State of the Art Non Linear Analysis

State-of-the-Art Review on Nonlinear Inelastic Analysis for Steel Structures

NRL Steel Lab., Sejong University

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CONTENTS

1. INTRODUCTION · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

· · · · · · · ·1

2. NONLIEAR INELASTIC ANALYSIS · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

· · · · · 3 2.1 Plastic-Zone Analysis· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

· · · · · · · ·4 2.2 Quasi-Plastic Hinge Analysis· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

· · · · · · 6 2.3 Elastic-Plastic Hinge Analysis· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

· · · · · · ·7 2.4 Notional-Load Hinge Analysis· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

· · · · · · 8 2.5 Refined-Plastic Hinge Analysis· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

· · · · · ·9

3. NONLINEAR INELASTIC EXPERIMENTS· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

· · · ·11 3.1 Kanchanalai’s Two-Bay Frames· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

· · · · ·12 3.2 Yarimci’s Three-Story Frames· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

· · · · · 12 3.3 Avery and Mahendran’s Large-scale testing of Steel Frame Structures· · · · · · · · · · · · · ·

13 3.4 Wakabayashi’s One-Quarter Scaled Test of Portal Frames· · · · · · · · · · · · · · · · · · · · · · ·

· 13 3.5 Harrison’s Space Frame Test· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

· · · · ·14 3.5 Kim’s 3D Frame Test· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 14

4. DESIGN USING NONLIEAR INELASTIC ANALYSIS· · · · · · · · · · · · · · · · · · · · · · · ·

· ·15 4.1 Design Format· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

· · · · · · · 15 4.2 Modeling Consideration· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

· · · · · · 16 4.2.1 Sections· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

· · · · · · · ·16 4.2.2 Structural members· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

· · · · · 17 4.2.3 Geometric imperfection· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

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· · · · · 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

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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

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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

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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).

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).

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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

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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

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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

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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

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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

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 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.

tE

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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.

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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.

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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

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 and maintained during the experiment. The

horizontal load was slowly increased until the test frame could not resist any more loads.

kN

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

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

i i nQ Rη γ φ≤∑ (1)

where nR = nominal resistance of the structural member, = force effect, iQ φ = resistance

factor, iγ = load factor corresponding to , iQ 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), ( nRφ ) 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

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 iQη γ∑ )

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

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 1,000ψ = for unbraced

frames and 1 1,000ψ = 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

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 and 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.

1B 2B

4.3.2 Resistance factor

19

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 400H . 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 : 360H

• Roof girder deflection : 240H

• Lateral drift : 400H for wind load

• Interstory drift : 300H 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

21

The limits for lateral unbraced lengths and compact sections are explicitly defined in AISC-LRFD

(1994).

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Attala, M. N., Deierlein, G. G., and McGuire, W. (1994). “Spread of plasticity: quasi-plastic-hinge

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Clarke, M. J., Bridge, R. Q., Hancock, G. J., and Trahair, N. S. (1992). benchmarking and verification

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23

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24

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25

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26

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TABLE 1. Summary of Tension Coupon Tests

Section Member number Specimen σy

ksi ,*y

×10-5,st

×10-5Est ksi

σult ksi

Elongation in 8 in, %

Flange 37.9 128 1140 442 62.4 28.2 Flange 37.7 127 1378 356 - 29.7

W8×17 (A36-70A)

C1A C1C Web 40.6 137 2450 345 61.7 32.9

Flange 48.5 164 1203 406 69.6 26.6 Flange 48.6 164 1062 399 69.9 27.2

M4×13 (A572-

73)

B1,B2 B3,B4 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

Frame Section Handbook

EI (kip-in2×104)

Measured EI

(kip-in2×104)

Handbook Mp

(kip-in)

Measured MP

(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

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 (cm2)

I (cm4)

Z (cm3)

Zp (cm3)

σy (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 (in)

D (in)

T (in)

E (ksi)

G (ksi)

σy(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

Height

( )mmH

Width

( )mmB

Thickness of Flange

( )mmt f

Thickness of Web

( )mmtw

Radius of Fillet

( )mmr1

Axial Area

( )2mm

Ag

Moment of Inertia about X

Axis

( )4610 mmI X

Moment of Inertia about Y

Axis

( )4610 mmI 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 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

(a) Experiment (b) Analysis (c) AISC-LRFD design (b)/(a) (c)/(a) P 681.8 680.9 510.2 0.9985 0.7483 H1 136.4 136.2 102.0 0.9984 0.7481 Test frame

3 H2 67.5 68.1 51.0 1.0083 0.7556

FIG. 1. Analysis and Design Method

28

FIG. 2. Interaction between A Structural System and Its Component Members

FIG. 3. Load-Deformation Characteristics of Plastic Analysis Methods

29

FIG. 4. Concept of Spread of Plasticity for Various Advanced Analysis Methods

FIG. 5. Model of Plastic-Zone Analysis

30

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

FIG. 8. Two-Bay Frame

FIG. 9. Axial Load-Deflection Behavior of Specimen

32

FIG. 10. Specimen for Three-Story Frame

FIG. 11. Lateral Load-Sway Behaviour of Frame C

33

FIG. 12. Schematic Diagram of Test Arrangement

FIG. 13. Sway Load-Deflection Curve for Test Frame 2

34

FIG. 14. One-Quarter Scaled Frames.(From Wakabayashi, M. And

Matsui, C., Trans. Arch. Inst. Jpn. 193,17,1972, With Permission)

35

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

FIG. 16. Harrison’s Space Frame(Harrison 1964)

FIG. 17. Load-Deflection for Harrison’s Space Frame

37

Horizontal load

①

④

H2

②

H1

③Base

2nd floor

Vertical load

P

P

P

Roof

P

2.20

m1.

76m

2.5m

3.0m

X

Z

Y

FIG. 18. Dimension and Loading Condition of Test Frame

X

Horizontal load

Z

2.20

m

3.0m

①

2.5m ④

②Y

③Base

2nd floor

1.76

m

Vertical load

P

P

P

Roof

P

H

FIG. 19. Dimensions and Loading Conditions of Test Frame in Main Test

38

0 10 20 30 4Ho

0rizontal displacement (mm)

0

40

80

120

160

200

Hor

izon

tal l

oad

(kN

)

Experiment(H1)Analysis(H1)Experiment(H2)Analysis(H2)

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