3. · · · · · 17
4.2.4 Load· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
· · · · · · · · 17
4.3 Design Consideration· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
· · · · · · ·18
4.3.1 Load-carrying capacity· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
· · · · · ·18
4.3.2 Resistance factor· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
· · · · · ·19
4.3.3 Serviceability limit· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
· · · · · · 19
4.3.4 Ductility requirement· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
· · · · · · 20
REFERENCES · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
· · · · · · · 21
1. INTRODUCTION
The steel design methods used in the U.S. are Allowable Stress Design (ASD), Plastic Design
(PD), and Load and Resistance Factor 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 structural 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 design 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 design
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
ii
4. design method.
In the current AISC-LRFD Specification (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 B1 and B2 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, inelasticity, 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 curve-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 interaction 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 required 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 B1 and B2 factors accounts only for
second-order effects 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 modes of a structural system subject to a
given load. This is because the LRFD interaction equation does not provide any information about
2
5. 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 development 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 capacity 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 inelastic 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 inelastic nonlinear
behavior of framed structures. The analysis and design principle using nonlinear inelastic analysis
are also addressed.
2. NONLINEAR INELASTIC ANALYSIS
3
6. Five different types of nonlinear inelastic analysis methods are discussed in the 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 the 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 the
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 fiber is assumed constant since the fibers are small enough. The stress state 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).
4
7. 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 requires 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 analysis 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 theory, 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. Although 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 trial-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 most 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).
5
8. 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 analysis 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 charts 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 requires 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 plastification under combined bending and axial force. Inelastic force-strain model of
6
9. 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 0.3Fy 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 equation 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 and 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 Αnonlinear inelastic analysis≅ of the SSRC task force report (1993), which states
ΑThe 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 elastic-
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
7
10. 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 formulation 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 account 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 inelastic 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 element 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-zone 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
0.5Py, 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
8
11. 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 method 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-Load 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 (1990) 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 distributed
plasticity not considered in frame analysis. The geometric imperfections adopted by EC3 are a
maximum out-of-plumbness of Ψ0 = 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. The 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 illustrated in Fig. 7. Liew' s research (1992) shows
9
12. 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 (1997), 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 member 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 account 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 yielding 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 plastic 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
10
13. effect of residual stresses and geometric imperfections. This model does not account for geometric
imperfections when P/Py 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 greater 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 Et 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 imperfections.
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 analyses.
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 elastic-plastic hinge analysis
without considering shear deformations. The material nonlinearity is considered by the tangent
modulus Et and the geometric nonlinearity is by a geometric stiffness matrix. Orbison's method,
however, underestimates 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.
11
14. 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 axial 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), Yarimci(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 observed.
12
15. 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 worst 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 members 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 tests 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 Frames 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 stiffness 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.
13
16. 3.4 Wakabayashi’s One-Quarter Scaled Test of Portal Frames
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 for 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 testing 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 points 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 equilateral 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
14
17. 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 caused 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 Kang(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 susceptible 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 load 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
frame. 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 curve 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
15
18. used. Load carrying capacities 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 ANALYSIS
4.1 Design Format
Nonlinear inelastic analysis follows the format of Load and Resistance Factor Design. In
AISC-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
η ∑ γ iQi ≤ φ Rn (1)
where Rn = nominal resistance of the structural member, Qi = force effect, φ = resistance
factor, γ i = load factor corresponding to Qi , η = a factor relating 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), ( φ Rn ) 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
16
19. 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), ( η ∑γ Q )
i i
represents the member forces in the LRFD method, but the applied load on the structural system in the
nonlinear inelastic analysis method.
4.2 Modeling Consideration
4.2.1 Sections
The AISC-LRFD Specification uses only one column curve for rolled and welded sections of
W, WT, and HP shapes, pipe, and structural 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
shapes 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 structural 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 elements (Kim and Chen, 1998). Two-element model adequately predicted the
17
20. 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 ψ = 2 1,000 for unbraced
frames and ψ = 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 pattern of geometric imperfections 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 nonlinear 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
18
21. highly 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
The elastic analysis method does not capture the inelastic redistribution of internal forces
throughout a structural system, since the first-order forces, even with the B1 and B2 factors,
account for the second-order geometric effect but not the inelastic redistributions of internal forces.
The 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 redistribution 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 factor
19
22. 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 dominent behavior is
flexure. When a structural system collapses by member buckling, the resistance factor of 0.85 is used
since the dominent 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 H 400 . The general limits on the interstory drift are
1 500 to 1 200 times the story height. 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 : H 360
• Roof girder deflection : H 240
• Lateral drift : H 400 for wind load
• Interstory drift : H 300 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.
20
23. The limits for lateral unbraced lengths and compact sections are explicitly defined in AISC-LRFD
(1994).
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TABLE 1. Summary of Tension Coupon Tests
Member σy ,* y ,st Est σult Elongation
Section Specimen
number ksi ×10-5 ×10-5 ksi ksi in 8 in, %
W8×17 Flange 37.9 128 1140 442 62.4 28.2
C1A
(A36- Flange 37.7 127 1378 356 - 29.7
C1C
70A) Web 40.6 137 2450 345 61.7 32.9
M4×13 Flange 48.5 164 1203 406 69.6 26.6
B1,B2
(A572- Flange 48.6 164 1062 399 69.9 27.2
B3,B4
73) Web 50.1 169 2228 323 69.5 26.7
C1B and C2B were not tested
,*y= Φy/E(E=29,500ksi)
TABLE 2. Measured Properties of Beam and Column Section
Handbook Measured Handbook Measured
Frame Section EI EI Mp MP
2 4
(kip-in ×10 ) (kip-in2×104) (kip-in) (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 br tr tw r1 Ag I S σy
26
29. frame (mm) (mm) (mm) (mm) (mm) (mm2) (106mm4) (104mm3) Flange Web
2 310UB32 298 149 8.0 5.5 13.0 4080 63.2 475 360 395
TABLE 4. Actual Section Properties of One-Quarter Scaled Frames
A I Z Zp σy
(cm2) (cm4) (cm3) (cm3) (t/cm2)
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 D T E G σy(ksi)
(in) (in) (in) (ksi) (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
Moment of Moment of
Thickness Thickness Radius of Axial Area
Height Width Inertia about X Inertia about Y
of Flange of Web Fillet
Axis Axis
H B tf Ag
tw r1
(mm )
IX IY
(mm) (mm) (mm) (mm) (mm) (10 mm ) (10 )
2
6 4 6
mm 4
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 Experimental 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
27
31. FIG. 2. Interaction between A Structural System and Its Component Members
FIG. 3. Load-Deformation Characteristics of Plastic Analysis Methods
29
32. FIG. 4. Concept of Spread of Plasticity for Various Advanced Analysis Methods
FIG. 5. Model of Plastic-Zone Analysis
30
33. 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
31
35. FIG. 10. Specimen for Three-Story Frame
FIG. 11. Lateral Load-Sway Behaviour of Frame C
33
36. FIG. 12. Schematic Diagram of Test Arrangement
FIG. 13. Sway Load-Deflection Curve for Test Frame 2
34
37. FIG. 14. One-Quarter Scaled Frames.(From Wakabayashi, M. And
Matsui, C., Trans. Arch. Inst. Jpn. 193,17,1972, With Permission)
35
38. FIG. 15. Horizontal Force-Displacement Behaviours of One-Quarter Scaled
Frame.(Two Story).(From Wakabayashi, M. And Matsui, C., Trans.
Arch.Inst. Jpn. 193,17,1972, With Permission)
36
39. FIG. 16. Harrison’s Space Frame(Harrison 1964)
FIG. 17. Load-Deflection for Harrison’s Space Frame
37
40. P
P
ad
Vertical lo
P
P
Roof
2.20m
H1
tal load
Horizon 2nd floor
Z
H2
①
1.76m
Y ②
③
X 2.5 Base
m ④
3.0m
FIG. 18. Dimension and Loading Condition of Test Frame
P
P
ad
Vertical lo
P
P
Roof
2.20m
H
ta l load
Horizon 2nd floor
Z
①
1.76m
Y ②
③
X 2.5 Base
m ④
3.0m
FIG. 19. Dimensions and Loading Conditions of Test Frame in Main Test
38
41. 200
160
Horizontal load (kN)
120
80
Experiment(H1)
Analysis(H1)
Experiment(H2)
40 Analysis(H2)
0
0 10 20 30 40
Horizontal displacement (mm)
FIG. 20. Comparison of Horizontal Load-Displacement Curves for Space Test Frame 2
FIG. 21. Horizontal Load-Displacement Curve for Test Frame (Column ②)
39
