analysis and design. If Type PR

construction is used, the effect of connection flexibility

must be taken into account in the analysis and design

procedures (see Fig. 7).

Currently, the most commonly used approach to

describe the moment-rotation curve of a semi-rigid con-

nection is to curve fit the experimental data with simple

expressions based on large data base [24–26]. Several

curve fitted semi-rigid connection models have been pro-

posed but the most notable is the three-parameter power

model proposed originally by Richard [27], modified by

Kishi and Chen [28] and Kishi et al. [29], and refined by

Wu [30].

The influence of partially restrained connections on

structural analysis is not only in changing the moment

distribution among beams and columns, but also in

increase frame drift and accordingly the increase of the

P-delta effect in the frame analysis. Extensive work has

been made in recent years on connection modeling [31],practical procedures for implementation [32], and guide-

lines on proper selection of model parameters for PR con-

struction [33]. Details of these developments can be found

in the comprehensive books by Chen [34], and by Chen,

Goto and Liew [32], among others.

A one-day workshop on frames with partially

restrained connections was organized and conducted by

the Structural Stability Research Council [35] in Atlanta,Georgia on September 23, 1998. The subsequent work-

shop proceedings covered various aspects of the state-

of-the-art progress on the practical analysis for partially

restrained frame design. This included codes, database,

modeling, classifications, analysis/design, and design

tables and aids. A comprehensive design guide with design

examples was also provided in a companion SSRC book-

let entitled ‘‘Practical Analysis for Partially RestrainedFrame Design’’ by Chen and Kim [33].

6 Practical advanced analysis for seismicdesign

Based on the response of structures in the Northridge

earthquake in 1994, significant changes were made to

the 1997 LRFD seismic provisions [36], especially the

beam-to-column connection design in moment frames

[37]. These changes are conceptually trying to achieve

both ‘‘structural fuse concept’’ and ‘‘performance-based

design’’ (see Fig. 8). Structural fuse concept was designedfor controlled failure or excessive deformations at pre-

selected locations. This type of design will help in the

repair of major structures with least cost and quick recov-

ery for business [38]. The performance-design is for differ-

ent levels of performance of structural components. To

achieve the required performance with adequate safety

and economy, advanced analytical procedures are neces-

sary for such an evaluation process.

Advanced analysis can directly address the overall sys-

tem response including second-order effects, gradual

Fig. 6 Rotational deformation of semi-rigid connection

Fig. 7 AISC-LRFD classification of flexibility of beam-to-column connections [4]

194 Wai Fah CHEN

yielding, inelastic force redistribution and semi-rigid con-

nections. Thus, the advanced analysis can address the

seismic design very well by providing answers to the crit-

ical questions. It is the most efficient tool in evaluating the

code besides practical design. A comprehensive report on

the development of practical advanced analysis method

for seismic steel frame design along with an evaluation

of the major impact of the 1997 LRFD steel seismic code

revision in the USA was made in the doctoral thesis by

Chen [39] along with its subsequent publication [40].

A comparative study of the LRFD methods with the

practical advanced analysis for steel building design was

reported in a recent thesis by Hwa [41]. It was found that

similar results were obtained since the advanced analysis

method was calibrated against the LRFD code. This pro-

vides structural engineers with more confidence in adopt-

ing the advanced analysis. As a demonstration of its

capability for a much wider application of the advanced

analysis method, both performance-based fire resistance

design and seismic design were also studied in some depth

by Hwa [41].

7 Summary

This paper describes the state-of-the-art progress of the

second-order inelastic analysis method for steel

building design now known as the ‘‘advanced analysis’’.

Extensive developments have been made in recent years

to make the advanced analysis work and practical in

engineering practice. To this end, a design method based

on the requirements put forward by the AISC-LRFD

specifications was developed and refined to achieve both

simplicity in use and, as much as possible, a realistic

representation of actual behavior. This practical

advanced analysis method for steel building design has

reached such a level of maturity that is now ready for

general use in engineering design.

Research works on advanced analysis are now in full

swing to develop nonlinear procedures and software for

practical use in design office. The theory and approaches

for advanced analysis of plane frames composed of mem-

bers of compact sections, fully braced out-of-plane, have

been well developed, verified and coded by the American

Institute of Steel Construction in the 2005 AISC

Specifications as well as others around the world.

8 Concluding remarks

Unlike the older second-order analysis used in seismic

and static design which is used for supplemental check-

ing only, the concept of advanced analysis can now be

used for primary design and member size framing.

Recent collapses of several shallow roof domes in

Russia, Germany and Poland clearly show the deficiency

of the current practice. The advanced analysis can be

easily applied to the design of these special or other

conventional structures.

For advanced analysis to achieve its full potential as a

tool for practical design for steel building structures,

future work is needed to integrate the effects of local buck-

ling of cross sections, lateral-torsional buckling of mem-

bers, flexibility of connections, and 3-D member behavior

into the advanced analysis. Furthermore, other structural

elements such as concrete floor slab, composite joints and

walls could also be included in the analysis for a more

realistic representation of the actual behavior of a steel

building structure during its life cycle performance and

service. Good progress has been made on some aspects

of this advancement, but much more remains to be done

(see for example, Ref. [42]).

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