Refined Plastic Hinge Analysis of Steel Frame Structures ...eprints.qut.edu.au/61751/1/1369433001502247.pdf · Advances in Structural Engineering Vol. 3 No. 4 2000 309 Refined Plastic

Advances in Structural Engineering Vol. 3 No. 4 2000 309

Refined Plastic Hinge Analysis ofSteel Frame Structures

Comprising Non-CompactSections

II: VerificationP. 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. Arefined plastic hinge method suitable for practical advanced analysis of steel framestructures comprising non-compact sections is presented in a companion paper.The method implicitly accounts for the effects of gradual cross-sectional yielding,longitudinal spread of plasticity, initial geometric imperfections, residual stresses,and local buckling. The accuracy and precision of the method for the analysis ofsteel frames comprising non-compact sections is established in this paper by com-parison with a comprehensive range of analytical benchmark frame solutions. Therefined plastic hinge method is shown to be more accurate and precise than theconventional individual member design methods based on elastic analysis and speci-fication equations.

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

1. INTRODUCTIONThe formulation of a refined plastic hinge method suit-able for practical advanced analysis of steel frame struc-tures comprising non-compact sections is presented ina companion paper. In order to establish the validity,accuracy and reliability of the refined plastic hingemethod for the analysis of steel frames comprising non-compact sections, the benchmark frames presented byAvery and Mahendran (1998a) were analysed with bothAS4100 and AISC LRFD parameters. In this paper, theresults of these refined plastic hinge analyses are com-

pared with those obtained using the shell finite elementdistributed plasticity model (Avery and Mahendran,2000a), assumed to be analytically “exact” due to theexplicit modelling of local buckling deformations, dis-tributed plasticity, imperfections and residual stresses.

Each benchmark frame was also designed in accord-ance with the AS4100 (SAA, 1990) and AISC LRFD(AISC, 1995) specifications, using a conventional sec-ond-order elastic method of analysis. The results of allrefined plastic hinge analyses, distributed plasticityanalyses, and specification designs are compared using

Refined Plastic Hinge Analysis of Steel Frame Strctures Comprising Non-Compact Sections. II: Verification

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

tabulated summaries of ultimate load capacities, nor-malised strength curves, and load-deflection curves. Arepresentative selection of these tables and charts is pre-sented in this paper. Additional results and compari-sons are provided by Avery (1998).

2. DISTRUBUTED PLASTICITYMODELAvery and Mahendran (1998a) used a shell distributedplasticity model to develop the analytical benchmarksolutions used in this paper. The model used AbaqusS4R5 shell elements in order to explicitly model localbuckling deformations and spread of plasticity effects.The Abaqus classical metal plasticity model was usedfor all analyses. Local imperfections were included inall non-compact sections based on appropriate buck-ling eigenvectors obtained from an elastic bucklinganalysis of the model. The magnitudes of the local flangeand web imperfections were taken as the assumed fab-rication tolerances (SAA, 1990). Out-of-plumbness andout-of-straightness member imperfections wereincluded in sway and non-sway frames, respectively.Their magnitudes were taken as the erection and fabri-cation tolerances for compression members specifiedin AS4100 (SAA, 1990). The residual stress distributionrecommended by ECCS (1984) for hot-rolled I-sectionswas used.

The accuracy of the distributed plasticity model wasfirst established by conducting two series of compari-sons. The first series involved the use of results pub-lished by Vogel (1985) for three frames comprising com-pact sections: a Portal frame, a Gable frame and a six-storey frame. The second series involved the use of re-sults from three large scale experiments of frames com-prising non-compact sections. These single storey, sin-gle bay experimental frames comprised of hot-rolledand welded I-sections, and cold-formed rectangularhollow sections, and had fixed base connections andrigid joints. They could be classified as sway frameswith full lateral restraint. Both series of comparisonsindicated that the distributed plasticity shell finite ele-ment model is accurate and reliable for second-orderinelastic analysis of steel frame structures comprisingnon-compact sections with full lateral restraint, and iscapable of explicit modelling of local buckling defor-mations and associated yielding. The model was there-fore used to develop a comprehensive range of analyti-cal benchmarks comprising non-compact I-sections(hot-rolled), suitable for the verification of simplifiedmethods of analysis. Full details of the distributed plas-ticity model are provided by Avery and Mahendran

(2000a) whereas those of the large scale experimentalframes are provided by Avery and Mahendran (2000b).

3. VERIFICATION OF THEREFINED PLASTIC HINGEMETHODA total of six series of benchmark frames (129 frames)provided by Avery and Mahendran (1998a) were usedin the verification of the refined plastic hinge method.These frames are representative of a variety of typicalframe configuration and parameters. They were ana-lysed using the refined plastic hinge method with bothAS4100 and AISC LRFD parameters, and the resultswere compared with those from both the distributedplasticity analyses, ie. analytical benchmark results fromAvery and Mahendran (1998a), and specification de-signs to the AS4100 and AISC LRFD standards. A sum-mary of the comparisons for each series is provided inthis section, and the performance of the refined plastichinge model is evaluated.

3.1 Modified Vogel FramesThree non-compact benchmark frames were developedby modifying the original Vogel frames (1985) by re-ducing the web and flange thicknesses of each sectionby 30% and by increasing the yield stress from 235 to350 MPa (see Figures 1 to 3). Further details of thesemodified Vogel benchmark frames are presented byAvery and Mahendran (1998a).

1. Portal frame. Single bay, single storey sway framewith fixed column base restraints.

2. Gable frame. Single bay, single storey sway framewith pinned column base restraints.

3. Six storey frame. Two bay, six storey sway framewith fixed column base restraints.

These benchmarks represent typical rectangular singlestorey and multi-storey frames in which each memberis a non-compact I-section bent about its major axis.

The ultimate load factors obtained from the refinedplastic hinge (RPH) analyses using AS4100 and AISCLRFD model parameters are presented in Table 1. Notethat four elements were used for each beam member inthe gable and six storey frames due to the significanttransverse loads. The refined plastic hinge model accu-rately predicts the ultimate capacity of both the modi-fied Vogel portal and gable frames. Using AS4100 modelparameters, the refined plastic hinge is 2.3 to 4.9% con-servative compared with the finite element benchmark

P. Avery and M. Mahendran


Figure 1. Sway load-deflection curves for the modified Vogel portal frame.

Figure 2. Sway load-deflection curves for the modified Vogel gable frame.



Figure 3. Sway load-deflection curves for the modified Vogel six storey frame.

Table 1. Comparison of ultimate load factors for modi-fied Vogel frames

solutions. The AISC LRFD model parameters produceslightly higher capacities but are not more than 1%unconservative.

The refined plastic hinge load-deflection curves forthe modified Vogel frames are illustrated in Figures 1to 3, and compared with the finite element benchmarkcurves. These comparisons demonstrate that the refinedplastic hinge model does not accurately model the ini-tial stiffness due to the simplified method of implicitlyaccounting for initial imperfections using the tangentmodulus function. The rate of stiffness reduction dueto material yielding and local buckling also appears tobe overestimated by the combined effects of the para-bolic flexural stiffness reduction parameter and tangentmodulus function. These errors are generally conserva-tive, and seem to be more significant in the case of thesix storey frame, with the refined plastic hinge analy-ses being 10.5 to 17.8% conservative compared withthe benchmark solution. This error may also be par-tially attributed to the approximate method used to

model the distributed member loads as lumped nodalloads.

3.2 Benchmark Series FramesThe single bay, single storey benchmark series framescomprising non-compact I-sections were chosen inorder to investigate a range of parameters that couldinfluence the behaviour of steel frame structures com-prising members with non-compact cross-sections (seeFigure 4(a)). These parameters included section slen-derness (k

f, Z

e/S), column member slenderness (L

c/r),

vertical to horizontal load ratio (P/H), and column tobeam stiffness ratio ( ). For this purpose, three ideal-ised Australian non-compact hot-rolled I-sections(360UBi44.7, 310UBi32.0 and 250UBi25.7), and sixother reduced sections obtained through the reductionsof web and flange thicknesses and yield stress were used.This gave three sets of section slenderness with k

f and

Ze/S values ranging from 0.802 to 0.943 and 0.887 to

1.0, respectively. Three sets of column slendernessesand two beam to column stiffness ratios (L

c/r from 24.2

to 56.9 and from 0.54 to 2.31) were obtained by chang-ing the column height and frame width and the analysisconsidered three vertical to horizontal load ratios (P/H= 3, 15 and 100). Further details of these benchmarkseries frames and analytical results are given in Averyand Mahendran (1998a).



Figure 4(a). Sway load-deflection curves for Series 1 frame. P/H = 100.

Figure 4(b). Sway load-deflection curves for Series 1 frame. P/H = 15.



Figure 4(c). Sway load-deflection curves for Series 1 frame. P/H = 3.

Figure 5. Vertical load-deflection curves for Series 1 frame.



Series 1: 54 Fixed base sway portal frames(major axis bending)Benchmark series 1 included frames with a range ofthree section slendernesses (represented by k

f, Z

e/S),

three column member slendernesses (represented by Lc/

r), three vertical to horizontal load ratios (representedby P/H), and two column to beam stiffness ratios (repres-ented by ). The refined plastic hinge sway load-de-flection curves for three of the series 1 portal framesare illustrated in Figures 4 (a) to (c), and compared withthe finite element benchmark solutions. A typical verti-cal load-deflection curve for one of the series 1 portalframes is shown in Figure 5.

Three normalised strength curves are presented inFigures 6 (a) to (c). These curves illustrate the ultimatestrength of a benchmark frame for the complete rangeof vertical and horizontal loads, normalised with respectto the axial squash load (P

y) and horizontal load (H’)

required to generate an elastic bending moment equalto the plastic moment capacity at the location of maxi-mum moment, respectively. H’ is defined in Equation(1) as a function of the plastic moment capacity (M

p),

column length (Lc), and stiffness ratio (s

r).

(1)

The stiffness ratio (sr) is a function of the frame con-

figuration and the beam-column stiffness ratio ( ). Forexample, the stiffness ratio for fixed base sway portalframes can be expressed as:

(2)

The normalised ultimate loads obtained from the re-fined plastic hinge (RPH) analyses using AS4100 andAISC LRFD model parameters are summarised inTable 2. This table also contains comparisons betweenthe specification design and finite element analysis(FEA) ultimate loads, specification design and refinedplastic hinge ultimate loads, AS4100 and AISC LRFDspecification design loads, and AS4100 and AISC LRFDrefined plastic hinge ultimate loads. The AS4100 refinedplastic hinge model is, on average, 6.5% conservativecompared with the finite element benchmark model forseries 1 frames. The maximum unconservative error is1.6%. The AS4100 refined plastic hinge model providesa mean capacity increase of 5.9% compared with the

Table 2. Statistical analysis of benchmark series 1results

conventional elastic analysis design procedure usingAS4100 specification equations, and a maximum cap-acity increase of 19.5%. The maximum capacityincrease should be even higher in more complex struc-tures with greater redundancy and scope for inelasticredistribution. The coefficients of variation indicate thatthe AS4100 refined plastic hinge model provides a sig-nificantly more uniform safety compared with theAS4100 design specification equations. These resultsindicate that the AS4100 refined plastic hinge model issuitably accurate and precise for fixed base sway framescomprising non-compact I-sections with major axisbending. However, the AISC LRFD refined plastic hingemodel is, on average, 0.6% unconservative and has anunacceptable maximum error of 10.3%.

The influence of the frame parameters (column sec-tion slenderness, column member slenderness, verticalto horizontal load ratio, and column to beam stiffnessratio) on the mean non-dimensional ultimate capacityis summarised in Table 3. The results shown in this tabledemonstrate that the accuracy of the refined plastic hingemodel is not sensitive to parametric variation for series1 frames. However, the mean capacity ratios do increaseslightly as the section slenderness increases.

Based on the results of the benchmark series 1 analy-ses, the following observations can be made regardingthe performance of the refined plastic hinge model:

1. The refined plastic hinge model does not accuratelyor consistently model the initial stiffness of thebenchmark frames. This inaccuracy is clearlyillustrated by the load-deflection curve compari-sons provided in Figures 4(a) to (c). The initialstiffness is overestimated for frames with high P/H ratios, and underestimated for frames with low



Figure 6(a). Comparison of strength curves for Series 1 frame. Lc/r = 24.2.

Figure 6(b). Comparison of strength curves for Series 1 frame. Lc/r = 40.3.



Figure 6(c). Comparison of strength curves for Series 1 frame. Lc/r = 56.4.

Table 3. The effect of parametric variation on theaccuracy of the refined plastic hinge model for bench-

mark series 1

P/H ratios. The overestimation of the initial stiff-ness is likely to contribute to an unconservativeultimate load prediction for very slender frameswith high P/H ratios but should not have a signifi-cant influence on the capacity of frames with typi-cal low to medium column slenderness. The erro-

neous initial stiffness predicted by the refined plas-tic hinge model is clearly due to the simplifiedmethod of implicitly accounting for initial mem-ber imperfections using the tangent modulus func-tion. The effect of initial out-of-plumbness imper-fections is most significant in sway frames withhigh P/H ratios, but does not have a significantinfluence on the sway stiffness of frames subjectto larger horizontal loads because the displacementdue to the horizontal load is relatively large in com-parison to the magnitude of the initial imperfec-tion. Furthermore, as the sway displacementincreases the stiffness reduction due to initial out-of-straightness imperfections gradually declines.These effects are not accounted for by the tangentmodulus function that assumes that the stiffnessreduction due to initial member imperfections isindependent of the P/H ratio and sway displace-ment. This explains the discrepancy between theinitial stiffness predicted by the refined plastichinge and finite element models.

2. The refined plastic hinge model’s rate of stiffnessreduction due to material yielding and local buck-ling appears to be overestimated by the combinedeffects of the parabolic flexural stiffness reduc-



tion parameter and tangent modulus function. Thisis particularly apparent for frames with high P/Hratios, which develop significant axial compres-sion forces resulting in lower tangent moduli inthe column members prior to failure. As the tan-gent modulus is derived from the column membercapacity equations, it alone will suffice to reducethe stiffness of a compression member to induceinstability at the appropriate load. However, theflexural stiffness reduction parameters are also afunction of the axial force, further reducing thestiffness and causing premature instability failure.This approximation is always conservative, but itseffects are difficult to quantify due to the interac-tion of the various other simplifications inherentin the refined plastic hinge model.

3. Initial yield is premature in the refined plastic hingemodel due to the use of

iy = 0.5 in the flexural

stiffness reduction function. For hot-rolled I-sec-tions, the ideal initial yield varies linearly from

iy

= 0.7 for pure compression to iy = 0.63 (typical)

for pure bending. The premature initial yield isillustrated by the load-deflection curves (Figures4 (a) to (c)). The refined plastic hinge model clearlycommences non-linear behaviour prior to the FEAinitial yield. The effect of the approximation (

iy

= 0.5) is always conservative, and contributes tothe overestimation of the rate of stiffness reduc-tion previously discussed.

4. Inelastic redistribution ductility is overestimated bythe refined plastic hinge model, particularly whenAISC LRFD parameters are used. This indicatesthat the hinge softening function underestimates therate of reduction in the section capacity followingthe formation of a plastic hinge in a non-compactsection when the AISC LRFD effective sectionproperties are used. This is demonstrated by theload-deflection curves for frames with low P/H ra-tios (Figure 4(c)). Single bay, single storey frameswith medium to high P/H ratios do not exhibit sig-nificant inelastic redistribution prior to failure,therefore the effects of hinge softening becomesincreasingly significant as the P/H ratio decreases.The significant difference between the shapes ofthe AISC LRFD and AS4100 refined plastic hingestrength curves for frames with P/H ratios less than0.1 (see Figures 6 (a) to (c)) clearly indicates theeffect of the AISC LRFD model’s increased ductil-ity on the predicted ultimate capacity. This substan-

tial difference between the AS4100 and AISCLRFD refined plastic hinge models suggests thatthe excessive ductility is primarily caused by anunconservative AISC LRFD prediction of theeffective section modulus rather than an inappro-priate hinge softening function. The AISC LRFDrefined plastic hinge model may therefore introducean unconservative error, which becomes significantin frames with potential for substantial inelasticredistribution in non-compact sections.

5. The AISC LRFD effective section properties areunconservative in some cases. However, theincreased capacity associated with thisunconservative error is nullified by the conserva-tive error associated with the excessive rate of stiff-ness reduction, which is particularly significant forframes with higher slenderness and/or P/H ratios.The refined plastic hinge model therefore moreaccurately predicts the ultimate capacity whenAISC LRFD parameters are used. However, insome instances the unconservative error associatedwith the AISC LRFD effective section propertiesexceeds the conservative error caused by theexcessive rate of stiffness reduction. The AISCLRFD refined plastic hinge model can thereforeoverestimate the ultimate capacity by as much as10%, while the AS4100 refined plastic hinge modelwas never more than 1.6% unconservative forbenchmark series 1. Furthermore, the mean RPH/FEA ultimate load ratio increases with increasingsection slenderness for both AS4100 and AISCLRFD models (see Table 3). This suggests that thespecification effective section property equationsdo not consistently predict the same section cap-acity as the FEA model. This can be attributed tothe use of the large local imperfection based onthe fabrication tolerance in the FEA model.

6. The reduction in axial stiffness is overestimatedby the tangent modulus function and is illustratedclearly by the vertical load-deflection curve (Fig-ure 5). This error is due to the use of the sametangent modulus for flexural and axial stiffness,which is based on compression member capacityequations and therefore has no rational basis foruse to model axial stiffness. However, this con-servative error does not significantly affect theframe capacity as the flexural stiffness reductionis generally much more significant than the axialstiffness reduction.



Figure 7. Comparison of strength curves for Series 2 frame.

7. The load-deflection curves for the benchmark se-ries and modified Vogel frames do not indicate theeffects of hinge softening. Hinge softeningdecreases the stiffness of the frame, and thus re-duces the slope of the load-deflection curve. How-ever, the solution method used in this formulationdoes not permit unloading of the frame, so theanalysis terminates before the slope of the frame’sload-deflection curve becomes negative. This doesnot mean that hinge softening does not occur. In-determinate frames will not necessarily commenceto unload when one of its hinges is softening. Therewill be an initial redistribution of load until suffi-cient hinges are formed.

Series 2: 12 Pinned base sway portal frames(major axis bending)Benchmark series 2 included frames with a range oftwo section slendernesses (Z

e/S), two member

slendernesses (Lc/r), and three vertical to horizontal load

ratios (P/H). Typical normalised strength curves arepresented in Figure 7. A summarised comparison of thebenchmark series 2 ultimate loads obtained from theRPH analyses, FEA, and specification design calcula-tions is presented in Table 4. The AS4100 refined plas-tic hinge model is, on average, 10.7% conservative com-pared with the finite element benchmark modelfor series 2 frames. The accuracy and precision of the


AS4100 refined plastic hinge model is very similar tothe AS4100 specification design equations. This isexpected, as the pinned base series 2 frames have littlescope for inelastic redistribution. The series 2 resultsindicate that the AS4100 refined plastic hinge model issuitably accurate and precise for pinned base swayframes comprising non-compact I-sections with majoraxis bending. The AISC LRFD refined plastic hingemodel is, on average, 9.4% conservative and has a maxi-mum unconservative error of 0.4% and is therefore alsosuitable for frames of this type. An investigation of theinfluence of the frame parameters demonstrated that theaccuracy of the refined plastic hinge model is not sen-sitive to parametric variation for series 2 frames.



Figure 8. Sway load-deflection curves for Series 3 frame.




Series 3: 36 Leaned column sway portalframes (major axis bending)Benchmark series 3 included frames with a range oftwo section slendernesses (k

f, Z

e/S), two member

slendernesses (Lc/r), three vertical to horizontal load

ratios (P/H), and three vertical column load ratios (P1/

P2). The refined plastic hinge sway load-deflection

curves for a typical series 3 portal frame is illustrated inFigure 8, and compared with the finite element bench-mark solution. Typical normalised strength curves arepresented in Figure 9.

A summarised comparison the benchmark series 3ultimate loads obtained from the RPH analyses, FEA,and specification design calculations is presented inTable 5. The AS4100 refined plastic hinge model is, onaverage, 2.3% conservative compared with the finiteelement benchmark model for series 3 frames. Themaximum unconservative error is 6.6%, slightly inexcess of the recommended 5% limit (Liew et al., 1993).The AS4100 refined plastic hinge model provides amean capacity increase of 7.6% compared with the con-ventional elastic analysis design procedure usingAS4100 specification equations, and a maximum cap-acity increase of 33.5%. As expected, the maximumcapacity increase is greater than for series 1 and 2 dueto the greater scope for inelastic redistribution in theleaned column frames with non-uniform column load-ing.

The coefficients of variation indicate that the AS4100refined plastic hinge model provides a significantly moreuniform safety compared with the AS4100 design speci-fication equations. These results indicate that theAS4100 refined plastic hinge model is reasonablyaccurate and precise for leaned column sway framescomprising non-compact I-sections with major axisbending and non-uniform column loading. However,the AISC LRFD refined plastic hinge model is,on average, 4.7% unconservative and has an unaccept-able maximum error of 16.5%. An investigation of theinfluence of the frame parameters demonstrated that theaccuracy of the refined plastic hinge model is not par-ticularly sensitive to parametric variation for series 3frames.

Series 4: 12 Pinned base non-sway portalframes (major axis bending)Benchmark series 4 included frames with a range oftwo section slendernesses (k

f, Z

e/S), two member

slendernesses (Lc/r), and three load combinations (w =

0, P + w, and P = 0). The refined plastic hinge sway


load-deflection curves for a typical series 4 portal frameare illustrated in Figure 10, and compared with the fi-nite element benchmark solution. All horizontal deflec-tions were measured at mid-height of the left hand col-umn. Typical normalised strength curves are presentedin Figure 11.

A summarised comparison of the benchmark series4 ultimate loads obtained from the RPH analyses, FEA,and specification design calculations is presented inTable 6. The AS4100 refined plastic hinge model is, onaverage, 3.1% conservative compared with the finiteelement benchmark model for series 4 frames. Themaximum unconservative error is 6.9%, slightly inexcess of the recommended 5% limit. The AS4100refined plastic hinge model provides a mean capacityincrease of 4.8% compared with the conventional elas-tic analysis design procedure using AS4100 specifica-tion equations, and a maximum capacity increase of15.0%.

The refined plastic hinge model’s coefficient of vari-ation is greater than for previous series, but still indi-cates slightly greater precision than the AS4100 designspecification equations. These results indicate that theAS4100 refined plastic hinge model is reasonablyaccurate and precise for pinned base non-sway (i.e.,braced) frames comprising non-compact I-sections withmajor axis bending. However, the AISC LRFD refinedplastic hinge model is, on average, 1.8% unconservativeand has an unacceptable maximum error of 17.4%.

An investigation of the influence of the frameparameters indicated than the accuracy of the refinedplastic hinge model is moderately sensitive to the sec-tion slenderness and load combination parameters forseries 4 frames. The mean capacity ratio is greater formore slender sections and frames with significant axialcompression column loads. Conversely, the capacity of



Figure 10. Sway load-deflection curves for Series 4 frame.






braced frames with dominant beam distributed loadingis conservatively predicted by the refined plastic hingemodel. This may be due to the approximate modellingof the distributed loads as lumped nodal loads.

Series 5: 6 Pinned base sway portal frames(minor axis bending)Benchmark series 5 included frames with a range oftwo section slendernesses (k

f, Z

e/S), and three vertical

to horizontal load ratios (P/H). Typical normalisedstrength curves are presented in Figure 12. A summa-rised comparison of the benchmark series 5 ultimateloads obtained from the RPH analyses, FEA, and speci-fication design calculations is presented in Table 7. The


AS4100 refined plastic hinge model is, on average, 4.1%conservative compared with the finite element bench-mark model for series 5 frames with a maximumunconservative error of 5.5%. Unlike previous series,the AS4100 refined plastic hinge model is more con-servative than the AS4100 specification design equa-tions for series 5 frames. However, as series 5 consistsof only six analyses (compared with 114 analyses forseries 1 to 4), no definite conclusions can be drawnregarding this result. The series 5 results suggest thatthat the AS4100 refined plastic hinge model may besuitably accurate and precise for pinned base swayframes comprising non-compact I-sections with minoraxis bending. Further investigation is required to jus-



tify this observation. The accuracy and precision of theAISC LRFD refined plastic hinge model is similar tothe AS4100 refined plastic hinge model for series 5.

4. CONCLUSIONSA concentrated plasticity model for the advanced analy-sis of steel frame structures has been presented in a com-panion 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. Inthis paper, the accuracy and precision of the new modelhas been extensively tested using the analytical bench-marks presented by Avery and Mahendran (1998a). Astatistical analysis of the combined results of bench-mark series 1 to 5 (a total of 120 analyses) is providedin Table 8.

The refined plastic hinge model based on the AS4100specifications is suitable for all of the frame typesinvestigated, and is significantly more accurate and pre-cise than the conventional individual member designmethod based on elastic analysis and specification equa-tions. On average, the refined plastic hinge model withAS4100 parameters is 5.1% conservative. The maxi-mum unconservative error is 6.9%, slightly greater thanthe 5% recommended maximum error, but still reason-able if a 0.9 capacity reduction factor is used. Therefined plastic hinge model can allow the design cap-acity to be increased by up to 33.5%, mainly due to theconsideration of inelastic redistribution. The refinedplastic hinge model based on the AISC LRFD specifi-cations is too unconservative in some situations (up to17.4%), and therefore is not recommended for generaluse.

Further research has been conducted in an attempt toimprove the accuracy and precision of the refined plas-tic hinge model by using more accurate model param-eters obtained from distributed plasticity analyses of astub beam-column model. The results of this researchare presented by Avery and Mahendran (1998b).

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.

Table 8. Statistical analysis of combined benchmarkseries 1–5 results

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

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

Avery, P. (1998), “Advanced analysis of steel frames comprisingnon-compact sections”, PhD thesis, School of Civil Engineer-ing, Queensland University of Technology, Brisbane, Australia.

Avery, P. and Mahendran, M. (1998a), “Analytical benchmarksolutions for steel frame structures comprising non-compact sec-tions”, Physical Infrastructure Centre Research Monograph 98–3, Queensland University of Technology, Brisbane, Australia.

Avery, P. and Mahendran, M. (1998b), “Pseudo plastic zone analy-sis of steel frame structures comprising non-compact sections”,Physical Infrastructure Centre Research Monograph 98–5,Queensland University of Technology, Brisbane, Australia.

Avery, P. and Mahendran, M. (2000a), “Distributed plasticity analy-sis of steel frame structures comprising non-compact sections”,Engineering Structures, 22(8), 901–919.

Avery, P. and Mahendran, M. (2000b), “Large scale testing of steelframe structures comprising non-compact sections”, Engineer-ing Structures, 22(8), 920–936.

Liew, J. Y. R. (1992), “Advanced analysis for frame design”, PhDdissertation, School of Civil Engineering, Purdue University, WestLafayette, IN, USA.

SAA (1990), “AS4100–1990 Steel Structures”, Standards Associa-tion of Australia, Sydney, Australia (www.standards.com.au).

NOTATIONA

e, A

g= effective and gross cross-section areas

H = applied horizontal loadH’ = applied horizontal load that would produce a

maximum first-order elastic bending momentequal to M

p

Ib

= second moment of area of beam section withrespect to the axis of in-plane bending

Ic

= second moment of area of column section withrespect to the axis of in-plane bending

kf

= form factor for axial compression member =A

e/A

g

Lb,

Lc

= lengths of beam and column members



Mp

= plastic moment capacity = yS

P = applied vertical loadP

1, P

2= left and right hand column applied verticalloads

Py

= squash load = yA

g

r = radius of gyration with respect to the axis ofin-plane bending

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

sr

= stiffness ratio used to calculate the normalisedhorizontal load

w = applied beam distributed loadZ

e= effective section modulus with respect to theaxis of in-plane bending

iy= force state parameter corresponding to initialyield= column to beam stiffness ratio = (I

c/L

c)/(I

b/L

b)

o= member out-of-plumbness imperfection

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 ...eprints.qut.edu.au/61751/1/1369433001502247.pdf · Advances in Structural Engineering Vol. 3 No. 4 2000 309 Refined Plastic