Journal of Constructional Steel Research 72 (2012) 254–260

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Journal of Constructional Steel Research

Bending analysis of partially restrained channel-section purlins subjectedto up-lift loadings

Chong Ren a, Long-yuan Li b,⁎, Jian Yang a

a School of Civil Engineering, University of Birmingham, Birmingham, UKb School of Marine Science and Engineering, University of Plymouth, Plymouth, UK

⁎ Corresponding author. Tel.: +44 1752 586 180; faxE-mail address: Long-yuan.Li@plymouth.ac.uk (L. Li)

0143-974X/$ – see front matter © 2012 Elsevier Ltd. Aldoi:10.1016/j.jcsr.2012.01.001

a b s t r a c t

a r t i c l e i n f o

Article history:Received 14 October 2011Accepted 3 January 2012Available online 28 January 2012

Keywords:Cold-formed steelChannelBendingUplift loadingRoof-purlin system

Cold-formed steel section beams are widely used as the secondary structural members in buildings tosupport roof and side cladding or sheeting. These members are thus commonly treated as the restrainedbeams either fully or partially in its lateral and rotational directions. In this paper an analytical model is pre-sented to describe the bending and twisting behaviour of partially restrained channel-section purlins whensubjected to uplift loading. Formulae used to calculate the bending stresses of the roof purlins are derivedby using the classical bending theory of thin-walled beams. Detailed comparisons are made between the pre-sent model and the simplified model proposed in Eurocodes (EN1993-1-3). To validate the accuracy of thepresent model, both available experimental data and finite element analysis results are used, from whichthe bending stress distributions along the lip, flange and web lines are compared with those obtained fromthe present and EN1993-1-3 models.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Thin-walled, cold-formed steel sections are widely used in build-ings as sheeting, decking, purlins, rails, mezzanine floor beams, latticebeams, wall studs, storage racking and shelving. Among these prod-ucts, purlins and rails are the most common members, widely usedin buildings as the secondary members supporting the corrugatedroof or wall sheeting and transmit the force to the main structuralframe. Roof purlins and cladding rails have been considered to bethe most popular products and account for a substantial proportionof cold-formed steel usage in buildings.

In the UK, most common sections are the zed, channel and sigmashapes, which may be plain or have stiffened lips. The lips are smalladditional elements at free edges in a cross section, and so added toprovide the structural efficiency under compressive loads [1]. Roofpurlins and sheeting rails are usually restrained against lateral move-ment by their supported roof or wall cladding. Such restraints reducethe potential of lateral buckling of the whole section, but do not nec-essarily eradicate the problem [2]. For example, roof purlins are gen-erally restrained against lateral displacement by the cladding, butunder wind uplift, which induces compression in the unrestrainedflange, lateral-torsional buckling is still a common cause of failure[3]. This occurs due to the flexibility of the restraining cladding andto the distortional flexibility of the section itself, which permits lateral

: +44 1752 586 101..

l rights reserved.

movement to occur in the compression flange even if the other flangeis restrained.

Several researchers have investigated the behaviour of the roofpurlins with partial restraints provided by their supported claddingor sheeting. For example, Lucas et al. investigated the interactionbetween the sheeting and purlins using finite element analysismethods [4,5]. Ye et al. presented several examples to demonstratethe influence of sheeting on the bending [6], local and distortionalbuckling behaviour [7] of roof purlins. Vieira et al. provided simplifiedmodels to predict the longitudinal stresses when the channel-sectionpurlin is subjected to uplift loading [8]. The lateral-torsional bucklingof purlins subjected to downwards and/or upwards loadings has alsobeen discussed by several researchers [9–13]. Analytical models havebeen developed to predict the critical loads of lateral-torsional bucklingand the influence of sheeting on the lateral-torsional buckling behav-iour of roof purlins [12–14]. Experimental tests have also been per-formed on both bridged and unbridged zed- and channel-sectionpurlins under uplift loads [15,16]. Calculation models for predictingthe rotational restraint stiffness of the sheeting have been proposed re-cently [17,18]. Design specifications for the purlin-sheeting systemhavebeen provided in Eurocodes [3].

In this paper an analytical model is presented to describe thebending and twisting behaviour of the partially restrained channel-section purlins when subjected to uplift loading. The classical bendingtheory of thin-walled beams is used to calculate the bending stressesof the roof purlins. In order to validate the model, both availableexperimental data and finite element analysis results are used, fromwhich the bending stress distributions along the lip, flange and

255C. Ren et al. / Journal of Constructional Steel Research 72 (2012) 254–260

web lines are compared with those obtained from the present andEN1993-1-3 models.

2. Analytical model

Consider a channel section that is partially restrained by the sheet-ing on its upper flange. When the member is subjected to a uniformlydistributed uplift load acting on the middle line of the upper flange,the restraint of the sheeting to the member can be simplified as a lat-eral restraint and a rotational restraint. For most types of sheeting thelateral restraint is sufficiently large and therefore the lateral displace-ment at the upper flange-web junction may be assumed to be fullyrestrained. The rotational restraint, however, is dependent on thedimensions of sheeting and purlin, number, type and positions ofthe screws used in the fixing. If the stiffness of the rotational restraintprovided by the sheeting is known, then the purlin-sheeting systemmay be idealized as a purlin with lateral displacement fully restrainedand rotation partially restrained at its upper flange-web junction asshown in Fig. 1.

Let the origin of the coordinate system (x,y,z) be the centroid ofthe channel cross-section, with x-axis being along the longitudinaldirection of the beam, and y- and z-axes taken in the plane of thecross-section, as shown in Fig. 1. According to the bending and torsiontheory of beams [1,19], the equilibrium equations, expressed in termsof displacements, are given as follows,

EIzd4vdx4

¼ qy ð1Þ

EIyd4wdx4

¼ qz ð2Þ

EIwd4ϕdx4

−GITd2ϕdx2

þ kϕϕ ¼ zkqy−yqqz ð3Þ

where v and w are the y- and z-components of displacement of thecross-section defined at the shear centre, ϕ is the angle of twistingof the section, E is the modulus of elasticity, G is the shear modulus,Iy and Iz are the second moments of the cross-sectional area abouty- and z-axes, Iw is the warping constant, IT is the torsion constant,kϕ is the per-unit length stiffness constant of the rotational spring,qy and qz are the densities of the uniformly distributed loads in y-and z-directions, zk is the vertical distance from the shear centre tothe force line qy, and yq is the horizontal distance from the shear cen-tre to the force line qz.

Fig. 1. Analytical model used for a channel-section purlin-sheeting system.

Using Eq. (1) to eliminate qy and Eq. (2) to eliminate w in Eq. (3),it yields,

EIwd4ϕdx4

−GITd2ϕdx2

þ kϕϕ−zkEIzd4vdx4

¼ −yqqz ð4Þ

Note that, the lateral displacement restraint applied at the upperflange-web junction requires,

zkϕþ v ¼ 0 ð5Þ

Using Eq. (5) to eliminate the angle of twisting, ϕ, in Eq. (4), ityields,

Iz þIwz2k

!d4vdx4

−GITEz2k

d2vdx2

þ kφEz2k

v ¼ yqqzEzk

ð6Þ

Let

a0 ¼ kϕEz2k

ð7Þ

a1 ¼ GITEz2k

ð8Þ

a2 ¼ Iz þIwz2k

ð9Þ

With the use of Eqs. (7)–(9), Eq. (6) can be rewritten into,

a2d4vdx4

−a1d2vdx2

þ a0v ¼ yqqzEzk

ð10Þ

Eq. (10) is a fourth-order differential equation, which, for givenboundary conditions, can be solved analytically.

3. Calculation of bending stresses

The longitudinal stress at any point on the cross-section generatedby the two displacement components and warping can be calculatedas follows [1],

σx x; y; zð Þ ¼ −Eyd2vdx2

−Ezd2wdx2

þ E �ω−ωð Þd2ϕdx2

ð11Þ

where ω is the sectorial coordinate with respect to the shear centreand �ω is the average value of ω. The first term in the right handside of Eq. (11) is the stress generated by the deflection of the beamin horizontal direction, the second term is the stress generated bythe deflection of the beam in vertical direction, and the third term isthe warping stress.

Using Eq. (5) to eliminate ϕ, Eq. (11) can be rewritten into,

σx x; y; zð Þ ¼ −Ezd2wdx2

−E yþ �ω−ωzk

� �d2vdx2

ð12Þ

Eq. (12) indicates that the total longitudinal stress in the beam canbe decomposed into two parts. One is the stress that is generated byload qz when the beam is considered to be fully restrained in rotationand can be calculated as follows,

σx1 x; y; zð Þ ¼ −Ezd2wdx2

ð13Þ

Fig. 2. Bending stress calculated in EN1993-1-3. Upper: Compression part consisting oflip and flange plus 1/5 of web length. Lower: Stress distribution due to the bendingabout z* axis.

4 6 8 10 12 14

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Beam length, m

Mom

ent c

orre

ctio

n fa

ctor

, κR

Mom

ent c

orre

ctio

n fa

ctor

, κR

Present solution, kφ=10 N

EN1993-1-3, kφ=10 N

EN1993-1-3*, kφ=10 N

Present solution, kφ=100 N

EN1993-1-3, kφ=100 N

EN1993-1-3*, kφ=100 N

Present solution, kφ=1000 N

EN1993-1-3, kφ=1000 N

EN1993-1-3*, kφ=1000 N

4 6 8 10 12 14

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Beam length, m

(a)

(b)

Fig. 3. Comparison of moment correction factors between the present and EN1993-1-3models. In EN1993-1-3 K is calculated using Eq. (40), while in EN1993-1-3* K=k

ϕ/h2 is

used, which ignores the section distortion in the spring stiffness used in EN1993-1-3model. (a) (h=120 mm, b=50mm, c=15mm, t=1.5 mm, a=b/2). (b) (h=400 mm,b=100 mm, c=30mm, t=2.5 mm, a=b/2).

256 C. Ren et al. / Journal of Constructional Steel Research 72 (2012) 254–260

The other is the stress that is generated by the lateral deflection ofthe beam and can be calculated as follows,

σ x2 x; y; zð Þ ¼ −E yþ �ω−ωzk

� �d2vdx2

ð14Þ

For a simply supported beam in both y- and z-directions the bend-ing stress σx1 can be expressed as follows,

σ x1l2; y; z

� �¼ zMy;max

Iyð15Þ

where My;max ¼ qzl2

8is the largest moment of the beam bent about

y-axis and l is the length of the beam. By solving the differentialEq. (10) analytically and then substituting the solution into Eq. (14), ityields,

(1) Rotation partially restrained case, when kϕ≠0

σ x2l2; y; z

� �¼ yþ �ω−ω

zk

� � A β21 þ β2

2

� �2β1β2 A2 þ B2

� �0B@

1CA yqqz

2zka0ð16Þ

(2) Rotation free case, when kϕ=0

σ x2l2; y; z

� �¼ yþ �ω−ω

zk

� �8yqzkEMy;max

l2GITsech

l2

ffiffiffiffiffia1a2

r� �−1

ð17Þ

where A, B, β1 and β2 are the constants defined as follows,

A ¼ sinβ2l2

sinhβ1l2

ð18Þ

B ¼ cosβ2l2

coshβ1l2

ð19Þ

β1 ¼ 12

ffiffiffiffiffia0a2

rþ 14a1a2

� �12 ð20Þ

β2 ¼ 12

ffiffiffiffiffia0a2

r−1

4a1a2

� �12 ð21Þ

For given dimensions of a section, one can use Eqs. (16) or (17) tocalculate the longitudinal stress at any coordinate point (y, z).

4. Comparison with EN1993-1-3

In the Eurocodes [3] the bending stress of the channel-section pur-lin of depth h, flange width b, lip length c, and thickness t, with lateraldisplacement fully restrained and rotation partially restrained at theupper flange-web junction is calculated based on the two parts ofbending. One is the beam bent only in the plane of the web, inwhich case the stress is calculated exactly the same as that given inEq. (15). The other is the compression part of the section, consistingof the lip and flange plus 1/5 of web height from the compressionflange, bent about an axis parallel to z-axis, in which case the stressis calculated as follows,

σx2l2; y; z

� �¼ k�RkhMy;max

Ifzy� ¼ k�RkhMy;max

Ifzyþ �yð Þ ð22Þ

10 20 30 40 50 60 701

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

h2/(bc)

I eq/I fz

y = -9.03/x 2 + 3.47/x + 1.133

Fig. 4. Ratio of the equivalent second moments of cross-section area used in the present(Ieq) and EN1993-1-3 (Ifz) models for 60 channel-sections.

0 20 40 60 80 100

-2

-1

0

1

2

3

4

Lip, flange and 1/5 of web lines, mm

Str

ess

ratio

, σx2

/abs

(σx1

) S

tres

s ra

tio, σ

x2/a

bs(σ

x1)

Present solutionEN1993-1-3

(a)

0 20 40 60 80 100 120 140 160 180 200-2

-1

0

1

2

3

4

Lip, flange and 1/5 of web lines, mm

Present solutionEN1993-1-3

(b)

Fig. 5. Bending stress distribution along the lip, flange and web lines (x-axis starts fromthe tip of lip and ends at the 1/5 of the web length). (a) Section of h=120 mm,b=50 mm, c=15 mm, t=1.5 mm and a=b/2. (b) Section of h=400 mm,b=100 mm, c=30 mm, t=2.5 mm, and a=b/2.

257C. Ren et al. / Journal of Constructional Steel Research 72 (2012) 254–260

where kR⁎ is the moment correction factor considering the influence ofthe rotation restraint, defined as

κ�R ¼ 1−0:0225R

1þ 1:013Rð23Þ

R ¼ Kl4

π4EIfzð24Þ

K ¼4 1−ν2� �

h2 hþ bmodð ÞEt3

þ h2

kφ

0@

1A

−1

ð25Þ

kh ¼ −yq2zk

ð26Þ

Ifz is the second moment of the cross-section area of the compres-sion part about z*-axis as defined in Fig. 2, �y is the horizontal distancebetween z- and z*-axes, v is Poisson's ratio, bmod=a is for cases wherethe equivalent lateral force (−khqz) bringing the purlin into contactwith the sheeting at the purlin web, or bmod=2a+b is for caseswhere the equivalent lateral force (−khqz) bringing the purlin intocontact with the sheeting at the tip of the purlin flange, and a is thehorizontal distance from the web line to the sheeting-purlin fixingpoint (in the present case a=b/2).

For the convenience of comparison, Eq. (16) is now rewritten intothe following format

σx2l2; y; z

� �¼ kRkhMy;max

Ieqyþ �ω−ω

zk

� �ð27Þ

in which,

kR ¼ 8Ieqa0l

2

A β21 þ β2

2

� �2β1β2 A2 þ B2

� �0B@

1CA ð28Þ

Fig. 6. (a) Finite element mesh employed for the analysis of a channel section purlin.(b) The deformed shape of the partially restrained purlin under uplift loading(h=200 mm, b=75 mm, c=20 mm, t=2.0 mm, a=b/2, kϕ=300N).

258 C. Ren et al. / Journal of Constructional Steel Research 72 (2012) 254–260

Ieq ¼14

Iz þIwz2k

!ð29Þ

The comparison of Eqs. (22) and (27) shows that, there are three dif-ferences between the presentmodel and the EN1993-1-3model for cal-culating the longitudinal stress. The first is the moment correction

0 50 100 150-5

0

5Present solutionEN1993-1-3FEA

0 50 100 150

-2

0

2

0 50 100 150

-2

0

2

Lip, flange and half-web lines, mm

(a) L=4000mm

0 50 100 150-5

0

5Present solutionEN1993-1-3FEA

0 50 100 150-1.5

-1

-0.5

0

0 50 100 150-1.5

-1

-0.5

0

Lip, flange and half-web lines, mm

(c) L=8000mm

Fig. 7. Bending stress distribution along the lip, flange and web lines (x-axis starts from tt=2.0 mm, a=b/2). Top: kϕ=0. Middle: kϕ=300N. Bottom: kϕ=750N.

factor used in calculating the lateral bending moment. The second isthe second moment of cross-section area used to calculate the bendingstress. The third is the lateral coordinate used to calculate the lateralbending stress (or the position of the neutral axis).

Fig. 3 shows the comparisons of the moment correction factorscalculated from the present and EN1993-1-3 models. It can be seenfrom the figure that, the moment correction factor decreases with

0 50 100 150-5

0

5

0 50 100 150-2

-1

0

1

0 50 100 150-1.5

-1

-0.5

0

0.5

Lip, flange and half-web lines, mm

0 50 100 150-5

0

5

0 50 100 150-1.5

-1

-0.5

0

0 50 100 150-1.5

-1

-0.5

0

Lip, flange and half-web lines, mm

(b) L=6000mm

(d) L=12000mm

he tip of lip and ends at the half of web length, h=200 mm, b=75 mm, c=20 mm,

0 20 40 60 80 100 120 140 160

-1.2

-1.1

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

Lip, flange and half-web lines, mm

Present solutionEN1993-1-3FEARousch&Hancock test

(a)

(b)

0 1000 2000 3000 4000 5000 6000 7000-30

-25

-20

-15

-10

-5

0

5

Dis

plac

emen

t, m

m

Beam Length, mm

Present solutionEN1993-1-3FEARousch&Hancock test

Fig. 8. (a) Bending stress distribution along the lip, flange and web lines (x-axis startsfrom the tip of lip and ends at the half of the web length). (b) Lateral deflection atlower flange-web junction (h=202 mm, b=76.7 mm, c=20.8 mm, t=1.51 mm).

259C. Ren et al. / Journal of Constructional Steel Research 72 (2012) 254–260

the increase of either beam length or the stiffness of rotational spring.The rate of the decrease is found to be faster in a small section than ina large section. It can also be observed from the figure that the mo-ment correction factor calculated using the EN1993-1-3 model ishigher than that calculated using the present model, particularlywhen the stiffness of the rotational spring is very small or very large.

Note that the main difference in the moment correction factorsbetween the present and EN1993-1-3 models is the torsional rigidity[21]. In the present model the torsional rigidity is included, whichmakes the moment correction factor smaller, particularly for caseswhere the stiffness of the rotational spring is small. In the EN1993-1-3 model, the torsional rigidity is not included. However, its springstiffness, that is Eq. (25), takes into account the influence of the sec-tion distortion, which makes the moment correction factor larger,particularly for the case where the stiffness of the torsional spring islarge. This is why more difference in the moment correction factorsbetween the two models is found in the case where the stiffness ofrotational spring is either very large or very small. If the section dis-tortion is ignored in the spring stiffness in EN1993-1-3 model (thatis, EN1993-1-3* plotted in Fig. 3 and the corresponding results arerepresented by the plus, x-mark, and star symbols), then the momentcorrection factor predicted by EN1993-1-3 model is found to behigher for the small stiffness of rotational spring but lower for thelarge stiffness of rotational spring than that calculated using the pre-sent model.

Fig. 4 shows the variation of the ratio of the equivalent secondmoment of cross-section area Ieq used in the present model and thatIfz used in EN1993-1-3 model for 60 channel-sections of differentsizes produced by a UKmanufacturer [20]. The justification of the def-inition of Ieq can be found in [21]. The figure shows that for all sectionsthe second moment of cross-section area employed in the EN1993-1-3 model is smaller than the equivalent second moment used in thepresent model. This indicates that the bending stress calculatedusing EN1993-1-3 model will be higher. Note that the increase inweb length involved in Ifz does not simply scale down or scale upthe bending stress in the flange. This is because although it increasesthe value of Ifz, it also changes the position of the neutral axis and thusalters the stress distribution pattern in the flange. In literature there issome argument on how to choose suitable web length to be includedin the beam-column model to calculate the out-of-plane bendingstress. In the Australia design codes [16], for example, it is suggestedto use 35% of the total web length, instead of the 20% of the total weblength as used in EN1993-1-3, in the beam-column model.

The third difference between the present and EN1993-1-3 modelsis the coordinates used to calculate the bending stress in Eqs. (22) and(27). EN1993-1-3 model does not take into account the warpingstress, whereas the present model does. Fig. 5 shows a comparisonof the bending stresses generated by the lateral displacement due tothe lateral load -khqz along the lip, flange and web lines, obtainedfrom the two models, which shows a combined influence of the sec-ond moment of cross-section area and the coordinate. For the sim-plicity of comparison, the correction factor in both models is notapplied (that is, kR=kR⁎=1) and the stresses are normalised byusing the maximum value of σx1. It can be seen from the figure thatthe EN1993-1-3 model predicts higher stresses in the lip, web, andmost part of the flange than the present model does. It is only in thesmall part near the flange–web junction of the large section wherethe stress predicted by the EN1993-1-3 is slightly lower than thatpredicted by the present model. This implies that the EN1993-1-3model is more conservative in predicting the bending stresses.

5. Validation of the present model

The present model is validated by using both finite element anal-ysis results and available experimental data. The finite element modelemployed here is very similar to what we used for the zed-section

[21]. Fig. 6 shows a typical mesh employed in the analysis and thedeformed shape of the partially restrained channel-section purlinunder a uniformly distributed uplift load (a half of the beam). Theexperimental data are taken from Hancock and his colleagues reports[15,16], who described a series of tests on simply supported channelsection purlins with screw-fastened sheeting under wind uplift load-ing, performed in a vacuum test rig.

Fig. 7 shows the detailed comparison of the total bending stressesobtained from the finite element analysis, the present model and theEN1993-1-3 model. It can be seen from the figure that, the resultspredicted by the present model agree very well with the finite ele-ment analysis results. While for most of cases the results providedby the EN1993-1-3 model are over conservative, particularly whenthe stiffness of the rotational spring is very small. Since the stressesplotted in Fig. 7 are normalised by the maximum bending stress inthe web plane, the variation of the stress along the compressionflange reflects the contribution of the bending stress due to lateral de-flection. It can be found from the figure that, the lateral bendingreduces the bending stress (or alters the stress direction from com-pression to tension) near the flange and lip junction but increasesthe bending stress near the flange and web junction. The extent of thereduction or increase in stresses is dependent on the stiffness of therotational spring. The larger the stiffness of the rotational spring, thesmaller the stress contributed by the lateral bending. It is noticed fromFig. 7 that, the contribution of the lateral bending is very significant to

260 C. Ren et al. / Journal of Constructional Steel Research 72 (2012) 254–260

the total longitudinal bending stress, particularly when the stiffness ofthe rotational spring is small.

Fig. 8 shows the comparison of total bending stresses and lateraldeflections of the compression flange obtained from different models.Again, the good agreement of the present model with the experimen-tal data demonstrates that the linear bending model with taking intoaccount the warping torsion can provide a good prediction for thebending stresses of the partially restrained channel-section purlinsubjected to uplift loading. In contrast, the bending stress providedby EN1993-1-3 model is likely over-predicted.

6. Conclusions

This paper has presented an analytical model which can describethe bending and twisting behaviour of the partially restrainedchannel-section purlins when subjected to uplift loading. Formulaeto calculate the bending stresses of roof purlins have been derivedusing the classical bending theory of thin-walled beams. Detailedcomparisons of bending stresses obtained from the finite elementanalysis, available experimental data, the present model and theEN1993-1-3 model are provided, which demonstrates that the linearbending model with taking into account warping torsion can providegood prediction for the bending stresses of the sheeting-purlin sys-tem. The results obtained from the present model have also shownthat the longitudinal stress induced by the lateral bending is signifi-cant for channel-section purlins. This additional stress may changethe failure modes from lateral-torsional buckling to local or distor-tional buckling.

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