Lateral–torsional buckling of Delta hollow flange beams under moment gradient

Lateral–torsional buckling of Delta hollow flange beams undermoment gradient

Amin Mohebkhah n, Mojtaba G. AzandarianiStructural Eng. Div., Faculty of Civil and Architectural Engineering, Malayer University, Malayer, Iran

a r t i c l e i n f o

Article history:Received 24 September 2014Received in revised form12 October 2014Accepted 13 October 2014

Keywords:Lateral–torsional bucklingFinite element methodOff-shear center loadingDelta hollow flange beamMoment-gradient factorLongitudinal stiffener

a b s t r a c t

Lateral–torsional buckling (LTB) resistance of an I-beam depends on the minor axis moment of inertia ofits compression flange and torsional rigidity of the section. A practical solution to increase lateralbuckling of I-beams is to add two Delta stiffeners between the compression flange and web plates toform a hollow compression flange. Although these beams known as Delta hollow flange beams (DHFBs)have been introduced about half a century ago for bridge construction, however, their LTB behavior hasnot been investigated, yet. This paper develops a three dimensional finite-element model using ABAQUSfor the inelastic nonlinear flexural–torsional analysis of DHFBs and uses it to investigate the effects ofunbraced length and central off-shear center loading (located at center, top flange and bottom flange) ontheir moment gradient factor in different behavioral zones. It was found that the Cb factor and momentresistance curve given by AISC-LRFD in Specification for structural steel buildings [2] are not accurate forintermediate (inelastic) and short plastic DHFBs leading to an unconservative design. Therefore, amodified moment resistance equation is proposed to be used instead of the code equation in inelasticzone for the investigated load cases in this paper.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

I-beams subjected to flexure have much greater strength andstiffness in the plain in which the loads are applied (major axis)than in the plane of minor axis. Unless these members areproperly braced against lateral deflection and twisting, they aresubjected to failure by lateral–torsional buckling (LTB) prior totheir in-plane failure.

The critical buckling moment in the case of pure bending isgiven by [1]

Mocr ¼ πL

ffiffiffiffiffiffiffiffiffiffiffiEIyGJ

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þW2� �r

ð1Þ

where

W ¼ πL

ffiffiffiffiffiffiffiffiffiECw

GJ

sð2Þ

L is the unbraced length which is the span length in this case. Ascan be seen in Eq. (1), LTB moment resistance of an I-beam withconstant unbraced length depends on lateral bending stiffness and

torsional stiffness of the beam cross section. A practical procedureto improve lateral stability of I-beams is to use longitudinal Deltastiffeners adjacent to their compression flange as shown in Fig. 1.Welding Delta stiffeners between the compression flange and webplates forms a two-cell triangular hollow flange increasing itstorsional stiffness as well as LTB moment resistance of the originalI-beam. This kind of beams is called Delta girder. Delta girderswere proposed and utilized first for bridge girders construction byHadley [7]. Using Delta stiffeners not only increases global LTBresistance of the beams, but also postpones premature localbuckling of their slender constituent plates (compression flangeand web plates) in noncompact and slender sections.

Ref. [3] studied flexural capacity of cold-formed hollow flangesteel beams under pure bending considering web lateral distor-tional buckling using finite element method. Dong and Sause [5]developed a finite element model for hollow tubular flange girders(HTFGs) to investigate the effects of stiffeners, initial geometricimperfections, and residual stresses on their flexural capacity.They found that the design formulas from the AASHTO LRFDBridge Design Specifications overestimate the lateral–torsionalbuckling strength for HTFGs. Hassanein et al. [8] investigated LTBof hollow tubular flange plate girders (HTFPGs) with slenderstiffened webs using FEM. They found that the inelastic LTBdominates the failure mode for higher spans of HTFPGs comparedto equivalent I-section plate girders. It was as well found that the

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http://dx.doi.org/10.1016/j.tws.2014.10.0110263-8231/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ98 851 2232346; fax: þ98 851 2221977.E-mail addresses: [email protected] (A. Mohebkhah),

[email protected] (M.G. Azandariani).

Thin-Walled Structures 86 (2015) 167–173

AISC and EC3 provide highly conservative predictions for theHTFPGs under pure bending. Hassanein and Kharoob [9] studiedflexural strength of HTFPGs with slender stiffened webs undermid-span concentrated loads. They found that unlike the case ofconventional beams with solid webs, the moment-gradient factorCb of HTFPGs is significantly influenced by the girder geometry andslenderness.

The effect of moment gradient on the critical moment caneasily be accounted for by the use of the moment gradient factor Cb.The nominal moment for an elastic beam is estimated asMn¼CbMocr. The moment gradient factor is mainly depends onthe load conditions of the beam. Apart from the load cases,however, it has been shown by Mohebkhah [10] that beamslenderness, cross-section geometry and web perforations (e.g.castellated beams) may also affect the moment gradient factor. Theequations for Cb given in the literature have been derived fromelastic LTB analysis of doubly symmetric I-beams. The results ofmany studies on evaluating the moment-gradient factor have beensummarized in Ziemian [15]. Samanta and Kumar [13] using anelastic finite element model, considering just the geometric non-linearity, studied the overall buckling resistance of monosym-metric I-beams with slender webs. They found that in case of shortbeams, distortional buckling is predominant and so flexural–torsional buckling solutions may give overestimated critical loads.

Therefore, they concluded that the Cb factor values depend on thelength–height ratio of the analyzed beams.

Despite the aforementioned studies on LTB of doubly sym-metric hollow flange beams, LTB of I-beams with Delta stiffeners(i.e. singly symmetric Delta hollow flange beams- HDFBs) hasnot been investigated. The purpose of this paper is to investi-gate lateral–torsional buckling and the effects of unbraced lengthon the moment gradient factor in different behavioral zones ofI-beams with Delta stiffeners. For this purpose a finite-elementmodel based on the commercial software package ABAQUS [1] isdeveloped for the nonlinear inelastic LTB analysis of I-beamshaving Delta stiffeners with a wide variety of slenderness. Then,it is used to investigate the applicability of the Cb factors given inthe literature for the simply supported inelastic beams under thecentral point load cases applied at shear center, bottom and topflanges.

2. Description of finite element modeling

To investigate the inelastic flexural–torsional buckling behaviorof I-beams with Delta stiffeners, a nonlinear inelastic finite-element model is developed based on the specifications andassumptions given in the following sections.

2.1. Modeling assumptions and material properties

The nonlinear computations were performed using the com-mercial finite element software package ABAQUS [1]. ABAQUS [1]has the ability to consider both geometric and material nonlinea-rities in a given model. Large displacement effects were accountedfor by utilizing the nonlinear geometry option in ABAQUS. A4-node doubly curved shell element with reduced integrationS4R [1] from the ABAQUS element library were used to modelthe web, flanges, Delta and the end supports stiffeners. The S4Relement is suitable for complex buckling behavior and has sixdegrees of freedom per node; 3 translational and 3 rotational thatprovides accurate solutions to most applications.

The models’ flanges were modeled with four elements acrossthe width and 6 elements were used throughout the web height.The same material properties were used for all the constituentplates. The steel is modeled as a J2 material with bilinear isotropichardening and an elastic-perfectly plastic stress-strain curve isassumed. In the elastic region, a typical value for the modulus ofelasticity (E¼200,000 MPa) and nominal yield stress value of235 MPa are specified. The Poisson’s ratio was set to 0.3 in thisstudy. Residual stresses were not considered in this work, thoughit may be relevant in this type of analysis.

List of symbols

bf width of flangeCb moment gradient factorCw warping section constantd depth of sectionE Young’s modulus of elasticityEst Strain hardening modulusFy yield stress of steelG shear modulus of elasticityIy second moment of area about y-axis

J torsional section constantL unbraced length of the beamMb LTBmoment resistance under point load at bottom flangeMc LTB moment resistance under point load at shear centerMo LTB moment resistance in uniform momentMocr elastic buckling uniform momentMt LTB moment resistance under point load at top flangetf thickness of flangetw thickness of webW dimensionless beam parameterZ plastic section modulus

Fig. 1. A typical Delta hollow flange beam (DHFB) section.

A. Mohebkhah, M.G. Azandariani / Thin-Walled Structures 86 (2015) 167–173168

2.2. Loads, boundary conditions, geometry and solution procedure

Simply supported I-beams having longitudinal Delta stiffenerswith different unbraced lengths are chosen under pure bending andcentral concentrated loading conditions in order to determine theirmoment capacities and the moment gradient factors. According toFig. 1, the dimensions of the considered Delta hollow flange beam(HDFB) cross-section are considered as: h¼400 mm, d¼100 mm,bf¼180 mm, t1¼13.5 mm, t2¼27 mm and tw¼ts¼8.6 mm. To pre-vent web local buckling and yielding, all beams were provided withtransverse bearing stiffeners at support and concentrated loadpoints. Also, it should be mentioned that in finite element analysisof the beams subjected to pure bending, the end moments wereapplied as couples acting on the centroids of top Delta-shapedflange and bottom flange to prevent web local yielding.

The nonlinear analysis solution was achieved using A RIKSbuckling method [1] which uses a geometrically nonlinear load–displacement inelastic solution procedure. Geometric imperfec-tions are also accounted for the analysis. To obtain a suitableimperfections pattern, an Eigenvalue analysis is performed todetermine the buckled mode shape of the beam when subjectedto the applied loads. Such analysis is a linear elastic analysisperformed with the load applied within the step. A scalingcoefficient of 2 mm is chosen to represent the maximum valueof geometric imperfections and then to generate a new geometrywith an imperfection pattern, the Eigen shape is added to theoriginal geometry.

3. Validation of the modeling technique

In this part the accuracy of the finite element model of thebeams are investigated. To this end, the comparison was per-formed using the available results of a test program on theultimate load capacity of Delta hollow flange plate girders(DHFPGs) as reported by Hadley [7]. In the test program, fourDHFPGs were tested under concentric or eccentric point loads(with respect to the girders’ web) at top flange. The dimensions ofthe DHFPGs’ cross-section and boundary conditions of the testedspecimens have been shown in Fig. 2. Table 1 summarizes thecharacteristics and material properties of each specimen. TheDHFPGs’ cross-sections at each support points were preventedfrom moving laterally and from twisting. All specimens hadbearing stiffeners at supports. However, transverse stiffenersunder point loads were used just for specimens SP1 and SP2.The point load in specimen SP4 was applied eccentrically with

respect to the girder’s web at the Delta stiffener-top flangejunction (i.e. patch loading with 181 mm eccentricity). Young’smodulus and Poisson’s ratio was set to 210 GPa and 0.3, respec-tively. Residual stresses were not considered in this validation.Table 1 also shows a comparison between the ultimate loadcapacities of the specimens obtained from the tests and the finiteelement analysis. It can be seen that the agreement is goodbetween the experimental load capacities and the loads obtainedby the nonlinear finite element analyses. Deformed geometry andfailure mode of specimen SP-2 (i.e. diagonal shear buckling of webplate and formation of tension field action phenomenon) has beenshown in Fig. 3 which is identical to the failure mode observed inthe experiments.

4. Parametric study

After validating the finite element model in the previoussection, a nonlinear analysis was performed looking at the factorsthat affect the value of Cb factor on DHFBs. The factors are thebeam’s overall slenderness and transverse load height from thesection centroid. To this end, the loading condition of a centralconcentrated loading (located at shear center, top flange andbottom flange) is taken into account. The DHFB section is a two-cell hollow section which its torsional constant J and warpingconstant Cw differ from that of conventional thin-walled I-beams.The torsional constant of the section can be computed as follows:

J ¼ 13

∑m

j ¼ 1ljt3j þ JΔ ð3Þ

where, lj and tj are the length and thickness of the jth plateelement connected to the top Delta hollow flange and JΔ is the

P

152 mm 152 mm L/2 L/2

Fig. 2. Boundary conditions and dimensions of the Delta hollow flange plate girders tested by Hadley [7].

Table 1Characteristics and ultimate load capacity of the test specimens [7] in comparisonwith the FEA results.

Specimen L(mm)

Yieldstress(MPa)

Ultimatestress(MPa)

Ultimatestrain (%)

P(Test)(kN)

P(FEM)

(kN)Difference(%)

SP1 6401 227.5 413.7 21 1423.4 1450 1.8SP2 6401 227.5 496.4 21 1490.1 1558 4.6SP3 5791 227.5 413.7 21 1112 1135 2.1SP4 5791 227.5 413.7 21 889.6 918 3.2

A. Mohebkhah, M.G. Azandariani / Thin-Walled Structures 86 (2015) 167–173 169

torsional constant of the closed Delta hollow flange given as:

JΔ ¼ 4A2Hds=t

ð4Þ

in which, A is the area surrounded by the closed Delta hollowflange section. Since the hollow flange is a symmetrical crosssection, then the applied torsion is taken by the outer wall of Deltaflange and the intermediate web plate remains unstressed(Timoshenko and Gere, [14]). Therefore, ignoring the inner webplate, we have A¼bfd/2 and

Hds=t ¼ b=t1þ2c=ts which results in:

JΔ ¼ 4A2

ðbf =t1Þþð2c=tsÞð5Þ

To compute the warping constant Cw of the HDFB section, theapproximate formula given in Ref. [8] is adopted as follows:

Cwffi ∑n

j ¼ 1ðCwjþx2j Ixj þy2j Iyj Þ ð6Þ

where Ixj and Iyj are the moment of inertia of each constituentplates with respect to horizontal and vertical axis of centroid,respectively; xj and yj are the horizontal and vertical distance fromglobal centroid of DHFB section to the centroid of Delta and bottomflanges as shown in Fig. 1.; Cwj is the warping constant of the jthplate element. The values of the cross sectional properties for theconsidered DHFB section in Section 2.2 are obtained as follows:Ix¼390,298,621 mm4, Iy¼25,929,180 mm4, J¼8502,650 mm4, andCw¼8.73�1011 mm6. Detailed derivation of Cw for DHFBs using theapproximate formula Eq. (6) has been given in Appendix. Furthermore,the yield and plastic bending moment capacity of the section arecomputed as 435 and 505 kNm, respectively. Using the above cross-sectional properties, the limiting unbraced lengths Lp and Lr areestimated for the DHFB section as 2300 mm and 14,000 mm,respectively.

The DHFB cross-section was analyzed with different unbracedlengths and load heights under uniform moment and concen-trated loading conditions (totally 120 models) to estimate themoment-gradient factor in different behavioral zones.

5. Discussion of the results

Plots of the FEM beam moment resistance curves versus theunbraced length under pure bending and concentrated loading atshear center are shown in Fig. 4. The theoretical beam momentcapacity curve of the beam under pure bending (Mo(AISC)) is alsoplotted on the same figure which is in good agreement with theFEM moment capacity curve (Mo(FEM)). The deformed geometry ofmodel M5 (defined in Table 2) under pure bending has beenshown in Fig. 5 which fails due to lateral–torsional buckling mode.

As it can be seen in Fig. 4, the moment capacity of DHFB withL¼3 m under concentrated load at shear center is less than itscorresponding moment capacity under pure bending. This unusualbehavior may be due to the presence of high shear forces in shortbeams with respect to the bending moments. In other words, shortbeams fail mainly due to shear yielding/buckling instead of flexuralplastification or lateral buckling modes. Similar behavior has beenobserved previously for hollow tubular flange plate girders byHassanein and Kharoob [9].

The moment resistance variation of model M8 (for example)under pure bending versus shear center lateral displacement isalso shown in Fig. 6. As can be seen, after the maximum moment,the moment resistance decreases as the lateral displacementincreases indicating an unstable post-buckling behavior for DHFBs.

The FEM moment-gradient factors are computed in Table 2 andcompared to the corresponding factor (i.e. 1.35) given in the AISC-360 Specifications. It is seen that for all ranges of unbraced lengththe Cb factor given by AISC-LRFD is always larger than thoseobtained using the nonlinear finite element method. And also, thevalue of Cb is not constant in the whole range of beam’s slender-ness, while AISC gives a constant value of 1.35 for any range ofslenderness. It can be seen that, the ratio between the AISC Cbfactor and the factors computed using FEM analysis decreases andapproaches to unity as the unbraced length increases. In otherwords, for long elastic beams, both the computed and AISC Cbvalues coincide with each other. The use of AISC Cb factor for theelastic DHFB section, overestimates the moment resistance up to1.15, 1.18 and 1.23 times the FEM results for bottom flange, shearcenter and top flange loading conditions, respectively. Similar Plotsof the FEM beam moment resistance curves for concentratedloading at bottom and top flanges are shown in Figs. 7 and 8. Itcan be seen that, the FEM moment resistance curves of the DHFBunder moment gradient (PL/4- FEM) are located between the AISCmoment resistance curves (i.e. Mo and PL/4).

Elastic LTB load depends on the lateral flexural and warpingrigidities, i.e. EIy and ECw. However, as the unbraced length isdecreased for a beam, the extreme fibers of both tension andcompression flanges plastify reducing the flexural and warpingrigidities to the effective rigidities (EIy)e and (ECw)e [12]. In the caseof inelastic beams under non-uniform moment distribution, yield-ing takes place in the high moment region along the beam span.Due to the sever yielding of this region, a semi-plastic hinge formsagainst out-of-plane bending increasing the likelihood of the LTBin comparison to the same beam loaded in the elastic zone. On theother hand, it is evident that the inelastic LTB resistance of the

Fig. 3. Deformed geometry and failure mode of specimen SP-2.0

100

200

300

400

500

600

0 10 20 30 40 50

Mn

(KN

.m)

L (m)

Mo (Theory)

PL/4 (FEM)

Mo (FEM)

AISC

Fig. 4. Nominal moment resistance curves under pure bending and concentratedload at shear center.

A. Mohebkhah, M.G. Azandariani / Thin-Walled Structures 86 (2015) 167–173170

Table 2Cb factors for the analyzed DHFBs.

Model L (m) Mo, FEM (kN m) PL/4 (kN m) Cb, FEM (1) Cb, AISC (2) Ratio (2)/(1)

Bot. flange Shear center Top flange Bot. flange Shear center Top flange Diff. location Bot. flange Shear center Top flange

M1 3 497 505 444 436 1.02 0.90 0.90 1.35 1.32 1.5 1.5M2 4 485 498 486 459 1.03 1.00 0.95 1.35 1.31 1.35 1.42M3 5 480 495 493 475 1.03 1.03 0.99 1.35 1.31 1.31 1.36M4 6 475 492 491 479 1.04 1.03 1.00 1.35 1.30 1.31 1.35M5 7 469 491 489 483 1.05 1.04 1.03 1.35 1.29 1.30 1.31M6 8 462 490 483 478 1.06 1.04 1.03 1.35 1.27 1.30 1.31M7 9 458 489 480 475 1.07 1.05 1.04 1.35 1.26 1.28 1.30M8 10 453 486 477 472 1.07 1.05 1.04 1.35 1.26 1.28 1.30M9 11 444 484 474 471 1.09 1.07 1.06 1.35 1.24 1.26 1.27M10 12 434 479 471 466 1.10 1.08 1.07 1.35 1.23 1.25 1.26M11 13 428 477 468 464 1.11 1.09 1.08 1.35 1.22 1.24 1.25M12 14 422 473 465 459 1.12 1.10 1.09 1.35 1.20 1.23 1.24M13 15 402 472 460 442 1.17 1.14 1.10 1.35 1.15 1.18 1.23M14 20 307 389 372 363 1.27 1.21 1.18 1.35 1.06 1.11 1.14M15 25 250 327 312 304 1.31 1.25 1.22 1.35 1.03 1.08 1.11M16 30 205 269 260 254 1.31 1.27 1.24 1.35 1.03 1.06 1.09M17 35 177 232 226 220 1.31 1.28 1.24 1.35 1.03 1.05 1.09M18 40 159 209 204 200 1.31 1.28 1.26 1.35 1.03 1.05 1.07M19 45 137 182 178 176 1.33 1.30 1.28 1.35 1.01 1.04 1.05M20 50 122 169 165 164 1.38 1.35 1.34 1.35 0.98 1.00 1.01

Fig. 5. Deformed geometry and failure mode of the analyzed model M5 under purebending.

050

100150200250300350400450500

0 25 50 75 100 125 150

Mn

(KN

.m)

Lateral Displacmente (mm)

Fig. 6. Moment–lateral displacement relationship for model M8 under purebending.

0

100

200

300

400

500

600

0 10 20 30 40 50

Mn

(KN

.m)

L (m)

Mo (AISC)

PL/4 (FEM)

Mo (FEM)

PL/4 (AISC)

Fig. 7. Nominal moment resistance curves under pure bending and concentratedload at bottom flange.

0

100

200

300

400

500

600

0 10 20 30 40 50

Mn

(KN

.m)

L (m)

Mo (AISC)

PL/4 (FEM)

Mo (FEM)

PL/4 (AISC)

Fig. 8. Nominal moment resistance curves under pure bending and concentratedload at top flange.

A. Mohebkhah, M.G. Azandariani / Thin-Walled Structures 86 (2015) 167–173 171

beam is limited to the plastic moment capacity for short beams inwhich no LTB occurs. That is why the inelastic Cb factors are lessthan those for elastic beams.

Another important conclusion from Table 2 is that, load heightparameter does not significantly influence the moment gradientfactor for DHFBs. Effect of load height on elastic lateral–torsionalbuckling (LTB) of I-beams has been investigated numerically byTimoshenko and Gere [14]. However, Nethercot and Rockey [11]based on the solutions given in Timoshenko and Gere [14],proposed some approximate equations which can be used toestimate the factor Cb for such cases. The values of Cb for a centralpoint load applied are given as follows ([4,6,15]):

Cb ¼1:35B for load at bottom flange1:35 for load at shear center1:35=B for load at top flange

8><>: ð7Þ

in which the value of B is given as 1þ0.649 W–0.180 W2. Due tohigh torsional rigidity (GJ) of the studied DHFB section, the beamparameter W approaches to 0.054 resulting in B equal to 1.035 formodel M16. Therefore, based on Eq. (7), the factor Cb is computedas 1.30, 1.35 and 1.40, for load at bottom flange, shear center andtop flange positions, respectively, which are close to the FEMresults for the longest DHFB (i.e. Model M20 with L¼50 m).

6. Proposed AISC moment capacity curve for DHFBs undermoment gradient

To estimate the nominal moment capacity of elastic DHFBsunder pure bending, an equation for the variation of Cb factor (obtainedin the previous section) versus unbraced length should be derived. Tothis end, the ratio of FEM to AISCmoment gradient factors (Rf, reductionfactor) given in Table 2 for loading at shear center is plotted in Fig. 9.The figure also shows a power regression equation (r2¼0.93) for thereduction factor. Hence, for any value of unbraced length in elastic zone,the proposed average moment gradient factor for shear center loading(Cbps) can easily be computed as follows as a function of unbracedlength:

Cbps ¼ 1:35Rf ¼ 0:631L0:114 ð8Þ

Similar reduction factors can be obtained for other load heights.For a beam that buckles inelastically, AISC-LRFD [2] proposes alinear transition equation from the end of the elastic region to theplastic moment and scales it with a constant moment gradientfactor, Cb, for all ranges of inelastic beam’s slenderness as follows:

Mn ¼ Cb Mp� Mp�Mr� � L�Lp

Lr�Lp

� � rMp ð9Þ

However, referring to Fig. 4, it can be seen that the nominalmoment resistance curve of the DHFB for inelastic zone (i.e. forLp!L!Lr) approaches almost linearly to the moment capacitycurve under pure bending as the unbraced length decreases.Therefore, the predicted moment capacities of inelastic beamsusing AISC-LRFD provisions are unconservative. In order to over-come this problem, the equations given by AISC-LRFD should bemodified slightly. To this end, for the moment gradient cases it isproposed to use a straight-line transition equation between thepoints (Lp, Mp) and (Lr, CbMr) for inelastic zone as follows:

Mn ¼Mp� Mp�CbpsMr� � L�Lp

Lr�Lp

� �rMp ð10Þ

Eq. (10) is proposed to be used instead of Eq. (10) for theestimation of flexural resistance of inelastic DHFBs under momentgradient. The Cbps in Eq. (10) is computed based on Eq. (8) forunbraced length L¼Lr.

Utilizing the proposed moment gradient factor Cbps and Eqs. (1)and (8), the moment capacity curve of the DHFB under midspanconcentrated loading at shear center is plotted in Fig. 10. The FEM andoriginal AISC moment capacity curves are also plotted on the samefigure for comparison. It can be seen that the moment resistancecurve predicted by the proposed AISC formula (i.e. Eq. (10)) is in goodagreement with the FEM capacity curve.

7. Conclusions

The nonlinear inelastic lateral–torsional buckling of DHFBswith a wide variety of overall slenderness under moment gradientwas studied by means of the finite element method. The main aimwas to investigate the effects of unbraced length and central off-shear center loading on the moment gradient factor in differentbehavioral zones of steel beams. It was found that the Cb factorsgiven in [2] are valid just for long-span DHFBs (i.e. elastic beams).The FEM predicted Cb factors for inelastic and short plastic DHFBsare lower than the Cb factors given by [2]. Therefore, a modifiedmoment gradient factor (Cbp) as a function of unbraced length wasproposed for such beams. Furthermore, it was observed that theAISC moment capacity curve is unconservative for inelastic DHFBs.Therefore, a straight-line transition equation proposed betweenthe points (Lp, Mp) and (Lr, CbpMr) to be used instead of thecorresponding AISC equation for DHFBs under the moment gra-dient case considered in this paper. However, owing to the limitednumber of beams studied herein, these findings are not yetconclusive and further numerical and experimental studies must

Rf= 0.631L0.114

R² = 0.927

0

0.2

0.4

0.6

0.8

1

10 20 30 40 50

Rf

L (m)

Shear Center

Power (Shear Center)

Fig. 9. Variation of reduction factor versus unbraced length for the elastic DHFBsunder shear center loading.

0

100

200

300

400

500

600

0 10 20 30 40 50 60

Mn

(KN

.m)

L (m)

Mo (AISC)

PL/4 (AISC- Proposed)

PL/4 (AISC)

PL/4 (FEM)

Fig. 10. Comparison of the FEM, original AISC and proposed AISC moment capacitycurves of the DHFB under concentrated loading at loading at shear center.

A. Mohebkhah, M.G. Azandariani / Thin-Walled Structures 86 (2015) 167–173172

be carried out to validate the results and the applicability of theproposed transition equation for other moment gradient cases.

Appendix

Derivation of Cw for DHFBs

Since warping for narrow rectangular sections and hollowsections composed of rectangular plate elements are usuallynegligible [4], so Cwj may be taken equal to zero for bottom andDelta hollow flange of DHFBs. Also, for bottom and Delta flanges,xj’s are equal to zero due to the section symmetry about y-axis.Therefore, Eq. (6) for computing Cw reduces to:

Cwffi ∑2

j ¼ 1ðy2j Iyj Þ ¼ a2IΔyþb2If y

¼ a2t1b

3f

12þ b3f12

� twsin θ

!þb2

t2b3f

12

in which IΔy and If y are the moment of inertia of Delta and bottomflanges with respect to vertical axis of centroid, respectively; a andb are the vertical distances from global centroid of DHFB section tothe centroid of Delta and bottom flanges as shown in Fig. 1. Thevalue of Cw for the considered DHFB cross-section given in Section2.2 is computed as follows:

b¼ y¼ ∑Aiyi∑Ai

¼ ðht3wÞðh=2ÞÞþ ðbf t1ÞðhÞþ2ctsðh�d=2Þbf ðt1 þ t2Þþhtw

� 189 mma¼ h�b�d=3¼ 177:7 mm

) Cwffi177:72 13:5� 1803

12þ1803

12� 8:60:67

!þ177:7227� 1803

12

¼ 8:73� 1011 mm6

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