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Design rules for out-of-plane stability of roller bent steel arches with FEMR.C. Spoorenberg a, , H.H. Snijder a, J.C.D. Hoenderkamp a, D. Beg ba Eindhoven University of Technology, Faculty of Built Environment, P.O. Box 513, 5600 MB Eindhoven, The Netherlandsb University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova 2, 1000 Ljubljana, Slovenia

a b s t r a c ta r t i c l e i n f o

Article history:Received 14 May 2012Accepted 25 July 2012Available online xxxx

Keywords:Roller bent archOut-of-plane bucklingFinite element modelingDesign rules

This paper describes a numerical investigation into the out-of-plane buckling behavior of freestanding rollerbent steel arches. As roller bent arches have structural imperfections which differ considerably from those of hot-rolled or welded sections, speci c attention is paid to their inclusion in the numerical model. Sensitivityanalyses are performed to assess the in uence of the imperfections due to roller bending on the out-of-planebuckling response. The accuracy of the nite element model is checked by comparing the results with earlierperformed experiments as presented in a related paper. The nite element model is able to replicate thestructural behavior displayed by the experiments with good accuracy. A database is created with elastic

plastic buckling loads for a large number of freestanding roller bent arches. The numerical data is analyzedand presented in a so-called imperfection parameter diagram from which imperfection parameter curvesare derived. The imperfection parameter curves are substituted into the European column curve formulation,leaving the original column curve formulation unaffected but extending its applicability to the out-of-planebuckling response of roller bent arches. The column curve with proposed imperfection parameter expres-sions can be used to check the out-of-plane buckling response of a roller bent steel arch with knownnon-dimensional slenderness.

2012 Elsevier Ltd. All rights reserved.

1. Introduction

The application of roller bent steel has seen a steady increase inthe construction industry over the past decades. Ease of manufactur-ing make roller bending a suitable method for achieving curved struc-tures. Roller bent steel is often applied in circular arch structureswhere its primary function lies in carrying the acting loads to theabutments. The loads are resisted by means of a combination of com-pression and bending, making the member susceptible to buckling.When local buckling is not considered, arch instability can besubdivided into three different categories: snap-through buckling(Fig. 1(a)), in-plane buckling ( Fig. 1(b)) and out-of-plane buckling(Fig. 1(c)). The latter occurs when an arch has no lateral bracingand is considered freestanding . This paper presents a study of thestructural performance of freestanding circular roller bent steelarches by means of the nite element method, for which out-of-plane buckling is the governing failure mode. The performance of the nite element model is veri ed through comparison with exper-imental results as reported in a related paper, La Poutr et al. [1].

1.1. Previous studies on out-of-plane arch buckling

The earliest theoretical studies on out-of-plane arch buckling onlyconsidered elastic buckling where material non-linearities and im-perfections were ignored. Valuable contributions were published byTimoshenko and Gere [2] and Vlasov [3] who provided formulae toapproximate the elastic out-of-plane buckling load of freestandingarches. Further re nements to calculation procedures for approxi-mating the elastic buckling load were proposed by Vacharajittiphanand Trahair [4], Yoo [5] and Rajasekaran and Padmanabhan [6].

The necessity to include material non linearities and imperfec-tions to obtain an accurate representation of out-of-plane bucklingbehavior of arches was recognized in Japan by the end of the 1970s.Research studies included experiments conducted on arches withsquare hollow sections by Sakimoto et al. [7] and welded I-sectionsby Sakata and Sakimoto [8] supplemented with nite element analy-ses by Komatsu and Sakimoto [9], Sakimoto and Komatsu [10] andSakimoto and Komatsu [11] . These Japanese research studies culmi-nated in design rules. For the calculation of the slenderness, thearch was treated as a straight column under uniform compressionwith identical cross-section, where the arch length corresponded withthe column length. Column curves were proposed by Sakimoto et al.[12] and Sakimoto and Sakata [13] to allow a check of the out-of-planearch stability. Their applicability was limited to arches with square hol-low sections and rise-to-span ratios between 0.1 and 0.2.

As the Japanese design provisions treated the out-of-plane archbuckling case identically to that of a column, the rise-to-span ratio

Journal of Constructional Steel Research 79 (2012) 9 21

Corresponding author at: Eindhoven University of Technology, Faculty of BuiltEnvironment, Den Dolech 2, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.Tel.: +31 40 247 2948; fax: +31 40 245 0328.

E-mail address: r.c.spoorenberg@bwk.tue.nl (R.C. Spoorenberg).

0143-974X/$ see front matter 2012 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.jcsr.2012.07.027

Contents lists available at SciVerse ScienceDirect

Journal of Constructional Steel Research

http://dx.doi.org/10.1016/j.jcsr.2012.07.027http://dx.doi.org/10.1016/j.jcsr.2012.07.027http://dx.doi.org/10.1016/j.jcsr.2012.07.027mailto:r.c.spoorenberg@bwk.tue.nlhttp://dx.doi.org/10.1016/j.jcsr.2012.07.027http://www.sciencedirect.com/science/journal/0143974Xhttp://www.sciencedirect.com/science/journal/0143974Xhttp://localhost/var/www/apps/conversion/tmp/scratch_9/Unlabelled%20imagehttp://dx.doi.org/10.1016/j.jcsr.2012.07.027http://localhost/var/www/apps/conversion/tmp/scratch_9/Unlabelled%20imagemailto:r.c.spoorenberg@bwk.tue.nlhttp://dx.doi.org/10.1016/j.jcsr.2012.07.027

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of the arch was considered to be of minor importance. However,earlier theoretical studies revealed that the rise-to-span ratio canhave a signi cant effect on arch buckling. This was recognized byPapangelis and Trahair [14] who performed experiments on archbuckling. These experiments were used to validate an in-house niteelement code developed by Pi and Trahair [15] from which designrules for arch buckling were developed and proposed. Pi and Trahair[16] stated that for pin-supported arches subjected to radial loadingand simply-supported arches subjected to uniform bending, theAustralian column curves were considered suitable for the design of arches failing by out-of-plane buckling. For out-of-plane xed archesa similar approach was used; Pi and Bradford [17] . According to Piand Trahair [16] and Pi and Bradford [17] the arch slenderness wasde ned as the square root of the ratio between the plastic capacityand out-of-plane elastic buckling load, taking implicitly into accountthe geometric properties of the arch. For arches subjected to verticalloading, interaction formulae were proposed to check the out-of-plane stability. These interaction formulae are analogous to those of a beam-column failing by elastic plastic buckling. The interactionformulae were valid for out-of-plane simply supported arches, Piand Trahair [18] and out-of-plane xed arches, Pi and Bradford [17].

1.2. Scope and aims

It is clear that the out-of-plane buckling behavior of steel archeshas received large attention, comprising analytical, numerical andexperimental studies. However, investigations involving materialnon-linearities and imperfections were limited to either weldedbox-sections or wide- ange sections for which the in uence of theroller bending process on the structural properties of the arch wasnot taken into account. Earlier experiments and nite element analy-ses by the authors have shown the in uence of the roller bending

process on the structural properties of wide ange sections. Usingthe existing design rules to check the out-of-plane buckling responseof freestanding arches without taking into account the in uence of the roller bending process can lead to either conservative orunconservative designs. The main goal of this paper is two-fold: pro-viding numerical modeling techniques for roller bent steel membersand suggesting design rules for out-of-plane buckling of arches with

nite element analyses. The European column curve formulation asdescribed in EN 1993-1-1 [19] (Eurocode3) will be adapted to includethe out-of-plane buckling of arches. The results from 3 different niteelement analysis types will be used to express the numerical data inthe column curve diagram to arrive at a design rule. Arches can besubjected to a wide range of loading types. In general a distinctionis made between symmetric and unsymmetric loads. The present -nite element study is limited to the former one. Design rules are pro-posed for a total of four load cases as shown in Fig. 2. The archgeometry is shown in Fig. 2(a), where S is the arch length, R the radi-us, 2 is the subtended angle, L is the span and f is the rise. Archeswith two opposite end moments ( Fig. 2(a)) and a radially distributedload ( Fig. 2(b)) are rather academic load cases. These load cases servefor comparison with other load cases, as the internal forces are limit-ed to uniform bending or uniform compression for an arch under twoopposite end moments or a radially distributed load, respectively.Arches with a central point load ( Fig. 2(c)) or uniformly distributedload ( Fig. 2(d)) display a combination of internal bending momentsand compressive forces. During the loading phase, the loads will un-dergo no directional change and are hence termed gravity loading.The present study is limited to circular I-section arches made fromeither steel grade S235 or steel grade S355 bent about their majoraxis through the roller bending process.

With the exception of arches subject to two opposite end mo-ments, all arches are pin-supported, preventing outward spreading

(a) Snap-through (b) In-plane buckling (c) Out-of-plane buckling

F F F F

Fig. 1. Global instability phenomena for arches.

S

2

M M

q

F q

(a) Two opposite endmoments - Uniform bending

(b) Radially distributed load - Uniformcompression

(c) Central point load (d) Uniformly distributed load

L

f R

Fig. 2. Load cases under investigation.

10 R.C. Spoorenberg et al. / Journal of Constructional Steel Research 79 (2012) 9 21

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of the support and introducing compressive action in the arch-rib.Arches under uniform bending need to be simply supported, aspin-ended support conditions would induce a small normal force.No in-plane xed boundary conditions are considered. For all archesthe torsional degree of freedom is xed at the support (fork support).From preliminary nite element analyses it was found that archeswith out-of-plane pin-ended conditions provided little resistance toout-of-plane buckling. In order to compensate for this and hence

consider arches which can sustain suf cient loads before failing byout-of-plane buckling all arches are xed out-of-plane at the supportsand warping deformations are prevented (restrained warping). Thebehavior of the supports with respect to the local coordinate systemof the arch is outlined in Section 2.3 .

2. Finite element modeling

The general purpose nite element package ANSYS v.11.0 is usedto perform the numerical computations. All analyses are performedin the implicit environment.

2.1. Types of analysis

For each arch con guration three types of analysis are necessaryto obtain the buckling response and to plot the nite element resultsin the column curve diagram: Linear Buckling Analysis (LBA), Materi-al Non-linear Analysis (MNA) and Geometrical Material Non-linearImperfect Analysis (GMNIA). An additional Geometrical Non-linearImperfect Analysis (GNIA) was performed to obtain the load corre-sponding to the onset of yielding. It should be noted that a LBAproduces a simpli ed representation of buckling behavior of arches,while GNIA and GMNIA are able to account for more complex buck-ling behavior of arches. The loads will be represented by the so-calledload ampli ers . The load ampli er is the load divided by the appliedloading, resulting in a load parameter independent of the type of loading (point load, uniformly distributed load or end moments).For example, for a GMNIA the ultimate load ampli er for an arch sub- ject to a central point load can be obtained as follows:

ult F ult =F : 1

Where:

ult is the ultimate load ampli er obtained by GMNIA;F ult is the maximum central point load obtained from the

load-de ection characteristics;F is the central point load applied on the arch.As F is always equal to unity, F ult is equal to ult . Analogous to ult ,

cr and pl can be computed from F cr and F pl , respectively.

2.2. Model

The arch is meshed with shell elements. The selected SHELL181element is a four-node shell element which permits inclusion of residual stresses and is capable of handling large rotations and largestrains. A reduced integration scheme has been adopted, featuring asingle integration point over the surface and 5 integration pointsover the shell thickness to capture growth of plastic zones.

2.3. Boundary conditions and loading

The boundary conditions and loading types comply with those de-scribed in Section 1.2 . Fig. 3 shows the nite element model for apin-supported arch. Finite element models comprising shell elementsneed speci c attention with respect to their supports or boundary

conditions, i.e. enforcing zero displacements and/or zero rotations.

In the present nite element model Multi-Point-Constraint elements(MPC184 in ANSYS) were used to ensure a smooth introduction of the support reactions. The supports were applied at the mid-heightnode of the web. The MPC elements applied over the cross-sectionensured proper transfer of the displacements and rotations of themid-height node to the adjacent nodes. Restrained warping at thesupport is included when applying MPCs over the web and anges,thereby preventing separate rotation of the anges ( Fig. 4).

The number of MPC elements is related to the shell element distri-bution over the height of the web and width of the anges. Loads areapplied at the centroid of the cross-section. The element mesh shownin Fig. 3 and Fig. 4 is based on mesh re nement studies as presentedin Section 2.8 .

2.4. Imperfections

2.4.1. Residual stressesThe residual stress model proposed by Spoorenberg et al. [20] is

used to de ne the initial stress state in the roller bent steel memberprior to loading ( Fig. 5(a)). The compressive residual stresses and ten-sile residual stresses are denoted negative and positive respectively. Asthis residual stress model is presented in non-dimensional form, thenominal yield stress according to material speci cations is used to ob-tain stress values ( f y =235 N/mm 2 for S235 and f y =355 N/mm 2 forS355). The residual stress values are applied at the integration pointsof the shell element which coincide with the element centroid. As theresidual stress value in the integration point de nes the stress statefor the whole element, a step-wise pattern is obtained for the niteelement model ( Fig. 5(b)).

2.4.2. Geometric imperfectionsLa Poutr [21] measured the out-of-straightness of 12 full-scale

roller bent arches prior to testing. Lateral crookedness, radial crook-edness and twist imperfections were measured on HE 100A archeswith an arch length S of 6 m ( Fig. 6). A substitute out-of-straightnesspatternwas derived thatrepresents themeasuredgeometric deviationsin the arch with nearly identical out-of-plane elastic plastic bucklingbehavior. The substitute out-of-straightness is characterized by thelowestglobaleigenmode from a LBAandvariable amplitude.Thelowesteigenmode wasselected as it is regarded as the most detrimental shapeto de ne the out-of-straightness because it shows strong coherencewith the arch de ections at elastic plastic buckling.

In a nite element parametric study the amplitude is modi ed toachieve a best t to the measured out-of-straightness for each of the12 arches. This produced a total of 12 differentamplitudesSpoorenberg[22] . From this nite element study it was concluded that the lowestglobal eigenmode from a LBA with amplitude of S /1000 providesa good substitute to de ne the out-of-straightness of a roller bentarch. It is mentioned that the above suggested substitute out-of-straightness is only applicable to freestanding arches subject toout-of-plane buckling failure.

2.4.3. Material modeling During roller bending the stress strain curve of the material prior

to forming changes from a single bi-linear curve to several differentnon-linear curves over the cross-section. From the experimental ob-servations a prediction model was developed that de nes generallyapplicable material models for a roller bent wide ange section;Spoorenberg et al. [23] . The prediction model consists of two differentcomponents: the estimation of salient mechanical properties of rollerbent steel and the development of curves de ning the stress straincharacteristic. Depending on the steel grade (S235 or S355), bendingratio ( R/h) (ratio between arch radius R and nominal section height h)and yield stress of the material prior to forming ( f y;s ), the predictionmodel produces the 7 different stress strain curves for 9 zones of

the roller bent arch ( Fig. 7).

11R.C. Spoorenberg et al. / Journal of Constructional Steel Research 79 (2012) 9 21

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From tensile tests conducted on coupons taken from roller bentarches it was observed that the anges exhibit gradual yieldingbehavior. Therefore, the modi ed Ramberg Osgood model, which isused for stainless steel and aluminum members, was selected to de-

ne the stress strain curve from the onset of loading up to the ulti-mate tensile stress in the anges. A bi-linear material curve wasselected to approximate the material characteristics in the web. Theroller bent stress strain curves are employed for both compressionand tension, i.e. the tensile and compression regime have an identicalstress strain curve. For the GMNIA the stress strain curves from thisprediction model were converted to true-stress, true-strain curves toaccount for large strain effects. The material models were entered inthe nite element model by discrete stress strain values, resultingin a piece-wise curve for each of the 7 zones ( Fig. 8).

The materials' response to plastic straining is described by the VonMises yield criterion, Prandtl-Reuss ow rule and an isotropic hard-ening law. The difference between the nominal yield stress accordingto material speci cations and the measured yield stress in roller bentsteel is caused by two different phenomena. Firstly, the material isplastically strained during roller bending resulting in hardening ef-fects thereby changing the yield stress values. Secondly, prior to rollerbending, there exists already a difference between the nominal yieldstress ( f y) and the measured yield stress ( f y;s ), where measuredvalues usually exceed nominal ones. Differences between f y and f y;swere signi cant for steel grade S235 whereas for steel grade S355this difference less pronounced. Including the difference between f yand f y;s in the calibration process of the prediction model would

imply that the difference between nominal yield stress and measuredyield stress in roller bent steel is completely due to the roller bendingprocess. This, of course, is not correct. Therefore, the prediction modelwas calibrated on the difference between yield stress values obtainedfrom measurements of coupons taken from straight sections, servingas reference sections, and coupons taken from roller bent sections.Although calibration in this way gives a better representation of thechange in material properties due to roller bending, it imposessome drawbacks on the use of the prediction model. In most casesmeasured values of the yield stress prior to forming are absent andthe nominal yield stress must be utilized to de ne the stress straincurves with the prediction model. Using the nominal yield stresscan lead to conservative assumptions of the stress strain curves, es-pecially for steel grade S235. The best accuracy with the predictionmodel is achieved when measured values for the yield stress of thestraight steel are used.

2.5. Solution procedure

All analyses are load-controlled. As the GNIA, MNA and GMNIA aredominated by non-linear behavior the load is applied gradually inthese analyses. As the primary objective is to nd the ultimate loadrather than post-buckling behavior the Newton Raphson procedurewas selected in preference to the arc-length method to solve thenon-linear equilibrium equations. The convergence criterion for theout-of-balance load vector is equal 0.5%. In addition to the forcenorm check a convergence criterion of 0.5% was employed for the dis-

placement increments. For the LBA the Block Lanczos method is usedfor the eigenvalue extraction.

2.6. Load-de ection curves

For all GMNIA the ultimate load ult is de ned as the load levelcorresponding to the last converged substep. Typical load-de ectioncurves from GNIA and GMNIA are shown in Fig. 9 for HE 100A archeswith a radially distributed load with imperfections as given inSection 2.4 . The acting load is normalized with respect to cr asfound from a LBA executed on the same arch. The non-linear materialmodel used is based on the prediction model for steel grade with f y;s =235 N/mm 2 . In order to correctly express the GMNIA results inthe column curve diagram the plastic multiplier pl must be obtained

from MNA using the same non-linear material models. In addition, as

Support

Shell-elements

Ar c h s p a n L

Wide flange section bentabout the major axis

Multi point constraint elements (MPC184)

Arch rise f

y

x

z

x z

y

R

y

z x w

u

v

Coordinate system with correspondingdeformations and rotations

u=v=w===0

Supportu=v=w===0

Restrained warping deformations

Restraine dwarpingdeformations

Fig. 3. Finite element model with local coordinate system.

Arch MPC184 elementsalong flange width

MPC184 elementsover web height

Support

Number of MPC184 elements

8

4

4

4

4

Fig. 4. Modeling of restraints.

12 R.C. Spoorenberg et al. / Journal of Constructional Steel Research 79 (2012) 9 21

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geometrical non-linear effects are ignored the arch continues to resistloads during the MNA as plastic straining progresses over thecross-section. In order to make a proper estimate of the plastic multi-plier, the plastic collapse load is de ned as the intersection betweentwo tangents to the load-de ection curve. In Fig. 10 the load-de ection curves are shown for MNA performed on the same archesas selected for Fig. 9, where the acting loads are normalized withrespect to cr . The intersection between the two tangents de nesthe normalized plastic collapse load ( pl / cr ).

2.7. Column curve and imperfection parameter curve representation

As mentioned in Section 2.1 each arch con guration requires 3 dif-ferent analyses to be performed, allowing the ultimate load foundfrom the GMNIA to be plotted in a column curve diagram. The columncurve expression as presented in EN 1993-1-1 [19] (ECCS columncurve) relates the normalized exural buckling strength of columns to their non-dimensional slenderness :

1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 2p but 1:0: 2

Where:

0:5 1 2 : 3The generalized imperfection or imperfection parameter is given

by the following equation:

0 0 4where the factor is based on the typical cross-section. Dependingon

the column curve used, a value for is given in Table 1 . The parame-ter 0 is set at 0.2, irrespective of the column curve. For identicalnon-dimensional slenderness values, higher values of result in an

increase of and a decrease of .The load ampli ers obtained froma MNA and LBA, pl and cr , respectively, can be used to de ne thenon-dimensional slenderness for the arch:

ffiffiffiffiffiffiffiffiffiffiffiffiffi pl= cr q : 5The load ampli er ult can, together with pl , be used to plot the

reduction factor of the arch:

ult = pl: 6

Eqs. (5) and (6) allow the nite element results to be plotted in acolumn curve diagram. Another approach to devise a design rulebased on column curves is by plotting the same nite element resultsin an imperfection parameter diagram. When isolating the expressionfor from the column curve formulation by rewriting Eqs. (2) and (3)a different expression of can be obtained, which is only dependingon and :

1 2

2 1 2 : 7Eq. (7) can be used in conjunction with Eqs. (5) and (6) to allow

the nite element results to be expressed in an imperfection parame-ter diagram:

num ult pl

pl ult

2

pl cr 1

pl cr

: 8

Plotting nite element results in the imperfection parameter dia-gram, nding an expression for the imperfection parameter andreplacing Eq. (4) by the newly found expression in the column

curve formulation results in an accurate expression for the columncurve. Moreover, it leaves the original column curve expressionintact.

(a) Residual stress modelwith element mest

(b) Implementation of residualstress model in FE-code

Fig. 5. Residual stress model and nite element implementation.

wimpvimp

R

(b) elevation(a) top view imp

(c) side view

L L

f

Radial crookednessLateral crookedness Twist imperfections

S

Fig. 6. Geometric imperfections measured by La Poutr.

13R.C. Spoorenberg et al. / Journal of Constructional Steel Research 79 (2012) 9 21

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The difference between plotting the nite element results in thecolumn curve diagram using Eqs. (5) and (6) or in the imperfectionparameter diagram using Eq. (8) is illustrated in Fig. 11 with the niteelement data from Fig. 9 and Fig. 10 in addition to ECCS column curve a and d . It can be seen that plotting the nite element data in thecolumn curve suggests that ECCS column curve d might be appro-priate. However, looking at the imperfection parameter diagram itappears that the linear imperfection parameter expression of columncurve d has no strong coherence with nite element results forarches with higher slenderness values. Hence, it is assumed thatusing the approach the imperfection parameter will result in a moreaccurate expression for column curves to predict the elastic plasticbuckling response of steel arches.

2.8. Mesh re nement study

A mesh re nement study was performed such that the ner meshis part of the larger mesh. A HE 100A with a subtended angle of 90,arch length S of 3 m and out-of-plane xed supports with restrainedwarping is subjected to a radially directed uniformly distributed load.The arch is meshed with four different element distributions, wherethe number of elements over the ange width, web height and overthe arch length is varied. For each mesh a LBA and GMNIA areperformed to evaluate the normal force at the abutment at elasticbuckling ( N cr ) and at elastic plastic buckling ( N ult ), respectively. For

the GMNIA a bi-linear material model and no residual stresses areimplemented to rule out any additional discretization effects withrespect to the zonal material distribution and step-wise residualstress model. The geometric imperfections for the GMNIA are basedon Section 2.4.2 . The results of the mesh re nement study areshown in Table 2 . The difference in N cr and N ult for a speci c meshand mesh no.4 is given as well.

It can be seen that in general an increase in mesh density produces

and increases in both the elastic buckling force and ultimate force. Itis noted that most mesh re nement studies are characterized by a de-creasein elastic buckling load and ultimate load as the mesh is furtherre ned. The small difference between mesh no.3 and mesh no.4 re-veals that the discretization error has almost vanished. Mesh no.3will therefore be used for all computations. The adopted niteelement mesh is shown in Fig. 3.

3. Validation

Experiments as published in the related paper are selected for thevalidation study. Full-scale tests on circular roller bent steel archeswere conducted as described in a related paper by La Poutr et al.[1]. Load was applied by means of a wire hanger, which meant thatit remained directed to the center of the baseline of the arch through-out the entire loading stage (this type of load is also known as adirected load). It is assumed that close coherence between the exper-iments by La Poutr and the nite element model for the directed loadimplies that the model is also able to correctly produce the out-of-plane elastic plastic buckling response for arches under gravity load-ing. A full overview of the experimental plan by La Poutr is shown inTable 3 . Only the experimental results from the full-scale arches wereused for the evaluation of the nite element model.

The stress strain curve obtained from coupons taken from theroller bent arches and the geometric imperfections measured priorto testing by La Poutr in addition to the residual stress modelshown in Section 2.4.1 were used to de ne the initial state of the roll-er bent arch. The arch was equipped with a tension rod by means of a3D spar element (LINK8) to represent the tension rod (directed load-ing). The maximum vertical displacement imposed on the tension rodduring testing was also applied in the nite element model. Afterreaching this maximum, the imposed displacement was released,allowing the arch to de ect back. A comparison between the experi-mentally and numerically obtained load-de ection curves is shownin Fig. 12 for two tests. The maximum loads ( F ult ), the correspondingin-plane displacements ( wult ), out-of-plane displacements ( vult ) andtwists ( ult ) as obtained from the experiments are tabulated inTable 4 , and compared to nite element results. It can be seen that

1

2

1

6

7

34

5

3

Fig. 7. Zonal distribution of mechanical properties.

0

100

200

300

400

500

600

True strain [-]

T r u e s t r e s s

[ N / m m 2

]

True strain [-]

f y;s=235N/mm2, R / h=39.79 f y;s=235N/mm

2, R / h=39.79Zone 1 Zone 1

Prediction model Numerical input

0 0.005 0.01 0.015 0.020

100

200

300

400

500

600

0 0.05 0.1 0.15 0.2 0.25

T r u e s t r e s s

[ N / m m 2

]

Fig. 8. Example stress

strain curves for roller bent steel, initial stage (left) full stress

strain curve (right).

14 R.C. Spoorenberg et al. / Journal of Constructional Steel Research 79 (2012) 9 21

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8/11/2019 Out-Of-plane Stability of Roller Bent Steel Arches

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0

0.2

0.4

0.6

0.8

1

1.2

/

c r [ - ]

/

c r [ - ]

/

c r [ - ]

/

c r [ - ]

Central arch deflection v [mm]

HE 100A S =6m 2 =180 deg. R=1.91 m

Central arch deflection v [mm]

HE 100A S =4m2 =120 deg. R=1.91 m

HE 100A S =3m 2 =90 deg. R=1.91 m

HE 100A S =2m 2 =60 deg. R=1.91 m

ult / cr=0.70 ult / cr=0.60

ult / cr=0.29

ult / cr=0.48

cr=1.0 cr=1.0

cr=1.0 cr=1.0

GNIA GMNIA

0 30 60 90 1200

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40

Central arch deflection v [mm]0 10 20 30 40

Central arch deflection v [mm]0 10 20 30 40

0

0.2

0.4

0.6

0.8

1

1.2

0

0.2

0.4

0.6

0.8

1

1.2

Fig. 9. Load-de ection curves from GNIA and GMNIA normalized with respect to the elastic buckling load from LBA.

0

5

10

15

20

/

c r [ - ]

/

c r [ - ]

/

c r [ - ]

/

c r [ - ]

Vertical deflection u [mm] Vertical deflection u [mm]

HE 100A S =6m 2 =180 deg. R=1.91 m

HE 100A S =4m 2 =120 deg. R=1.91 m

HE 100A S =3m 2 =90 deg. R=1.91 m

HE 100A S =2m 2 =60 deg. R=1.91 m

pl / cr=14.98

pl / cr=2.82

pl / cr=0.38

pl / cr=1.10

cr=1.0 cr=1.0

cr=1.0 cr=1.0

0 25 50 75 1000

2

4

6

8

0 25 50 75 100

Vertical deflection u [mm] Vertical deflection u [mm]0 25 50 75 100 0 25 50 75 100

0

0.5

1

1.5

2

0

0.5

1

1.5

2

Fig. 10. Load de ection curves from MNA normalized with respect to the elastic buckling load from LBA.

15R.C. Spoorenberg et al. / Journal of Constructional Steel Research 79 (2012) 9 21

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8/11/2019 Out-Of-plane Stability of Roller Bent Steel Arches

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the nite element model is able to replicate the experimentalload-de ection behavior with moderate to good accuracy. Overall,

the nite element modelunderestimates the experimentally obtainedultimate loads.

4. Sensitivity analyses

Sensitivity analyses were conducted on HE 100A and HE 360Barches with a radially uniformly distributed load, which resulted inuniform compression in the arch-rib. An initial residual stress distri-bution from Fig. 5, a geometric imperfection using the lowest globalbuckling mode together with an amplitude of S /1000 and a bi-linearmaterial model with nominal yield stress were used to de ne thebaseline of the initial state of the roller bent member from whichvariation in imperfections was applied.

4.1. Residual stress

The group of arches covering a non-dimensional slendernessrange of 0 :6 3:8 was used for the sensitivity analyses. For eacharch a GMNIA was performed with two different residual stressmodels and one GMNIA was performed without inclusion of residualstresses, in addition to a GNIA. A typical hot-rolled residual stress pat-tern was used and the roller bending residual stress model as shownin Fig. 5. The various load de ection curves for a single arch geometryare shown in Fig. 13. The hot-rolled residual stress model is moredetrimental to the elastic plastic buckling load than the roller-bentresidual stress model. The results for all arches are presented in thecolumn curve diagram to show the sensitivity of residual stresses withrespect to thearch non-dimensionalslenderness.The in uenceof resid-ual stresses on the elastic plastic buckling load is most pronouncedfor arches with non-dimensional slenderness smaller than 1.5.

4.2. Geometric imperfections

The same set of arches as employed in Section 4.1 was selected toinvestigate the sensitivity of a change in imperfection amplitude onthe elastic plastic buckling response. Three different levels of ampli-tude together with the lowest eigenmode from the LBA were usedto de ne the out-of-straightness of the member: S /100, S /1000 andS /10000. For each arch geometry with a given amplitude, twodifferent analysis types were performed: GNIA and GMNIA. The

load-de ection curves from a single arch geometry with different am-plitudes are presented in Fig. 14. A change in imperfection amplitudechanges the load-de ection curves from the onset of loading and af-fects the buckling response to a considerable extent. The results of all GMNIA are presented in the column curve diagram. The in uenceof a change in amplitude is most signi cant within the range of 0:6 2:5.

4.3. Material modeling

Arches are equipped with different material models to monitortheir in uence on the elastic plastic buckling load. Each arch con g-uration is analyzed with different material properties: (1) nominalsteel properties ( f y =235 N/mm 2 for steel grade S235 and f y =355 N/mm 2 for steel grade S355) in conjunction with a bi-linear ma-terial law, resembling the material properties for straight hot-rolledsteel. The second material model (2) is featured by substituting ayield stress of the material prior to forming equal to nominal values( f y;s = f y) in prediction model to generate 7 different stress straincurves for steel grade S235 and for steel grade S355. This casewould resemble the situation when measured yield stresses forstraight steel are absent and one has to resort to nominal values.The third material model (3) is identical as (2) but with a yield stressfor the material prior to forming of f y;s =290 N/mm 2 and f y;s =370 N/mm 2 for steel grade S235 and S355 respectively. These valuesgive a better approximation of the measured values of the yield stresscompared to nominal values in straight steel prior to forming. Theresults from the analyses are shown in Fig. 15.

From the comparison between the material models (2) and (3) itcan be seen that the yield stress value of the material prior to rollerbending is of signi cant in uence on the results. When nominalyield stress values are substituted in the prediction model instead of yield stress values resembling measured properties from straightsteel, the calculated failure load is conservative. This effect is mostpronounced for steel grade S235. There exists a large difference in re-duction factors obtained with the prediction model for steel grade

Table 1Imperfection factor for column curves according to EN 1993-1-1 [19].

Column curve a 0 a b c d

Imperfection value 0.13 0.21 0.34 0.49 0.76

i m p e r

f e c t

i o n p a r a m e t e r

[ - ]

non-dimensional slenderness [-]

r e d u c t

i o n

f a c t o r

[ - ]

non-dimensional slenderness [-]

ECCS column curve d

ECCS column curve a

ECCS column curve d

ECCS column curve a

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 40

1.5

3

4.5

6

7.5

0 1 2 3 4

Fig. 11. Finite element results in column curve graph (left) and imperfection parameter diagram (right).

Table 2Mesh re nement.

Mesh no. No. of elements E lastic bucklingforce

Ultimate force

Flanges/web

Arch length Total N cr [N] Diff [%] N ult [N] Diff [%]

1 2 24 144 426808 15.28 317537 6.342 4 48 576 488955 2.94 335417 1.073 8 96 2304 501591 0.43 339160 + 0.034 16 192 9216 503772 339049

16 R.C. Spoorenberg et al. / Journal of Constructional Steel Research 79 (2012) 9 21

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8/11/2019 Out-Of-plane Stability of Roller Bent Steel Arches

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S235 and for steel grade S355. The degree of roundness of the stress

strain curve for S235 is higher than that of S355, resulting in prema-ture loss of stiffness and hence lower -values.

Furthermore, it should be noted that for an incidental case a valueof greater than 1.0 is obtained. This can be explained by the plasticmultiplier pl being determined from nite element analyses with abi-linear material model with nominal yield stress. Arches with lownon-dimensional slenderness values will have higher elastic plastic

Table 3Experimental program La Poutr et al. [1] of full-scale arches.

Testno.

No. of tests

Radius R[mm]

Angle2 []

ArchlengthS [mm]

Rise f [mm]

Span L[mm]

Rise-to-spanratio f /L [ ]

1A, 1B, 1C 3 1910 180 6000 1910 3820 0.502A, 2B 2 2149 160 6000 1775 4231 0.423A, 3B 2 2546 135 6000 1572 4705 0.334A, 4B 2 3125 110 6000 1333 5120 0.26

5A, 5B, 5 C 3 3820 90 6000 1119 5402 0.21

0

20000

40000

60000

80000

100000

120000

C e n

t r a l p o

i n t l o a d

F [ N ]

0

20000

40000

60000

80000

100000

120000

C e n

t r a l p o

i n t l o a d

F [ N ]

C e n

t r a l p o

i n t l o a d

F [ N ]

In-plane deflection w [mm] In-plane deflection w [mm]

wult, F ult (experimental) wult , F ult (experimental)

wult, F ult (numerical)

wult , F ult (numerical)

Test 1A Test 5A

Test 1A

Out-of-plane deflection v [mm]Out-of-plane deflection v [mm]

vult , F ult (numerical)

vult, F ult (numerical)

vult , F ult (experimental)vult, F ult (experimental)

Test 5A

0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 80

0 30 60 90 120 1500

20000

40000

60000

80000

100000

120000

C e n

t r a l p o

i n t l o a d

F [ N ]

0

20000

40000

60000

80000

100000

120000

0 30 60 90 120 150

C e n

t r a l p o

i n t l o a d

F [ N ]

Twist [] Twist []

ult , F ult (numerical)ult, F ult (numerical)

ult, F ult (experimental) ult , F ult (experimental)

Test 1A Test 5A0

20000

40000

60000

80000

100000

120000

C e n

t r a l p o

i n t l o a d

F [ N ]

0

20000

40000

60000

80000

100000

120000

0 5 10 15 20 0 5 10 15 20

Fig. 12. Load-de ection characteristics for tests 1A (left) and 5A (right).

Table 4Load-de ection results obtained from experiments and nu