State

of

the

Art Review on Nonlinear Inelastic
Analysis for Steel Structures
NRL Steel Lab., Sejong University
i
CONTENTS
1. INTRODUCTION
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1
2. NONLIEAR INELASTIC ANALYSIS
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3
2.1 Plastic

Zone Analysis
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4
2.2 Quasi

Plastic Hinge Analysis
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6
2.3 Elastic

Plastic Hinge Analysis
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7
2.4 Notional

Load Hinge Analysis
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8
2.5 Refined

Plastic Hinge Analysis
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9
3.
NONLINEAR INELASTIC EXPERIMENTS
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11
3.1 Kanchanalai
’
s Two

Bay Frames
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12
3.2 Yarimci
’
s Three

Story Frames
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12
3.3 Avery and Mahendran
’
s Large

scale
testing
of Steel Frame Structures
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13
3.4 Wakabayashi
’
s One

Quarter
Scaled Test of Portal Frames
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13
3.5 Harrison
’
s Space Frame Test
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14
3.5 Kim
’
s 3D Frame Test
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14
4. DESIGN USING NONLIEAR INELASTIC ANALYSIS
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4.1 Design Format
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15
4.2 Modeling Consideration
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16
4.2.1 Sections
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16
4.2.2 Structural members
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4.2.3 Geometric imperfection
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4.2.4 Load
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17
4.3 Design Consideration
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18
4.3.1 Load

carrying capacity
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18
4.3.2 Resistance factor
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19
4.3.3 Serviceability limit
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4.3.4 Ductility requirement
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20
REFERENCES
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21
1.
INTRODUCTION
The steel design methods used in the U.S. are Allowable Stress Design (ASD), Plastic Design
(PD), and Load and Resistance Fac
tor Design (LRFD). In ASD, the stress computation is based on a
first

order elastic analysis, and the geometric nonlinear effects are implicitly accounted for in the
member design equations. In PD, a first

order plastic

hinge analysis is used in the stru
ctural analysis.
Plastic design allows inelastic force redistribution throughout the structural system. Since geometric
nonlinearity and gradual yielding effects are not accounted for in the analysis of plastic design, they
are approximated in member des
ign equations. In LRFD, a first

order elastic analysis with
amplification factors or a direct second

order elastic analysis is used to account for geometric
nonlinearity, and the ultimate strength of beam

column members is implicitly reflected in the desi
gn
interaction equations. All three design methods require separate member capacity checks including
the calculation of the K

factor.
This design approach is marked in Fig.
1
as the indirect analysis and
2
design method.
In the current AISC

LRFD Specifica
tion (AISC, 1994), first

order elastic analysis or second

order elastic analysis is used to analyze a structural system. In using first

order elastic analysis, the
first

order moment is amplified by B
1
and B
2
factors to account for second

order effects.
In the
Specification, the members are isolated from a structural system, and they are then designed by the
member strength curves and interaction equations as given by the Specifications, which implicitly
account for the effects of second

order, inelastici
ty, residual stresses, and geometric imperfections
(Chen and Lui, 1986). The column curve and beam curve were developed by a curve

fit to both
theoretical solutions and experimental data, while the beam

column interaction equations were
determined by a cu
rve

fit to the so

called "exact" plastic

zone solutions generated by Kanchanalai
(1977).
In order to account for the influence of a structural system on the strength of individual
members, the effective length factor is used as illustrated in Fig. 2.
The
effective length method generally provides a good design of framed structures.
However, several difficulties are associated with the use of the effective length method as follows:
(1) The effective length approach cannot accurately account for the intera
ction between the
structural system and its members. This is because the interaction in a large structural system is too
complex to be represented by the simple effective length factor K. As a result, this method cannot
accurately predict the actual requ
ired strengths of its framed members.
(2) The effective length method cannot capture the inelastic redistributions of internal forces in a
structural system, since the first

order elastic analysis with B
1
and B
2
factors accounts only for
second

order effe
cts but not the inelastic redistribution of internal forces. The effective length
method provides a conservative estimation of the ultimate load

carrying capacity of a large structural
system.
(3) The effective length method cannot predict the failure mo
des of a structural system subject to a
given load. This is because the LRFD interaction equation does not provide any information about
3
failure modes of a structural system at the factored loads.
(4) The effective length method is not user

friendly for
a computer

based design.
(5)
The effective length method requires a time

consuming process of separate member capacity
checks involving the calculation of K

factors.
With the development of computer technology, two aspects, the stability of separate members
,
and the stability of the structure as a whole, can be treated rigorously for the determination of the
maximum strength of the structures. This design approach is marked in Fig. 1
as the direct analysis
and design method
(Kim and Chen, 1996a

b)
. The dev
elopment of the direct approach to design is
called “Advanced Analysis” or more specifically, “Second

Order Inelastic Analysis for Frame
Design.” In this direct approach, there is no need to compute the effective length factor, since
separate member capac
ity checks encompassed by the specification equations are not required. With
the current available computing technology, it is feasible to employ nonlinear inelastic analysis
techniques for direct frame design. This method has been considered impractical
for design office
use in the past.
Over the past 20 years, extensive research has been made to develop and validate several
nonlinear inelastic analysis methods.
The purpose of this paper is to review recent efforts to develop
various nonlinear inelasti
c analyses ranging from a simple elastic

plastic to
rigorous
plastic

zone
analysis for frame design.
Emphasis in this review is
design application of nonlinear inelastic
analysis.
This paper
also summarizes
reports of experimental studies to provide inel
astic nonlinear
behavior of framed structures.
The analysis and design principle using nonlinear inelastic analysis
are also addressed.
2. NONLINEAR INELASTIC ANALYSIS
4
Five different types of nonlinear inelastic analysis methods are discussed in t
he following:
(1) Plastic

zone method
(2) Quasi

plastic hinge method
(3) Elastic

plastic hinge method
(4) Notional

load plastic hinge method
(5) Refined

plastic hinge method
These different methods are based on the degree of refinement in representing th
e plastic
yielding effects. The plastic

zone method uses the greatest refinement while the elastic

plastic hinge
method allows a drastic simplification. The quasi

plastic hinge method is somewhere in between
these two methods. The notional

load plastic
hinge method and the refined

plastic hinge method are
an improvement on the elastic

plastic hinge method for
approximating real behavior of structures
.
The load

deformation characteristics of the plastic analysis methods are illustrated in Fig.
3
, while t
he
spread of plasticity is illustrated schematically in Fig.
4
.
2.1
Plastic

Zone Method
In the plastic

zone method, frame members are discretized into finite elements, and the cross

section of each finite element is subdivided into many fibers shown in Fig
.
5
. The deflection at each
division point along a member is obtained by numerical integration. The incremental load

deflection
response at each loading step, which updates the geometry, captures the second

order effects. The
residual stress in each fib
er is assumed constant since the fibers are small enough. The
s
tress
s
tate at
each fiber can be explicitly traced so the gradual spread of yielding can be captured. The plastic

zone
analysis eliminates the need for separate member capacity checks since it
explicitly accounts for
second

order effects, spread of plasticity, and residual stress. As a result, the plastic

zone solution is
known as an "exact solution." The AISC

LRFD beam

column equations were established in part
based upon a curve

fit to the "
exact" strength curves obtained from the plastic

zone analysis by
Kanchanalai (1977).
5
There are two types of plastic

zone analyses. The first involves the use of three

dimensional
finite shell elements in which the elastic constitutive matrix in the usual
incremental stress

strain
relations, is replaced by an elastic

plastic constitutive matrix when yielding is detected. Based on a
deformation theory of plasticity, the effects of combined normal and shear stresses may be accounted
for. This analysis requ
ires modeling of structures using a large number of finite three

dimensional
shell elements and numerical integration for the evaluation of the elastic

plastic stiffness matrix.
The three

dimensional spread

of

plasticity analysis when combined with second

order theory which
deals with frame stability is computational intensive and, therefore, best suited for analyzing small

scale structures, or if the detailed solutions for member local instability and yielding behavior are
required. Since a detailed anal
ysis of local effects in realistic building frames is not common
practice in engineering design, this approach is considered too expensive for practical use.
The second approach for second

order plastic

zone analysis is based on the use of beam

column theo
ry, in which the member is discretized into line segments, and the cross

section of each
segment is subdivided into finite elements. Inelasticity is modeled considering normal stress only.
When the computed stress at the centroid of any fiber reaches the
uniaxial normal strength of the
material, the fiber is considered to have yielded. Also, compatibility is treated by assuming that full
continuity is retained throughout the volume of the structure in the same manner as elastic range
calculations. Altho
ugh quite sharp curvature may exist in the vicinity of inelastic portions of the
structure, “plastic hinges” can never develop. In plastic

zone analysis, the calculation of forces and
deformations in the structure after yielding requires an iterative tria
l

and

error process because of the
nonlinearity of the load

deformation response, and the change in cross

section effective stiffness in
inelastic regions associated with the increase in the applied loads and the change in structural
geometry. Although mo
st plastic

zone analysis methods have been developed for planar analyses
(Clarke et al., 1992; White, 1985; Vogel, 1985; El

Zanaty et al., 1980; Alvarez and Birnstiel, 1967)
three

dimensional plastic

zone techniques are also available (Wang, 1988; Chen and
Atsuta, 1977).
6
A plastic

zone analysis that includes the spread of plasticity, residual stresses, initial
geometric imperfections, and any other significant second

order effects, would eliminate the need for
checking individual member capacities in the
frame. Therefore, this type of method is classified as
nonlinear inelastic inelastic analysis in which the checking of beam

column interaction equations is
not required. In fact, the member interaction equations in modern limit

states specifications were
developed, in part, by curve

fit to results from this type of analysis. In reality, some significant
behaviors such as joint and connection’s performances tend to defy precise numerical and analytical
modeling. In such cases, a simpler method of analysi
s that adequately represents the significant
behavior would be sufficient for engineering application.
Whereas the plastic

zone solution is regarded as an "exact solution," the method may not be
used in daily engineering design, because it is too intensive
in computation. Its applications are
limited to (ECCS, 1984):
(1) The study of detailed structural behavior
(2) Verifying the accuracy of simplified methods
(3) Providing comparison with experimental results
(4) Deriving design methods or generating cha
rts for practical use
(5) Applying for special design problems
2. 2
Quasi

Plastic Hinge Method
The quasi

plastic hinge method developed by Attala (1994) is an intermediate approach
between the plastic

zone and the elastic

plastic hinge methods. It req
uires less computation but its
results are very similar to those of plastic

zone method. For this reason, it is called a quasi

plastic
hinge method.
An element, developed from equilibrium, kinematic, and constitutive relationships, accounts
for gradual pl
astification under combined bending and axial force. Inelastic force

strain model of
7
the cross

section is developed by fitting nonlinear equations to data of the moment

axial force

curvature response. Using the inelastic cross

section model, flexibility
coefficients for the full
member are obtained by successive integrations along its length. An inelastic

element stiffness
matrix is obtained by the use of the incremental flexibility relationships.
Initial yield and full plastification surface are used to
analytically represent gradual yielding
effect of the cross

section. Ketter’s residual stress pattern (1955) is used to determine an initial yield
surface. Ketter’s pattern has peak compressive residual stresses at the flange tips equal to 0.3F
y
with
a
linear transition of stress from the flange tips to the web

joint and constant tensile stress through the
web. A fully plastic surface is generated by calibration to a plastic

zone solution (Sanz

Picon, 1992).
The parameters of the full plastification eq
uation are determined by a curve

fit procedure.
This method predicts strengths with an error less than 5% compared with the plastic

zone
method for a wide range of case studies. The accuracy of this method is thus compatible with the
plastic

zone method a
nd less computational effort is necessary.
However, it is difficult to extend this method to three

dimensional analysis since the
formulation is based on flexibility relationships.
As a result, it does not meet one of the
requirements of
nonlinear inelas
tic
analysis
of the SSRC task force report (1993), which states
The model should be readily extensible to three

dimensional analysis. That is, the framework of
the model should accommodate the formulation of three

dimensional elements.
Moreover, this
model does eliminate the necessity of the refined model through the cross

section but still requires
many elements along the member.
2. 3
Elastic

Plastic Hinge Method
A more simple and efficient approach for representing inelasticity in frames is the elas
tic

plastic hinge method. It assumes that the element remains elastic except at its ends where zero

length plastic hinges form. This method accounts for inelasticity but not the spread of yielding or
8
plasticity at sections nor the residual stress effect
between two plastic hinges.
The elastic

plastic hinge methods may be divided into; first

order and second

order plastic
analyses. For first

order elastic

plastic hinge analysis, the nonlinear geometric effects are neglected,
and not considered in the for
mulation of the equilibrium equations. As a result, the method predicts
the same ultimate load as conventional rigid

plastic analyses.
In second

order elastic

plastic hinge analysis, the deformed structural geometry is considered.
The simple way to acc
ount for the geometric nonlinearity is to use the stability function which enables
only one beam

column element per a member to capture the second

order effect. This provides an
efficient and economical method of frame analysis, and has a clear advantage
over the plastic

zone
method. This is particularly true for structures in which the axial force in component members is
small and the dominated behavior is bending. In such cases, second

order elastic

plastic hinge
analysis may be used to describe the in
elastic behavior sufficiently, assuming that lateral

torsional and
local buckling modes of failure are not prevented (Liew, 1992).
The second

order elastic

plastic hinge analysis is only an approximate method. When used
to analyze a single beam

column ele
ment subject to combined axial load and bending moment, it may
overestimate the strength and stiffness of the element in the inelastic range. Although elastic

plastic
hinge approaches provide essentially the same load

displacement predictions as plastic

z
one methods
for many frame problems, they may not be classified as
nonlinear inelastic
analysis methods in
general (Liew et al., 1994; Liew and Chen, 1991; White, 1993).
However, research by Ziemian (Ziemian et al., 1990; Ziemian, 1990) has shown that the
elastic

plastic hinge analysis can be classified as an advanced inelastic analysis since it is accurate for
matching the strength and load

displacement response of several building frames from plastic

zone
analysis. Many cases considered in Ziemian
s work, especially when the axial load is less than
0.5P
y
, are not sensitive benchmarks for determining the accuracy and the possible limitations of the
elastic

plastic hinge method. Therefore, suitable benchmark problems should be used to provide a
9
more
in

depth study of the qualities and limitations of second

order elastic

plastic hinge method
before it can be accepted as a legitimate tool in the design of steel structures.
For slender members whose dominant mode of failure is elastic instability, the m
ethod provides good
results when compared with plastic

zone solutions. However, for stocky members with significant
yielding, the plastic

hinge method over

predicts the actual strength and stiffness of members due to
the gradual stiffness reduction as the
spread of plasticity increases in an actual member (Liew and
Chen, 1991; Liew et al., 1991; White et al., 1991). As a result, considerable refinements must be
made before it can be used for analysis of a wide range of framed structures.
2. 4
Notional

Lo
ad Plastic

Hinge Method
One approach to advance the use of second

order elastic

plastic hinge analysis for frame
design is to specify artificially large values of frame imperfections (i.e., initial out

of

plumbness
)
.
This is the approach adopted by EC3 (1
990) for frame design using second

order analysis. In
addition to accounting for the standard erection tolerance for out

of

plumbness, these artificial large
imperfections intend to account for the effect of residual stresses, frame imperfections, and dis
tributed
plasticity not considered in frame analysis. The geometric imperfections adopted by EC3 are a
maximum out

of

plumbness of
Ψ
0
= 1/200 for an unbraced frame, but no maximum out

of

straightness value recommended for a braced member as shown in Fig.
6
.
The notional load plastic hinge approach is similar in concept to the “enlarged” geometric
imperfection approach of the EC3. T
he ECCS (1984, 1991), the Canadian Standard (
1989,
1994),
and the Australian Standard (1990) allow to use this technique. The notional

load approach uses
equivalent lateral loads to approximate the effect of member imperfections and distributed plasticity
.
In the ECCS, the exaggerated notional loads of 0.5 % times gravity loads are used to avoid over

predicting the strength of the member as does the elastic

plastic hinge method. The application of
these notional loads to several example frames is illustr
ated in Fig.
7
. Liew'
s research (1992) shows
10
that this method under

predicts the strength by more than 20% in the various leaning column frames
and over

predicts the strength up to 10% in the isolated beam

columns subject to the axial forces and
bending
moments. As a result, modification of this approach is required before it may be used in
design applications.
2. 5
Refined Plastic

Hinge Method
In recent work by Abdel

Ghaffar et al. (1991), Al

Mashary and Chen (1991), King, et al.
(1991), Liew and Chen
(1991), Liew et al. (1993a

b), White et al. (1991), Kim (1996), Kim and Chen
(1996),
Chen and Kim (1997),
Kim and Chen (199
7
)
, Kim et al (2000) and
among others, an inelastic
analysis approach, based on simple refinements of the elastic

plastic hinge model
, has been proposed
for plane frame analysis. It represents the effect of distributed plasticity through the cross

section,
assuming that the plastic hinge stiffness degradation is smooth. The inelastic behavior of the
member is modeled in terms of membe
r force instead of the detailed level of stresses and strains as
used in the plastic

zone analysis model. The principal merits of the refined

plastic hinge model are
that it is as simple and efficient as the elastic

plastic hinge analysis approach, and it
is sufficiently
accurate for the assessment of strength and stability of a structural system and its component members.
The refined plastic

hinge method is based on simple modifications of the elastic

plastic hinge
method. Two modifications are made to a
ccount for the gradual section stiffness degradation at the
plastic hinge locations as well as gradual member stiffness degradation between the two plastic hinges.
Herein, the section stiffness degradation function is adopted to reflect the gradual yieldi
ng effect in
forming plastic hinges. Then, the tangent modulus concept is used to capture the residual stress
effect along the member between two plastic hinges. As a result, the refined plastic

hinge method
retains the efficiency and simplicity of the p
lastic hinge method without overestimating the strength
and stiffness of a member.
In the recent work by Liew (1992), the LRFD tangent modulus is used to account for both the
11
effect of residual stresses and geometric imperfections. This model does not acc
ount for geometric
imperfections when P/P
y
is less than 0.39, because the LRFD tangent modulus is identical to the
elastic modulus in this range. As a result, the approach over

predicts the column strength by more
than 5% when KL/r of the column is greate
r than 85 for yield stresses at 36 ksi, and when KL/r of the
column is greater than 70 for yield stresses at 50 ksi. The LFRD E
t
may not be an appropriate model
to be used for nonlinear inelastic analysis (Kim, 1996; Kim and Chen, 1996).
The CRC tangent
modulus in Liew's work (1992) only accounts for the effect of residual
stresses. It over

predicts the strength of members by about 20% compared to the conventional
LRFD solutions, because the modulus does not account for the effect of geometric imperfecti
ons.
However, in the CRC tangent modulus model, different members with different residual stresses can
be incorporated since the effect of geometric imperfections is considered separately. As a result,
CRC tangent modulus is used in refined plastic analy
ses.
Second

order inelastic analysis methods for the three

dimensional structure have been
developed by Orbison (1982), Prakash and Powell (1993), Liew and Tang (1998)
, Kim et al (2001),
Kim and Choi (2001) and Kim et al (2001)
. Orbison's method is an ela
stic

plastic hinge analysis
without considering shear deformations. The material nonlinearity is considered by the tangent
modulus
t
E
and the geometric nonlinearity is by a geometric stiffness matrix. Orbison's method,
however, und
erestimates the yielding strength up to 7% in stocky members subjected to axial force
only. DRAIN

3DX developed by Prakash and Powell is a modified version of plastic hinge methods.
The material nonlinearity is considered by the stress

strain relationship
of the fibers in a section. The
geometric nonlinearity caused by axial force is considered by the use of the geometric stiffness matrix,
but the nonlinearity caused by the interaction between the axial force and the bending moment is not
considered. This
method overestimates the strength and stiffness of the member subjected to
significant axial force. Liew and Tang's method is a refined plastic
hinge analysis. The effect of
residual stresses is taken into account in conventional beam

column finite element
modelling.
12
Nonlinear material behavior is taken into account by calibration of inelastic parameters describing the
yield and bounding surfaces. Liew and Tang's method, however, underestimates the yielding strength
up to 7% in stocky member subjected to ax
ial force only.
Against this background, it can be concluded that the refined

plastic hinge method strikes a
balance between the requirements for realistic representation of frame behavior and for ease of use.
It is considered that in both theses respects
, the method is satisfactory for general practical use.
3. NONLINEAR INELASTIC EXPERIMENTS
Experimental studies to capture inelastic nonlinear behavior of framed structures are
summarized. The frames riviewed herein were tested by Kanchanalai(1977), Y
arimci(1966),
Avery(1999), Wakabayashi(1972), Harrison(1964) and Kim and Kang(2001).
3.1 Kanchanalai
’
s Two

Bay Frames
Three two

bay full

size frames were tested to verify the Plastic

zone analysis(Kanchanalai,
1977). The dimensions and members of Frame 2
among these frames are shown in Fig. 8. The
material properties of the members are summerized in Table 1. The frames were designed to behave
equivalently to a one

story two

bay and could be tested on the floor. Supports were provided only at
the top and
bottom of the interior column member. All frames were bent with respect to the week
axis in order to avoid out

of

plane buckling. In Frame
2,
all columns were loaded simultaneously up
to about 70kips, corresponding to points 2

11 in Fig. 9. Then, only
the axial load on the interior
column was increased up to point 17, where the frame reached its instability limit load of 233.6 kips.
Comparisons of the test results with the plastic zone theory are shown in Fig. 9. In
general
, good
agreements are observe
d.
13
3.2 Yarimci
’
s Three

Story Frames
An experimental research study was conducted at Lehigh University for three full

size frames
(Yarimci, 1966). Fig. 10 shows dimensions and loads conditions of Frame C among the three frames.
To investigate and compare
the mechanical properties of the members with nominal values, Yarimci
conducted a series of seven beam tests. The results of these tests are summarized in Table 2. The
beams were welded to the columns and designed so as to behave elastically in the wors
t loading
condition: the flexibility of the connections was eliminated from a factor which affects the strength of
the frames. The frames were sandwiched and supported laterally by two parallel auxiliary frames
preventing out

of

plane buckling. All membe
rs were bent in strong axis. The result of test is
shown in Fig. 11 for Frame C. The load deflection behavior at the first and third story is shown in
Fig. 11.
3.3 Avery and Mahendran
’
s Large

Scale Testing
of
Steel Frame Structures
A series of four te
sts was conducted by Avery and Mahendran(1999). Each of the four
frames could be classified as a two

dimensional, single

bay, single

story, large

scale sway frame with
full lateral restraint and rigid joints, as shown in Fig. 12.
In Frame 2, Non

compact
I

sections(310UB32.0) of Grade 300 steel(nominal yield
stress=320MPa)was used.
This section was
selected as one of the standard hot

rolled I

sections mostly affected by local buckling. The
dimensions, material properties, and section properties used in F
rames 2 are listed in Table 3. The
vertical and horizontal loads were applied simultaneously in a ratio of approximately four times
greater than the horizontal reaction measured by the load cell. The frame failed by in

plane instability
due to a reduced s
tiffness caused by yielding and P

Δ
effect. The horizontal reaction force and the
measured relative in

plane horizontal displacement of the right hand column for test Frame 2 are
related in Fig. 13.
14
3.4 Wakabayashi
’
s One

Quarter Scaled Test of Portal Fra
mes
Two

series of test were conducted for a one

story frame and a two

story frame by
Wakabayashi et al(1972). Configurations of the two

story frame are shown in Fig. 14. The
nominal dimensions of members are H

100
100
6
8 for columns and H

100
50
4
6 fo
r beams.
The specimens consist of rolled H

shapes. The connections were welded and stiffened to prevent
local buckling in the joint panels. To prevent the out

of

plane buckling, two of the same specimens
were set in parallel and connected at the joints
and the mid

length of the members. In the other
words, twin specimens were tested simultaneously. Measured Material and sectional properties of
members are listed in Table 4.
The vertical load was first applied at the top of four columns by a fixed tes
ting machine.
The
parallel
twin specimens were loaded simultaneously. Then, the horizontal load at the top of
frame was increased gradually. When the frame swayed by the horizontal loading jack followed a
horizontal movement so that vertical loading poi
nts could be kept on the center of the columns. The
loads
were measured by the load cells which were installed between the hydraulic jacks and the
specimen.
The load

deflection curves of the two

story frames are shown in Fig. 15. Comparisons of a
series
of test show the effects of axial force and stiffness of the beam on the frame behavior. The
larger the axial force in columns and the smaller the stiffness of the beam, the more unstable the
frames become.
3.5 Harrison
’
s Space Frame Test
The equilater
al triangular space frame depicted in Fig. 3 was tested by Harrison(1964)
in the
J.W.
Roderick Laboratory for Materials and Structures at the University of Sydney. Configuration of the
frame is shown in Fig. 16. Measured dimensions and
material properties
are listed in Table 5. A
15
horizontal load(H) is applied on the top of the column and a vertical load of 1.3H is applied at mid
span of the beam.
It can be seen from Fig. 17 that, compared to the experimental results, the plastic

zone
analysis predicted a
slightly stiffer response of the space frame under the applied loads. As the
column bases of the space frame were welded to steel plates clamped to steel joists(Harrison 1964),
the more flexible response measured in the laboratory test might have been ca
used by the flexibility of
the joist flanges.
3.6 Kim
’
s 3D Frame Test
Two

series of test were conducted for space steel frame subjected proportional loads shown
in Fig 18 and space steel frame subjected proportional loads shown in Fig.
19 by Kim and Ka
ng(2001).
Hot

rolled I

section was used for all three frames. Nominal dimension of
the section was H

150
×
150
×
7
×
10 commonly used in Korea. The dimensions and properties of the
section
are listed in Table 6. The section is compact so that it is not suscept
ible to local buckling.
For proportional loads test, The vertical loads were applied on the top of the four columns,
and the horizontal
loads were applied on the column
②
and
④
at the second floor level of the test
frame. The vertical loads were slowly increased until the system could not resist any more loads.
The horizontal loads were automatically increased according to the specified l
oad ratio for each test
frame controlled by the computer system.
For non

proportional loads test, The vertical loads were applied on the top of the four
columns, and the horizontal load was applied on the column
②
at the second floor level of the test
fr
ame. The vertical loads were first increased 680
kN
and maintained during the experiment. The
horizontal load was slowly increased until the test frame could not resist any more loads.
Fig. 20. and Fig. 21. show load

displacement cu
rve for test frames. The obtained results
from 3D non

linear analysis and AISC

LRFD method were compared with experimental data.
ABAQUS, one of mostly widely used and accepted commercial finite element analysis
program
, was
16
used.
Load carrying capacitie
s obtained by the experiment and AISC

LRFD method are compared
in Table 7 and 8. The results showed that the AISC

LRFD capacities were approximately 25 percent
conservative for frame subjected to
proportional
loads test and 28 percent conservative for non

proportional
loads test. This difference is
derived
from the fact that the AISC

LRFD approach does
not consider the inelastic moment redistribution, but the experiment includes the inelastic
redistribution effect.
4. DESIGN USING NONLINEAR INELASTIC AN
ALYSIS
4
.1 Design Format
Nonlinear inelastic analysis follows the format of Load and Resistance Factor Design. In
A
ISC

LRFD
(1994)
, 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 load
and resistance factor design has the format
i i n
Q R
(1)
where
n
R
= nominal resistanc
e of the structural member,
i
Q
= force effect,
= resistance
factor,
i
= load factor corresponding to
i
Q
,
= a factor relati
ng to ductility, redundancy, and
operational importance.
The main difference between current LRFD method and nonlinear inelastic analysis method is that the
right side of Eq. (
1
)
,
(
n
R
) in the LRFD method is the resistance or stre
ngth of the component of a
structural system, but in the nonlinear inelastic analysis method, it represents the resistance or the
17
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.
(1
), (
i i
Q
)
represents the member forces in the LRFD method, but the app
lied load on the structural system in the
nonlinear inelastic analysis method.
4
.
2
Modeling
C
onsideration
4
.
2
.1 Sections
The A
ISC

LRFD Specification uses only one column curve for rolled and welded sections of
W, WT, and HP shapes, pipe, and structur
al tubing
(AISC, 1994)
. The Specification also uses same
interaction equations for doubly and singly symmetric members including W, WT, and HP shapes,
pipe and structural tubing
, even though the interaction equations were developed on the basis of W
shape
s by Kanchanalai (1977).
The proposed analysis was developed by calibration with the LRFD column curve. To this
end, it is concluded that the proposed methods can be used for various rolled and welded sections
including W, WT, and HP shapes, pipe, and s
tructural tubing without further modifications.
4.2
.2 Structural members
An important consideration in making this nonlinear inelastic analysis practical is the
required number of elements for a member in order to predict realistically the behavior of
frames. A
sensitivity study of nonlinear inelastic analysis
for two

dimensional frames was
performed on the
required number of element
s
(Kim and Chen, 1998)
. Two

element model adequately predict
ed
the
18
strength of a
two

dimensional
member.
This rule may be
used for modeling a three

dimensional
member.
4.2.3 Geometric imperfection
The magnitudes of geometric imperfections are selected as
2 1,000
for unbraced
frames and
1 1,000
for braced frames.
To model a parabolic
out

of

straightness in the member
,
two

element model with maximum initial deflection at the mid

height of a member adequately
captures imperfection effects. It
is
concluded that practical nonlinear inelastic analysis is
computationally efficient. The pat
tern of geometric
imperfection
s is assumed to be the same as the
elastic first order deflected shape.
4.2
.
4
Load
1) Proportional loading
In the proposed nonlinear inelastic analysis, the gravity and lateral loads should be applied
simultaneously, since
it does not account for unloading. As a result, the method under

predicts the
strength of frames subjected to sequential loads, large gravity loads first and then lateral loads. It is,
however, justified for the practical design since the development of
the LRFD interaction equations
was also based on strength curves subjected to simultaneous loading and the current LRFD elastic
analysis uses the proportional loading rather than the sequential loading.
2) Incremental loading
It is necessary, in an non
linear inelastic analysis, to input each increment load (not the total
loads) to trace nonlinear load

displacement behavior. The incremental loading process can be
achieved by scaling down the combined factored loads by a number between 20 and 50. For a
19
h
ighly redundant structure, dividing by about 20 is recommended and for a nearly statically
determinate structure, the incremental load may be factored down by 50. One may choose a number
between 20 and 50 to reflect the redundancy of a particular structure
.
Since a highly redundant
structure has the potential to form many plastic hinges and the applied load (i.e. the smaller scaling
number) may be used.
4.3
Design Consideration
4
.3.1 Load

carrying capacity
T
he elastic analysis method does not capture t
he inelastic redistribution of internal forces
throughout a structural system,
since
the
first

order forces, even with the
1
B
and
2
B
factors,
account for the second

order geometric effect but not the inelas
tic redistributions of internal forces.
T
he method may provide a conservative estimation of the ultimate load

carrying capacity. Nonlinear
inelastic analysis, however, directly considers force redistribution due to material yielding and thus
allows smaller
member sizes to be selected. This is particularly beneficial in highly indeterminate
steel frames. Because consideration at force redistribution may not always be desirable, the two
approaches (including and excluding inelastic force redistribution) can
be used. First, the load

carrying capacity, including the effect of inelastic force redistribution, is obtained from the final
loading step (limit state) given by the computer program. Secondly, the load

carrying capacity
without the inelastic force red
istribution is obtained by extracting that force sustained when the first
member yield or buckled. Generally, nonlinear inelastic analysis predicts the same member size as the
LRFD method when force redistribution is not considered.
4.3.2 Resistance fac
tor
20
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 fa
ctors. When a structural system collapses
by forming plastic mechanism, the resistance factor of 0.9 is used since the dominent behavior is
flexure. When a structural system collapses by member buckling, the resistance factor of 0.85 is used
since the domi
nent behavior is compression.
4
.3.
3
Serviceability limit
According to the ASCE Ad Hoc Committee on Serviceability report (Ad Hoc Committee,
1986), the normally accepted range of overall drift limits for building is
1 750
to
1 250
times the
building height,
H
, with a typical value of
400
H
. The general limits on the interstory drift are
1 500
to
1 200
times the story heig
ht. Based on the studies by the Ad Hoc Committee (1986),
and
by Ellingwood (1989), the deflection limits for girder and story are selected as
Floor girder live load deflection :
360
H
Roof girder deflection :
240
H
Lateral drift :
400
H
for wind load
Interstory drift :
300
H
for wind load
At service load levels, no plastic hinges are allowed to occur in order to avoid permanent
deformations under service loads
.
4.3.4 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.
21
The limits for lateral unbraced le
ngths and compact sections are explicitly defined in AISC

LRFD
(1994).
REFERENCES
Abdel

Ghaffar, M., White, D. W., and Chen, W. F. (1991). “Simplified second

order inelastic
analysis for steel frame design
.
” Special Volume of Session on Approxi
mate Methods and
Verification Procedures of Structural Analysis and Design
,
Proceedings at Structures Congress 91,
ASCE, New York, 47

62.
Ad Hoc Committee on Serviceability, Structural serviceability (1986). A critical appraisal and
research needs, ASCE, J
. Struct. Eng., 112(12); 2646

2664.
Al

Mashary, F. and Chen, W. F. (1991). “Simplified second

order inelastic analysis for steel frames.”
J. Inst. Struct. Eng., 69(23), 395

399.
AISC (1994). Load and Resistance Factor Design Specification, American Institu
te of Steel
Construction, 2nd Ed., Chicago.
Alvarez, R. J. and Birnstiel, C. (1967). “Elasto

plastic analysis of plane rigid frames, school of
engineering and science.” Department of Civil Engineering, New York University, New York.
22
Attala, M. N., Deierlei
n, G. G., and McGuire, W. (1994). “Spread of plasticity: quasi

plastic

hinge
approach.” J. Struct. Engrg., ASCE, 120(8), 2451

2473.
Avery, P. and Mahendran, M. (2000). “Large

scale testing of steel frame structures comprising non

compact sections.” Engrg.
Struct., 22, 920

936.
Chen, W. F. and Atsuta, T. (1977). “Theory of beam

columns, vol. 2, space behavior and design.”
McGraw

Hill, New York, 732 pp.
Chen W.F. and Kim, S. E.(1997).
“
LRFD steel design using advanced analysis.
”
, CRC Press, Boca
Raton, Florid
a.
Chen, W.F. and Lui, E. M.(1986).
“
Structural stability

theory and implementation.
”
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19.
TABLE 1.
Summary of Tension Coupon Tests
Section
Member
number
Specimen
y
ksi
*
y
10

5
st
10

5
E
st
ksi
ult
ksi
Elongation
in 8 in,
W8
17
(A36

70A)
C1A
C1C
Flange
37.9
128
1140
442
62.4
28.2
Flange
37.7
127
1378
356

29.7
Web
40.6
137
2450
345
61.7
32.9
M4
13
(A572

73)
B1,B2
B3,B4
Flange
48.5
164
1203
406
69.6
26.6
Flange
48.6
164
1062
399
69.9
27.2
Web
50.1
169
2228
323
69.5
26.7
C1B and C2B were no
t tested
*
y
=
y
/E(E=29,500ksi)
TABLE 2. Measured Properties of Beam and Column Section
Frame
Section
Handbook
EI
(kip

in
2
10
4
)
Measured
EI
(kip

in
2
10
4
)
Handbook
M
p
(kip

in)
Measured
M
P
(kip

in)
C
12B16.5
310
271
742
845
C
10B15
203
190
576
635
C
6WF15
158
165
686
760
TABLE 3. Dimensions and Properties of Members
Test
Section
D
b
r
t
r
t
w
r
1
A
g
I
S
y
27
frame
(mm)
(mm)
(mm)
(mm)
(mm)
(mm
2
)
(10
6
mm
4
)
(10
4
mm
3
)
Flange Web
2
310UB32
298
149
8.0
5.5
13.0
4080
63.2
475
360 395
TABLE 4. Actual Se
ction Properties of One

Quarter
Scaled Frames
A
(cm
2
)
I
(cm
4
)
Z
(cm
3
)
Z
p
(cm
3
)
y
(t/cm
2
)
Column
21.8
391
77.4
88.5
2.64
Beam
10.6
177
35.0
40.6
3.04
TABLE 5. Dimensions and Material Properties of Equilateral Triangular Space Frame
L
(in)
D
(in)
T
(in)
E
(ksi)
G
(ksi)
y
(ksi)
Column Beam
All members
48
1.682
0.176
28800
11520
30.6 31.1
TABLE 6. Dimensions and
Properties
of Section H

150
150
7
10 Used in the Frame
H
eight
mm
H
Width
mm
B
Thickness
of Flange
mm
t
f
Thickness
of Web
mm
t
w
Radius of
Fillet
mm
r
1
Axial
Area
2
mm
A
g
Moment of
Inertia about X
Axis
4
6
10
mm
I
X
Moment of
Inertia about Y
Axis
4
6
10
mm
I
Y
Nominal
150
150
10
7
11
4014
16.40
5.63
Measured
Column
152.3
149.9
10.2
6.75

4053
17.20
5.74
Beam
149.1
150.0
9.2
6.50

3713
15.14
5.18
TABLE 7. Comparison of Experime
ntal and Design Load Carrying Capacity
(a) Experiment
(b) Analysis
(c) AISC

LRFD design
(b)/(a)
(c)
/(a)
P
612.0
612.0
443.5
1.0000
0.7247
H
169.2
175.5
122.6
1.0372
0.7246
TABLE
8
. Comparison of Experimental and Design Load Carrying Capacities
28
(a) Experiment
(b) Analysis
(c) AISC

LRFD design
(b)/(a)
(c)/(a)
Test frame
3
P
681.8
680.9
510.2
0.9985
0.7483
H
1
136.4
136.2
1 0 2.0
0.9 9 8 4
0.7 4 8 1
H
2
6 7.5
6 8.1
5 1.0
1.0 0 8 3
0.7 5 5 6
F I G. 1. A
n a l y s i s a n d De s i g n Me t h o d
29
FIG. 2. Interaction
between
A Structural System and Its Component Members
FIG. 3. Load

Deformation Characteristics of Plastic Analysis Methods
30
FIG. 4. Concept of Spread of Plasticity for Various Advanced A
nalysis Methods
FIG. 5. Model of Plastic

Zone Analysis
31
FIG. 6. Explicit Imperfection Model for Elastic

Plastic Analysis Recommended By ECCS
FIG. 7. Examples on Application of Notional Loads for Second

Order Elastic

Plasic Hinge
Analysis
32
F
IG. 8. Two

Bay Frame
FIG. 9. Axial Load

Deflection Behavior of Specimen
33
FIG. 10. Specimen for Three

Story Frame
FIG. 11. Lateral Load

Sway Behaviour of Frame C
34
FIG. 12. Schematic Diagram of Test Arrangement
FIG. 13. Sway Load

Deflection Curve for Test Frame 2
35
FIG. 14. One

Quarter Scaled Frames.(From Wakabayashi, M. And
Matsui, C., Trans. Arch. Inst. Jpn. 193,17,1972, With Permission
)
36
FIG. 15. Horizontal Force

Displacement Behaviours o
f One

Quarter Scaled
Frame.(Two Story).(From Wakabayashi, M. And Matsui, C., Trans.
Arch.Inst. Jpn. 193,17,1972, With Permission)
37
FIG. 16. Harrison
’
s Space Frame(Harrison 1964)
FIG. 17. Load

Deflection for Harrison
’
s Space Fra
me
38
Horizontal load
①
④
H2
②
H1
③
Base
2nd floor
Vertical load
P
P
P
Roof
P
2.20m
1.76m
2.5m
3.0m
X
Z
Y
FIG. 18. Dimension and Loading Condition of Test Frame
X
Horizontal load
Z
2.20m
3.0m
①
2.5m
④
②
Y
③
Base
2nd floor
1.76m
Vertical load
P
P
P
Roof
P
H
FIG. 19. Dimensions and Loading Conditions of Test Frame in Main Test
39
0
10
20
30
40
H
o
r
i
z
o
n
t
a
l
d
i
s
p
l
a
c
e
m
e
n
t
(
m
m
)
0
40
80
120
160
200
H
o
r
i
z
o
n
t
a
l
l
o
a
d
(
k
N
)
Experiment(H1)
Analysis(H1)
Experiment(H2)
Analysis(H2)
F
IG
.
20
.
Comparison of
Horizontal
L
oad

Displacement
C
urves for
Space T
est
F
rame
2
F
IG
.
21
.
Horizontal Load

Displacement Curve for Test Frame (Column
②
)
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