Refined Plastic Hinge Analysis of Steel Frame Structures ... · analysis and design of steel frame structures permits direct and accurate determina-tion of ultimate system strengths,

Advances in Structural Engineering Vol. 3 No. 4 2000 291

Refined Plastic Hinge Analysis ofSteel Frame Structures

Comprising Non-CompactSections

I: FormulationP. Avery and M. Mahendran

Physical Infrastructure Centre, School of Civil Engineering, Queensland University of Technology, Brisbane QLD 4000, Australia

ABSTRACT: Application of “advanced analysis” methods suitable for non-linearanalysis and design of steel frame structures permits direct and accurate determina-tion of ultimate system strengths, without resort to simplified elastic methods ofanalysis and semi-empirical specification equations. However, the application ofadvanced analysis methods has previously been restricted to steel frames compris-ing only compact sections that are not influenced by the effects of local buckling. Aconcentrated plasticity formulation suitable for practical advanced analysis of steelframe structures comprising non-compact sections is presented in this paper. Thisformulation, referred to as the refined plastic hinge method, implicitly accounts forthe effects of gradual cross-sectional yielding, longitudinal spread of plasticity, ini-tial geometric imperfections, residual stresses, and local buckling.

Keywords: Refined plastic hinge analysis, steel frame structures, advanced analysis, local buckling, Formulation

1. INTRODUCTIONA distributed plasticity model suitable for advancedanalysis of steel frame structures comprising non-com-pact sections was presented by Avery and Mahendran(2000). Although this model was shown to accuratelypredict the structural response, it is not practical forgeneral design use due to the computational resourcesrequired. The objective of the research described in thispaper was to develop a simpler method of analysis thatadequately captures the non-linear behaviour of steelframe structures comprising non-compact sections. Thissimplified method of analysis must therefore be able toadequately represent the effects of local buckling inaddition to the other significant factors such as materialyielding, second-order instability, residual stresses, andgeometric imperfections.

All factors relevant to compact sections not subjectto local buckling have been investigated by a numberof other researchers who have developed concentratedplasticity advanced analysis formulations for steel framestructures comprising only compact sections. Five ofthe most significant such formulations are: the refinedplastic hinge method (Liew, 1992; Liew et al., 1993),the notional load plastic hinge method (Liew et al.,1994), the hardening plastic hinge method (King andChen, 1994), the quasi plastic hinge approach (Attallaet al., 1994), and the springs in series method (Yau andChan, 1994). A summary and evaluation of each methodis presented by Avery (1998).

Comparison of the various techniques lead to theconclusion that the refined plastic hinge method bestlends itself to the modifications required to account for

Refined Plastic Hinge Analysis of Steel Frame Strctures Comprising Non-Compact Sections. I: Formulation

292 Advances in Structural Engineering Vol. 3 No. 4 2000

the effects of local buckling. These effects can be con-sidered as three distinct phenomena:

1. Reduction in the axial compression force and bend-ing moment section capacities due to stressescaused by local buckling.

2. Additional gradual reduction in cross-sectionalstiffness due to local buckling deformations andassociated yielding.

3. Softening of plastic hinges due to the progressionof local buckling.

The formulation of a frame element force-displace-ment relationship suitable for the advanced analysis ofsteel frames comprising non-compact sections and sub-ject to proportional loading is presented in this paper.The formulation is based on one of the concentratedplasticity methods, the refined plastic hinge methodoriginally developed by Liew (1992) for compact sec-tions. It implicitly accounts for the reduction in sectioncapacity, gradual stiffness reduction, and hinge soften-ing caused by local buckling by the application of sim-ple equations. Although the formulation is for non-com-pact sections, it is referred to as “refined plastic hingeanalysis” to indicate its relationship to the originalmethod. The concentrated plasticity model was veri-fied by comparison with the analytical benchmarks pro-vided by Avery and Mahendran (1998). These compari-sons are presented and discussed in a companion paper.

2. FORMULATION OF THEFRAME ELEMENT FORCE-DISPLACEMENT RELATIONSHIPStructural analysis requires the determination of un-known forces and displacements. A force-displacementrelationship can be developed using the equations ofequilibrium and laws of compatibility. The force-dis-placement relationship for a beam-column element (Fig-ure 1) with no plastic hinges using a local co-rotationalcoordinate system is given by:

(1)

With appropriate stiffness coefficients, this relationshipcan be applied to calculate either incremental or totalforces and displacements. For non-linear analysis, thestiffness matrix is a function of the total forces and/ordisplacements. The forces must therefore be applied

incrementally, with the appropriate tangent stiffnessmatrix calculated for each increment. The objective ofthis section is to establish procedures suitable for thedetermination of the tangent stiffness matrix of a beam-column element to be used for the advanced analysis oflaterally restrained two dimensional steel frame struc-tures comprising non-compact sections with rigid mem-ber connections and fixed or pinned supports. For classi-fication as advanced analysis, the tangent stiffness ma-trix must account for all factors that may significantlyinfluence the behaviour of a structure, including sec-ond-order (instability) effects, material properties (elas-tic stiffness, yield stress and post-yield behaviour), re-sidual stresses, geometric imperfections (member out-of-straightness and out-of-plumbness, and local imper-fections), and local buckling.

The effects of each of these factors on the tangentstiffness of a beam-column element are considered inthe following sections. The tangent stiffness formula-tion is based on the refined plastic hinge method (Liew,1992) and includes modifications to account for the ef-fects of local buckling. The use of the stability func-tions to account for second-order instability effects isdescribed. The techniques used to determine the sec-tion capacity and account for the gradual stiffness re-duction due to spread of plasticity (including residualstresses, initial geometric imperfections, and local buck-ling) are presented. The effects of hinge softening arealso considered.

2.1 Second-order EffectsThe formulation of second-order analysis is based onthe deformed configuration of the structure, and incor-porates both member chord rotation (P- ) and membercurvature (P- ) second-order effects. Accurate deter-mination of second-order effects is an essential featureof any advanced analysis formulation attempting to pre-dict instability failure. Chen and Lui (1987) demon-strated that by application of the slope-deflection equa-tions, the second-order elastic incremental force-dis-placement relationship for an elastic beam-column ele-ment in the local co-rotational co-ordinate system withno plastic hinges could be expressed as:

(2)

P. Avery and M. Mahendran


Figure 1. Beam-column element, showing local degrees of freedom.

The refined plastic hinge method is implemented usingthe simplified expressions for the stability functions (s

1,

s2) first proposed by Lui and Chen (1986):

(3)where:

(4)

Note that an axial tension force is taken as positive inEquation (4).

The stability functions are used to model second-order instability effects in the refined plastic hinge meth-od’s incremental force-displacement relationship. Asthese functions are obtained from the second-order dif-ferential equation describing the behaviour of an elas-tic beam-column, they do not accurately represent theinelastic second-order effects. Inelastic second-ordereffects (including those associated with local buckling)are approximately accounted for in the refined plastichinge method by the tangent modulus and flexural stiff-ness reduction factor, which are based on column mem-ber capacity curves and therefore include second-ordereffects.

2.2 Plastic Hinge FormulationConcentrated plasticity methods of analysis (such asthe refined plastic hinge method) account for materialyielding by the insertion of zero-length plastic hingesat the element ends. If the state of forces at any cross-section equals or exceeds its section capacity, a plastic

hinge is formed and slope continuity at that location isdestroyed. The force-displacement relationship of theelement containing the plastic hinge must therefore bemodified to reflect the change in element behaviour.The modified incremental force-displacement relation-ship of a beam-column element with a plastic hinge atend A can be expressed as:

(5)

A similar relationship can be obtained for an elementwith a plastic hinge at end B:

(6)

If plastic hinges form at both ends of the element, themodified incremental force-displacement relationshipcan be expressed as:

(7)

Note that the rotation corresponding to the location ofthe plastic hinge has been effectively removed from therelationship, and replaced with the change in moment



at the plastic hinge (M�

sc). This value is a function of the

increment in axial force (P�), and can be obtained from

the equations defining the section capacity. The pseudo-force terms (containing M

�

scAand M

�

scB) therefore allow in-

elastic force redistribution to be accurately representedwithout violation of the section capacity requirements.

2.3 Section CapacityThe original refined plastic hinge analysis (Liew, 1992)employs the AISC LRFD (AISC, 1995) bilinear inter-action equations to define the cross-section plasticstrength of members subjected to either major axis orminor axis bending. Note that the term plastic strengthis used to refer to the maximum section capacity, ignor-ing the possible reduction in capacity due to local buck-ling. The following equations are a simplified form ofEquation H1-1 (AISC, 1995), with the effective lengthtaken as zero for compression and bending:

(8)

Equation 8 is reasonably accurate for compact I-sec-tions subject to major axis bending, and is conservativefor minor axis bending. However, it is not an appro-priate definition of the section capacity for non-com-pact sections subject to local buckling effects. Thestresses associated with the local buckling deformationsthat occur in non-compact sections reduce the capacityof the section to resist an applied axial compressionforce and bending moment.

The effects of local buckling can be accounted for byusing the section capacity equations and provisions forlocal buckling provided in either the AISC LRFD (1995)or the AS4100 (SAA, 1990) specifications. Both ap-proaches will be considered and compared in this sec-tion.

The AS4100 section capacity is defined in Clauses5.2, 6.2, and 8.2 (SAA, 1990). The axial compressionand bending moment section capacities (N

s, M

s) are

determined using the effective area concept. The limit-ing slenderness ratios provided in AS4100 (Tables 5.2and 6.2.4) were established from lower bound fits tothe experimental local buckling resistances of plate el-ements in uniform compression. The effects of localbuckling are accounted for by the use of the form factor(k

f), normalised effective section modulus (Z

e/S), and

web slenderness ratio (w). The AS4100 section cap-

acity can be expressed in the same form as the AISCLRFD equations:

(9)

where:

(10)

The form factor (kf) represents the reduction in pure

axial compression section capacity, and is defined inClause 6.2 (SAA, 1990) as:

(11)

The ratio of the effective section modulus to plastic sec-tion modulus (Z

e/S) represents the reduction in pure

bending moment capacity, and can be obtained fromClause 5.2 (SAA, 1990):

(12)

Equation (9) is appropriate for either compact or non-compact sections subject to combined axial compres-sion and major axis bending. It may conservatively beused for minor axis bending if c

1 is taken as one, in

which case Equation 9 reduces to a simple linear inter-action equation.

The AISC LRFD section capacity is defined in Chap-ter H1, Appendix B5, and Appendix F1 (AISC, 1995).Equation (9) can also be used for the AISC LRFD sec-



tion capacity, with the parameters c1, c

2, c

3, k

f, and Z

e/S

derived from the provisions for local buckling providedin the AISC LRFD specification.

(13)

where:

(14)

The AISC LRFD form factor (Q) and nominal flexuralstrength (M

n) are defined in Appendices B5 and F1,

respectively.A comparison of the AISC LRFD and AS4100 sec-

tion capacity equations for compact sections is providedin Figure 2. The influence of section slenderness on theAS4100 section capacity of non-compact sections isillustrated in Figure 3. It illustrates the reduction in sec-tion capacity due to local buckling of non-compact sec-tions, as predicted by the AS4100 section capacity equa-tions (9 and 10). The AISC LRFD Equations (9) and(13) also provide a similar section capacity reduction.Application of these equations ensures that the forcestate corresponding to plastic hinge formation in non-compact sections is accurately modelled.

2.4 Gradual Yielding and DistributedPlasticityGradual yielding, distributed plasticity, and the associ-ated instability effects cannot be accurately representedby elastic-plastic hinge methods in which members areassumed to be fully elastic prior to the formation of theplastic hinges and subsequently remain fully elasticbetween hinge locations. Two functions are used in therefined plastic hinge formulation to approximatelyaccount for gradual yielding, distributed plasticity andthe associated instability effects: the tangent modulus(E

t) and the flexural stiffness reduction factor ( ). These

functions represent the distributed plasticity along thelength of the member due to axial force effects and thedistributed plasticity effects associated with flexure,respectively. In the original refined plastic hinge for-mulation (Liew, 1992), these functions accounted forresidual stresses, initial geometric imperfections, andinelastic second-order effects. If appropriate new func-tions are selected the tangent modulus and stiffnessreduction factor can also be used to implicitly accountfor the additional gradual yielding and spread of plas-ticity effects associated with local buckling in non-com-pact sections.

Tangent Modulus. The elastic modulus is replacedwith a tangent modulus (E

t) to represent the distributed

plasticity along the length of the member due to axialforce effects. The member inelastic stiffness, representedby the axial rigidity (E

tA) and the bending rigidity (E

tI),

is assumed to be a function of the axial force only. Thevalues E

tA and E

tI represent the properties of an effec-

tive core of the section. The tangent modulus can beevaluated from column member capacity curve specifi-cation equations and therefore implicitly includes theeffects of residual stresses, initial geometric imperfec-tions, and inelastic second-order effects.

Figure 4 illustrates the procedure used to evaluatethe tangent modulus using a member capacity columncurve. This procedure relies on the assumption that thetangent modulus of a compression member with a par-ticular non-dimensional axial load (p) and slenderness(

n) can be approximated by the tangent modulus of an

“equivalent stiffness” compression member with a mem-ber capacity equal to the axial load (p). This assump-tion is only strictly valid for the limiting (and thereforemost significant) case given by

n = ’

n. The procedure

used to evaluate the tangent modulus is described below:

1. The “equivalent stiffness” capacity of a memberwith a particular slenderness (

n) and applied non-

dimensional axial force (p) is obtained by extrapo-lating line EB to the intersection with the columncurve at C.

2. The slenderness corresponding to point C ( ’n) and

the non-dimensional Euler buckling load corre-sponding to this slenderness (p’

e) are given by the

axis intercepts D and F.

3. The non-dimensional tangent modulus (et = E

t/E)

is defined as the ratio p/p’e and is conveniently

independent of the actual member slenderness (n).

The original refined plastic hinge formulation (Liew,1992) offered a choice of two tangent modulus func-tions derived from the CRC column curve and the AISCLRFD column curve for members with compact cross-sections. The tangent modulus is intended to implicitlyaccount for the effects of initial geometric imperfec-tions, gradual yielding associated with residual stresses,and the associated instability. The tangent modulus func-tions recommended by Liew (1992) are appropriate forcompact hot-rolled I-sections but are not appropriatefor non-compact sections subject to local buckling eff-ects. The member instability associated with the local



Figure 2. Comparison of the AISC LRFD and AS4100 section capacity equations for compact sections.

Figure 3. Comparison of AS4100 section capacity equations for compact and non-compact sections with varying slenderness.



Figure 4. Tangent modulus calculation using column curve.

buckling deformations that occur in non-compact sec-tions causes a reduction in stiffness which must beincluded in the tangent modulus function.

The effects of local buckling on the tangent moduluscan be accounted for by using the compression mem-ber capacity equations and provisions for local buck-ling provided in either the AISC LRFD or the AS4100specifications. The AS4100 compression member cap-acity defined in Clause 6.3 (SAA, 1990) is based onexperimental testing of a range of compact and non-compact sections and is appropriate for either major orminor axis column buckling. It can be used for a widevariety of common section types (hot-rolled I-sections,welded I-sections, rectangular hollow sections, etc.) byselection of the appropriate member section constant(

b), from Table 6.3.3 (SAA, 1990). The effects of lo-

cal buckling are accounted for by the use of the formfactor (k

f) which is used to calculate the section cap-

acity (Ns), member slenderness ratio (

n), and member

section constant (b). The AISC LRFD compression

member capacity for sections subject to local bucklingis defined in Appendix B5, Clause 3d (AISC, 1995).

A comparison of the CRC, AISC LRFD, and AS4100compression member capacity curves for compact hot-rolled I-sections is provided in Figure 5. This figureindicates that the AS4100 column member capacityequation is more conservative than the CRC and AISC

LRFD equations for columns with intermediate and lowslenderness. This can be attributed to the differentmethods used to derive the equations: the AS4100 equa-tion was obtained by lower-bound curve fitting of ex-perimental results, while the AISC LRFD and CRCequations were derived from numerical and theoreticalmodels. Furthermore, the CRC equation does notinclude the effects of initial geometric imperfections.

Figure 6 illustrates the effect of section type on theAS4100 compression member capacity of compact sec-tions by the use of appropriate member section con-stants (

b). Higher member capacities are predicted for

sections with low residual compressive stresses such asstress-relieved compact rectangular hollow sections (

b

= –1) than for sections with high residual compressivestresses such as welded I-sections (

b = 0.5). The dif-

ference is particularly significant for columns with in-termediate slenderness.

The influence of section slenderness on the compres-sion member capacity for compact and non-compact I-sections is illustrated in Figure 7. This figure illustratesthe reduction in compression member capacity due tolocal buckling of non-compact sections, as predictedby the AS4100 equation for

b = 0.

Application of the AS4100 compression membercapacity equations ensures that the tangent modulusfunction can allow for the additional stiffness



Figure 5. Comparison of CRC, AISC LRFD, and AS4100 compression member capacity curves for compact hot-rolled I-sections.

Figure 6. Comparison of AS4100 compression member capacity curves for various types of compact sections.



Figure 7. Comparison of AS4100 compression member capacity curves for non-compact sections with varying section slendernesses(

b = 0).

reduction caused by local buckling of non-compact sec-tions in the refined plastic hinge analysis. The equa-tions can be used for a variety of different section typesincluding hot-rolled I-sections, welded I-sections, andcold-formed rectangular hollow sections.

As the AS4100 compression member capacity is pre-sented as a complex set of equations (see Clause 6.3.3),a simple equation expressing the tangent modulus as afunction of the non-dimensional axial force could notbe derived. The tangent modulus based on the AS4100column capacity equations was therefore calculatedusing the following procedure developed by Avery(1998):

1. For a given non-dimensional axial force (p),obtain the “equivalent stiffness” member slender-ness reduction factor ( ’

c) using Equation (15).

(15)

2. Calculate the corresponding slenderness ratio ( ’)using Equation (16).

(16)

3. Calculate the corresponding modified slendernessratio ( ’

n) using Equation (17). Note that the modi-

fied slenderness ratio ( ’n) is equal to the slender-

ness ratio ( ’) for all hot-rolled I-sections and somewelded sections (

b = 0). Equations (17) to (20)

are therefore not required for these sections.

(17)

where:

(18)

(19)

(20)

4. Using Equation (21), calculate the non-dimen-sional Euler buckling load (p’

e) corresponding to

the modified slenderness ratio determined from theprevious step.



(21)

5. Determine the tangent modulus for the given non-dimensional axial force using Equation (22).

(22)

6. Check the limiting values of the tangent modulususing Equations (23) and (24).

(23)

(24)

Due to the relative simplicity of the AISC LRFD com-pression member capacity equation, a simple equationcan be derived to express the AISC LRFD tangent modu-lus as a function of the normalised axial force and theform factor.

(25)

These equations provide an alternative to the AS4100tangent modulus for the refined plastic hinge analysisof steel frames comprising members with either com-pact or non-compact cross-sections. For sectionswhich are compact for pure axial compression (i.e., k

f

= 1), Equation (25) reduces to the equation used byLiew (1992) for the original refined plastic hingeanalysis.

A comparison of the reduced CRC, AISC LRFD, andAS4100 tangent modulus functions for compact I-sec-tions is provided in Figure 8. This figure indicates thatthe tangent modulus predicted by the AS4100 equationis more conservative than the corresponding reducedCRC and AISC LRFD equations for higher axial forces(p > 0.2), but less conservative for lower axial forces (p< 0.2). This can be attributed to the different methodsused to derive the compression member capacity equa-tions discussed previously.

The effect of section type on the AS4100 tangentmodulus function for compact sections is illustrated inFigure 9. Figure 9 indicates that the effect of the dif-ferent residual compressive stresses induced duringmanufacture or fabrication of various section typescan be accounted for by the tangent modulus functionbased on the AS4100 compression member capacityequation. The stiffness reduction is more gradual forsections with low residual compressive stresses suchas stress-relieved compact rectangular hollow sections(

b = –1) and significantly greater for sections with

high residual compressive stresses such as welded I-sections with flange thickness over 40 mm (

b = 1).

The additional stiffness reduction due to local buck-ling can also be partially accounted for by the use of agreater member section constant when k

f < 1 for cer-

tain section types.The influence of section slenderness on the AS4100

tangent modulus function for a hot-rolled I-section isillustrated in Figure 10. This figure indicates that theincreased rate of stiffness reduction due to local buck-ling of non-compact sections is represented by the tan-gent modulus function based on the AS4100 compres-sion member capacity equation. The stiffness reductiondue to local buckling becomes increasingly significantas the non-dimensional axial load increases.

Flexural Stiffness Reduction Factor. Distributed plas-ticity effects associated with flexure are represented byintroducing a gradual degradation in stiffness as yield-ing progresses and the section capacity at one or bothends is approached. The member stiffness graduallydegrades according to a prescribed function after theelement end forces exceed a predefined initial yieldfunction from the elastic stiffness to the stiffness associ-ated with the formation of plastic hinges at one or bothends.

To represent this gradual transition for the formationof a plastic hinge at each end of an initially elastic beam-column element, Liew (1992) described the refined plas-tic hinge element incremental force-displacement rela-tionship during the transition using a stiffness reduc-tion factor ( ):



(26)

where:�flp = local element incremental pseudo-force vector,

which accounts for the change in moment correspond-ing to a change in axial force at plastic hinge loca-tions.

The stiffness reduction parameter ( ) is equal to onewhen the element end (referenced by the subscript) iselastic, and zero when a plastic hinge has formed. Thegradual stiffness reduction is only associated with theflexural stiffness, and does not influence the axial stiff-ness. It can be seen that:

� When A =

B = 1, both ends are fully elastic. Equa-

tion (26) reduces to the incremental form of thesecond-order elastic force-displacement relation-ship (Equation (1)). All pseudo-force vector com-ponents are zero.

� When A = 1 and 1 >

B > 0, Equation (26) repres-

ents the state at which end A is elastic and end Bis partially yielded. All pseudo-force vector com-ponents are zero.

� When B = 1 and 1 >

A > 0, Equation (26) repres-

ents the state at which end B is elastic and end Ais partially yielded. All pseudo-force vector com-ponents are zero.

� When 1 > A > 0 and 1 >

B > 0, Equation (26)

accounts for partial plastification at both ends ofthe element. All pseudo-force vector componentsare zero.

� When A = 0 and

B > 0, Equation (26) accounts

for the formation of a plastic hinge at end A, whileend B is still elastic (

B = 1) or partially yielded (1

> B > 0). The pseudo-force vector (Liew, 1992) is

given by:

(27)

� When B = 0 and

A > 0, Equation (26) accounts

for the formation of a plastic hinge at end B, whileend A is still elastic (

A = 1) or partially yielded (1

> A > 0). The pseudo-force vector is given by:

(28)

� When A =

B = 1, plastic hinges have formed at

both ends of the element. The pseudo-force vec-tor is given by:

(29)

Liew (1992) considered several alternative functions tocalculate the stiffness reduction factor for the originalrefined plastic hinge formulation. The most appropriatefunction was established by comparison with plasticzone analytical benchmarks provided by Kanchanalai(1977). The flexural stiffness reduction parameter (was assumed to be a function of the combined axialforce and bending moment (represented by the force



Figure 8. Comparison of CRC, AISC LRFD, and AS4100 tangent modulus functions for compact hot-rolled I-sections.

Figure 9. Comparison of AS4100 tangent modulus functions for different types of compact sections.



Figure 10. Comparison of AS4100 tangent modulus functions for non-compact I-sections with varying section slenderness.

state parameter ), and declined according to a pre-scribed parabolic function following the initial yield.

The flexural stiffness reduction function used in theoriginal refined plastic hinge model (Liew, 1992) wasonly intended for compact sections. It was necessary tomodify this function to account for the effects of localbuckling. The following generalised function is there-fore proposed:

(30)

The symbol represents a force-state parameter thatmeasures the magnitude of the axial force and bendingmoment at the element end, normalised with respect tothe plastic strength. Note that the initial yield surface isdenoted by =

iy, the section capacity by =

sc, and

the plastic strength surface by = 1. The section cap-acity and plastic strength are identical for compact sec-tions (i.e.,

sc = 1). For non-compact sections,

sc repres-

ents the reduction in section capacity due to local buck-ling. Note that Equation (30) reduces to the originalform when

sc = 1 (i.e., for compact sections) and

iy =

0.5.The expressions for the force state parameter ( )

based on the AS4100 and AISC LRFD section capacityequations are defined in Equations (31) and (32).

(31)

where:

(32)

The force state parameter corresponding to the sectioncapacity (denoted by

sc) can also be calculated using

the AS4100 section capacity equations (Avery, 1998):

(33)



Equation (33) derived based on Equations (9) and (31)enables the calculation of

sc into a single step. For the

AISC LRFD section capacity equations, Equation (33)can still be used by using Q and M

n/M

p instead of k

f

and Ze/S, respectively, and the appropriate coefficients

(Equations (13) and (32)). The force state parametercorresponding to initial yield (denoted by

iy) can simply

be taken as 0.5 (as for the original refined plastic hingemodel) and assumed to be independent of the sectionslenderness. Equation (30) is graphically presented inFigure 11 for

iy = 0.5 and various section slendernesses.

Figure 11 demonstrates that the increased rate offlexural stiffness reduction due to local buckling of non-compact sections can be accounted for by the general-ised form of the parabolic flexural stiffness reductionfunction (Equation (30)) if the AS4100 or AISC LRFDsection capacity equations (31 to 33) for non-compactsections are used to determine the magnitude of the forcestate parameters.

2.5 Hinge softeningFollowing the formation of a plastic hinge a compactsection can maintain an axial force and bending mo-ment combination as defined by the plastic strengthequations ( = 1) as plastic deformation increases atthe hinge location until collapse of the structure. How-ever, non-compact sections subject to local bucklingexhibit hinge softening behaviour. Following the for-mations of a plastic hinge, a non-compact section cannot maintain the axial force and bending moment com-bination as defined by the section capacity equations( =

sc) as plastic deformation increases at the hinge

location. This hinge softening is due to the increasingstresses caused by increasing local buckling deforma-tions, resulting in a reduction in the effective sectioncore available to resist the applied axial force and bend-ing moment. Hinge softening reduces ductility, and canhave a moderately significant effect on the ultimate cap-acity of redundant framing systems.

The reduction in bending moment capacity withincreasing plastic rotation is illustrated in Figure 12.The curve labelled FEA was obtained from distributedplasticity finite element analysis of a stub beam-col-umn model with a non-compact cross-section (Avery,1998).

Hinge softening can be modelled within the estab-lished framework of the refined plastic hinge formula-tion by using negative values of the tangent modulusand the flexural stiffness reduction factor at the locationof a plastic hinge in a non-compact section. Severalapproaches were considered to find the most appropriate

function to define the rate of softening. Each approachwas investigated by comparison with the analyticalbenchmarks presented by Avery and Mahendran (1998).The simple equation shown below (Equation (34)) wasfound to approximately predict the rate of hinge sof-tening in typical steel sections as a function of the sec-tion slenderness. This equation is based on the simpli-fying assumption that the normalised softening modu-lus (e

s) is constant for a particular section, and was de-

rived from the element moment-rotation relationship(Equation (26)) for an element with a constant bendingmoment distribution. In the analysis it replaces Equa-tion (30) after the section capacity is reached. Avery(1998) provides the derivation of Equation (34).

(34)

where:

(35)

The method used to include the effects of hinge soften-ing is approximate. The verification of the model indi-cates that it is reasonably accurate for the types of frameand section considered in this study. However, furtherresearch is needed to develop an improved hinge-softening model and verify with some benchmarks thatare more sensitive to hinge softening effects.

3. ASSEMBLY AND SOLUTION OFTHE STRUCTURE FORCE-DISPLACEMENT RELATIONSHIPThe formulation of an element force-displacementrelationship for refined plastic hinge analysis includ-ing the effects of local buckling of non-compact sec-tions was presented in the previous section. The ele-ment relationships obtained using this formulation canbe transformed to a global Cartesian coordinate sys-tem, assembled to form the structure force-displacementrelationship and solved for unknown forces anddisplacements using the same procedures establishedby Liew (1992) for his implementation of the originalrefined plastic hinge analysis of steel frame structurescomprising only members of compact cross-section.

The new refined plastic hinge analysis program wasused to investigate the influence and sensitivity of anumber of analytical model parameters (Avery, 1998).Based on this study, a linear incremental solution methodis recommended, with a minimum of 100 load incre-ments and two elements per member.



Figure 11. Flexural stiffness reduction factor equations showing the effect of section slenderness for iy = 0.5.

Figure 12. Moment-rotation curve illustrating hinge softening behaviour of a non-compact section.



4. CONCLUSIONSA concentrated plasticity formulation for the advancedanalysis of steel frame structures has been presented inthis paper. The model is based on the refined plastichinge method (Liew, 1992), modified to account for theeffects of local buckling using simple equations derivedfrom the AS4100 and AISC LRFD specifications. Theaccuracy of the model is established in a companionpaper.

5. ACKNOWLEDGEMENTSThe authors wish to thank QUT for providing financialsupport through the QUT Postgraduate Research Award(QUTPRA) and the 1996 Meritorious Research ProjectGrants Scheme, and the Physical Infrastructure Centreand the School of Civil Engineering at QUT for provid-ing the necessary facilities and support to conduct thisproject.

REFERENCESAISC (1995), “Manual of Steel Construction, Load and Resistance

Factor Design. 2nd Edition”, American Institute of Steel Construc-tion, Chicago, IL, USA.

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NOTATIONA = cross-section areaA

e, A

g= effective and gross cross-section areas

ai, b

i= temporary variables used to solve cubicequation for ’

n

c’i

= constant used to define the plastic strengthc

i= constant used to define the section capacity

E = elastic moduluse

s= non-dimensional softening modulus = E

s/E

Es

= softening moduluse

t= non-dimensional tangent modulus = E

t/E

Et

= tangent modulusf

lp= local element pseudo-force vector

I = second moment of area with respect to the axisof in-plane bending

k = axial force parameter = p/EI, or local bucklingcoefficient

kf

= form factor for axial compression member =A

e/A

g

L = member length or length of element chordM = bending momentm = non-dimensional bending moment = M/M

p

MA,

MB= bending moments at element ends A and B

Miy

= bending moment defining the initial yieldm

iy= non-dimensional bending moment defining theinitial yield = M

iy/M

p

Mn

= AISC LRFD nominal flexural strengthM

p= plastic moment capacity =

yS

Mps

= bending moment defining the plastic strengthm

ps= non-dimensional bending moment defining theplastic strength = M

ps/M

p

Msc

= bending moment defining the section capacitym

sc= non-dimensional bending moment defining thesection capacity = M

sc/M

p

P = axial force or applied vertical loadp = non-dimensional axial force = P/P

y

Pe

= Euler buckling load =p

e= non-dimensional Euler buckling load = P

e/P

y

Piy

= axial force defining the initial yieldp

iy= non-dimensional axial force defining the initialyield = P

iy/P

y

Pps

= axial force defining the plastic strengthp

ps= non-dimensional axial force defining the plasticstrength = P

ps/P

y

Psc

= axial force defining the section capacity



psc

= non-dimensional axial force defining thesection capacity = P

sc/P

y

Pu

= ultimate applied vertical loadP

y= squash load =

yA

g

Q = AISC LRFD form factorq, r = temporary variables used to solve cubic

equation for ’n

S = plastic section modulus with respect to the axisof in-plane bending

s1, s

2= elastic stability functions

u = axial displacementZ = elastic section modulus with respect to the axis

of in-plane bendingZ

e= effective section modulus with respect to theaxis of in-plane bending= relative lateral deflection between member endsdue to member chord rotation= force state parameter

b= member section constant

c= member slenderness reduction factor

iy= force state parameter corresponding to initial

yield

sc= force state parameter corresponding to sectioncapacity= deflection associated with member curvaturemeasured from the member chord= flexural stiffness reduction factor

A, B= flexural stiffness reduction factors for elementends A and B= member slenderness ratio

c= AISC LRFD member slenderness ratio

n= modified compression member slendernessratio

s= section slenderness

sy, sp= section yield and plasticity slenderness limits

w= web slenderness ratio

wy= web yield slenderness limit= rotation of deformed element chord

A, B= rotations at element ends A and B= axial force normalised with respect to the Eulerbuckling load = P/P

e

y= yield stress

DR PHILIP AVERY was a research scholar at Queensland University of Technology during1995-98, where he completed his PhD in 1998. He also holds a Bachelors degree in CivilEngineering with a first class honours and a university medal from the same university. He iscurrently a Research Associate in the Department of Aeronautics at the Imperial College ofScience, Technology and Medicine, London.

DR MAHEN MAHENDRAN is an Associate Professor of Civil Engineering (Structures) andDirector of Physical Infrastructure Centre at Queensland University of Technology. Heobtained his BScEng degree with a first class honours from the University of Sri Lanka in 1980and his PhD from Monash University in 1985. He has since worked as an academic and appliedresearcher at various universities including James Cook University Cyclone Testing Station. Hiscurrent research interests include behaviour and design of profiled steel cladding systemsunder high wind forces, full scale behaviour of steel building systems and components, andbuckling and collapse behaviour of thin-walled steel structures

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Refined Plastic Hinge Analysis of Steel Frame Structures ... · analysis and design of steel frame structures permits direct and accurate determina-tion of ultimate system strengths,