elastic lateral-torsional buckling of restrained web-tapered i-beams

Computers and Structures 88 (2010) 1179–1196

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Computers and Structures

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Elastic lateral-torsional buckling of restrained web-tapered I-beams

A. Andrade a, P. Providência a, D. Camotim b,*

a Department of Civil Engineering, INESC Coimbra, University of Coimbra, FCTUC-Pólo II, Rua Luı́s Reis Santos, 3030-788 Coimbra, Portugalb Department of Civil Engineering and Architecture, ICIST/IST, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

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

Article history:Received 12 April 2010Accepted 18 June 2010Available online 17 July 2010

Keywords:Lateral-torsional bucklingWeb-tapered I-beamsRestrained beamsBracingCollocation methodCOLNEW

0045-7949/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.compstruc.2010.06.005

* Corresponding author. Tel.: +351 21 8418403; faxE-mail addresses: [email protected](A.Andrade),pro

[email protected] (D. Camotim).

This paper presents an investigation on the elastic lateral-torsional buckling behaviour of discretelyrestrained tapered beams. Through a model problem consisting of a doubly symmetric web-tapered I-section cantilever acted by a tip load, it is shown how the effects of linearly elastic or rigid restraintscan be included in the one-dimensional model previously developed by the authors to characterize theelastic buckling behaviour of thin-walled tapered beams. The restraints may have a translational, tor-sional, minor axis bending and/or warping character. The resulting self-adjoint eigenproblem is cast innon-dimensional form over a fixed reference domain, a process that leads to the identification of a com-plete set of independent non-dimensional parameters. It is then reformulated as a standard inhomoge-neous boundary value problem and solved numerically with the collocation package COLNEW. Aparametric study is carried out in order to examine in some detail (i) the effectiveness of different typesof restraint, (ii) the influence of the restraint stiffness and (iii) the interplay between these two aspectsand the degree of web tapering. This study provides insight into the peculiarities of the lateral-torsionalbuckling behaviour of tapered beams and some seemingly paradoxical results are given a physicalexplanation.

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1. Introduction

Tapered members are increasingly used in the constructionindustry because of their unique ability to combine efficiency,economy and aesthetics – the three ideals of structural art [1].Since the load-carrying capacity of laterally unsupported beams(either prismatic or tapered) is estimated on the basis of their (i)cross-section and (ii) elastic lateral-torsional buckling strengths,there is a clear need for methods capable of accurately predictingthe elastic critical load factor of a given tapered beam at a lowcomputational cost. Bearing this in mind, the authors developedand validated a one-dimensional model suitable for describingthe elastic lateral-torsional buckling behaviour of singly symmetrictapered thin-walled open beams [2,3]. But so far only isolatedbeams with idealized support conditions have been analysed. Suchconditions are seldom found in actual design practice – indeed,beams are usually connected to other elements that may contrib-ute significantly to their buckling strength, even when they arenot primarily intended for that specific purpose.

Since the pioneering work of Flint [4] (see also [5]) and Austinet al. [6], there have been extensive studies on the effect ofrestraints on the elastic lateral-torsional buckling behaviour of

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: +351 21 [email protected] (P.Providência),

prismatic beams – broad surveys on this subject, as well as detailedreferences to the literature, can be found in [7–11]. On the con-trary, there is little information available on the lateral-torsionalbuckling behaviour of tapered beams with elastic or rigidrestraints. In fact, to the authors’ best knowledge the only publica-tions on this subject are due to Butler [12] and to Bradford [13].The former reported an experimental investigation aimed atstudying the influence of lateral and torsional braces on the elasticbuckling strength of tip-loaded tapered cantilever I-beams – unfor-tunately, these experiments are insufficiently documented to be ofany real value. The latter extended the tapered beam-column finiteelement formulation of Bradford and Cuk [14] to include the effectsof continuous elastic restraints.

Using a model problem that is intended to ‘‘contain all thegerms of generality” (to quote David Hilbert), this paper showshow the presence of out-of-plane restraints can be accounted forin one-dimensional elastic buckling analyses of tapered beams.This model problem consists of a perfectly straight doubly sym-metric web-tapered I-section cantilever, acted by a tip load andexhibiting end restraints that may (i) have a translational,torsional, minor axis bending and/or warping character and (ii)be either linearly elastic or perfectly rigid.

The resulting self-adjoint eigenproblem is cast in non-dimensional form over a fixed reference domain, and a completeset of independent non-dimensional parameters is identified. Forits numerical solution, we use the collocation package COLNEW – this

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1180 A. Andrade et al. / Computers and Structures 88 (2010) 1179–1196

requires the previous reformulation of the eigenproblem as a stan-dard inhomogeneous boundary value problem. A parametric studyis then conducted in order to examine in some detail (i) the effec-tiveness of different types of restraint, (ii) the importance of the re-straint stiffness and (iii) the interplay between these two aspectsand the degree of web tapering. An in-depth discussion of the pe-culiar torsional behaviour of web-tapered I-beams provides aphysical basis for explaining some seemingly paradoxical results.

It is important to stress that the entire analysis is founded uponthe fundamental assumption that cross-sections do not distort intheir own planes. Ronagh and Bradford [15,16] relaxed thisassumption and considered the deformation of the web in theplane of the cross-section, an approach that allowed them to inves-tigate what they called ‘‘distortional buckling” (note that someauthors classify such a buckling phenomenon as ‘‘local–global”rather than ‘‘distortional” [17]).

Although only doubly symmetric web-tapered I-beams withdiscrete restraints are dealt with in this work, the considerationof other beam geometries (possibly with a single plane of symme-try) and the inclusion of continuous restraints are straightforwardmatters. Finally, the paper is not concerned with the forces devel-oping in the restraining elements, which are always assumed tohave adequate strength – such an analysis would necessarily re-quire the consideration of the imperfections of the restrainedbeam.

A preliminary and partial version of this work was reported in[18].

2. Model problem

Man muss immer mit den einfachsten Beispielen anfangen.

DAVID HILBERT

As a model problem, we consider the lateral-torsional bucklingof a perfectly straight doubly symmetric web-tapered I-section

ϕ

l

z

0λQ

y

Tk

Rxk

Rfk

ft0h

Rfk

0 λQ l

Fig. 1. Model

Fig. 2. Some typical bracing and sup

cantilever acted by a tip point load – see Fig. 1. For convenience,a fixed rectangular right-handed Cartesian reference system isadopted – in the beam undeformed configuration, (i) the x-axiscoincides with the centroidal (and shear centre) axis, (ii) the y-and z-axes correspond to the major and minor central axes of thecross-sections and (iii) the top flange corresponds to negative val-ues of the coordinate z. The end cross-sections lie initially in theplanes x = 0 (always taken as the supported section) and x = l.The flanges are uniform, with thickness tf and width b. The webhas constant thickness tw and exhibits a linearly varying height.The cross-section height, measured between flange mid-lines, isdenoted by h and its variation along the cantilever’s length, ex-pressed in terms of the web-taper ratio a ¼ hðlÞ

h0, reads

hðxÞ ¼ 1� ð1� aÞ xl

h ih0; ð1Þ

with 0 < a 6 1 (a = 1 is associated with a prismatic beam – note thata larger web-taper ratio corresponds to less taper). The flanges ex-hibit symmetrical slopes ±tgu with respect to the (x,y) plane, wheretgu = (1 � a)h0/(2l).

The cantilever is homogeneous and isotropic, made of a St. Ve-nant–Kirchhoff material characterized by the Young modulus Eand the shear modulus G. It is acted by a conservative tip point loadQ = Qk, where k is the unit vector directed along the z-axis. Theload acts initially on the plane y = 0, at the level z = zQ, and remainsparallel to the z-axis. Its magnitude Q is deemed proportional to asingle factor k – thus, we write Q = Qrefk, where Qref is a positive ref-erence magnitude.

More often than not, in actual engineering practice, such a can-tilever exhibits some kind of bracing at the tip. While the actualbracing arrangement may vary significantly (Fig. 2(a) shows sometypical details), it can usually be modelled by means of (i) a trans-lational spring in the y-direction with stiffness kT (0 6 kT 6 +1), lo-cated at a level z = zT, and/or (ii) a torsional spring with stiffnesskRx (0 6 kRx 6 +1). On the other hand, a cantilever is frequently just

0hαx

QzTz

b

y

z

2h

2h

wt

cosϕft

problem.

port details (adapted from [22]).

A. Andrade et al. / Computers and Structures 88 (2010) 1179–1196 1181

the overhanging portion of a beam extending beyond an endsupport, as illustrated in Fig. 2(b). In such case, the assumptionof a built-in support may lead to a considerable overestimationof the buckling strength (see the prismatic beam results reportedin [19,20]) – a more realistic description of the actual support con-ditions is achieved by considering, in each flange, equal rotationalsprings about the z-axis, with stiffness kRf (0 6 kRf 6 +1). They sim-ulate the warping and lateral bending restraints provided by theadjacent span, while the torsional rotation (about x) and the lateraldisplacement are kept fully prevented. In fact, this very simple andphysically meaningful procedure is closely related to the warpingspring concept introduced in [21] to analyze space frames madeof prismatic members with partial warping restraint. Note that ifkRf = kT = 0, the cantilever may rotate freely about the z-axis in a ri-gid-body mode – in order to rule out this possibility, kRf and kT

must satisfy 0 < kRf + kTl2 6 +1.Our first aim is to compute the elastic buckling load factors and

the corresponding buckling modes for this beam-load structuralsystem.

3. Mathematical model

Building on previous work by Wilde [23], the authors have re-cently developed and validated a one-dimensional model suitablefor describing the lateral-torsional buckling behaviour of taperedthin-walled open beams (symmetric with respect to the loadingplane) [2,3]. It is assumed that a beam resists bending and torsionby (i) membrane action of its walls and (ii) shear stresses that‘‘loop” around the cross-section, being zero at the mid-line. In or-der to describe the first of these resisting modes, we adopt the fol-lowing hypothesis:

(H1) The beam behaves as a membrane shell subjected to twointernal constraints (i.e. constitutive prescriptions restrict-ing the class of possible deformations [24, p. 451]), whichextend to the tapered case the classical Vlassov assumptions[25]:(H11) The projection of the mid-line of the cross-sections

on a plane orthogonal to the x-axis experiences nodistortion throughout the whole deformation pro-cess. This constraint implies that this model is notable to capture any local-plate or distortional buck-ling phenomenon – in other words, the beam isforced to buckle in a pure global mode.

(H12) Let b1 and b2 be orthogonal vectors that span the tan-gent plane, at a given point, to the mid-surface of theundeformed beam, with b2 tangent to the cross-sec-tion mid-line through that point. The shear strainbetween fibers originally along vectors b1 and b2 iszero – i.e. these fibers remain at right-angles afterdeformation.

The second resisting mode, which shall be referred to, even ifinadequately, as St. Venant torsion, is accounted for, in asimplified manner, by means of the following hypothesis:

(H2) The strain energy associated with St. Venant torsion is givenby the expression valid for prismatic bars, though regardingthe rigidity GJ as a function of the axial coordinate x. In par-ticular, this means that, for each x in [0, l], the effect of taperon the calculation of the cross-sectional property J(x) isneglected.

For simplicity, the pre-buckling deflections are neglected in thiswork, which means that the cantilever is assumed to remainstraight up until the onset of buckling. Indeed, in design applica-tions it is prudent to disregard the beneficial effect of the pre-buck-

ling deflections, as it may be significantly reduced by any pre-cambering of the member – concerning these effects, the inter-ested reader is referred to the papers [26,27], on prismatic beams,and [28], on tapered ones.

Based on the above assumptions, it can be shown that the en-ergy functional for lateral-torsional buckling analysis (i.e. the sec-ond variation, from a fundamental equilibrium state, of the totalpotential energy of the beam-load system), duly specialized todoubly symmetric beams and in the absence of elastic restraints,is given by (see [2] for details)

F½v;u; k� ¼ 12

Z l

0EI�zv

2;xxdxþ 1

2

Z l

0EI�x/2

;xxdxþZ l

0EI�xw/;xx/;xdx

þ 12

Z l

0GJ þ EI�w� �

/2;xdxþ

Z l

0Mf

y:ref v ;xx/dxk

þ 12

zQ Q ref k/ðlÞ2; ð2Þ

where (i) v and / denote the centroid displacement along the y-axisand the cross-section rotation about the x-axis, (ii) Mf

y:ref stands forthe first-order bending moment distribution associated with thereference load Qrefk, that is

Mfy:ref ðxÞ ¼ �ðl� xÞQ ref ð3Þ

and (iii) I�z ; I�x; I�w; I�xw and J are geometrical properties defined inreference [2]. For the particular beam geometry dealt with here,they read

I�zðxÞ ¼16

b3tf cos3 u ¼ I�z0; ð4Þ

I�xðxÞ ¼hðxÞ2

4I�z0 ¼

124

hðxÞ2b3tf cos3 u ¼ 1� ð1� aÞ xl

h i2I�x0; ð5Þ

I�xwðxÞ ¼1

12hðxÞh;xðxÞb3tf cos3 u ¼ 2

h;xðxÞhðxÞ I�xðxÞ

¼ �2lð1� aÞ 1� ð1� aÞ x

l

h iI�x0; ð6Þ

I�wðxÞ ¼16

hðxÞ2;xb3tf cos3 u ¼ 2hðxÞ;xhðxÞ

� �2

I�xðxÞ ¼4

l2 ð1� aÞ2I�x0; ð7Þ

JðxÞ ¼ 2b3

tf

cos u

� �3

þ hðxÞt3w

3¼ 1� h0t3

w

3J0ð1� aÞ x

l

� �J0; ð8Þ

where (�)0 indicates a geometrical property evaluated at x = 0. Itshould be noticed that

I�x;x ¼ I�xw; ð9Þ

I�x;xx ¼ I�xw;x ¼12

I�w: ð10Þ

The complete definition of F still requires the specification ofthe class of admissible functions v and /, a procedure that involvesthe characterisation of the following two properties: (i) thesmoothness these functions must exhibit and (ii) the boundaryconditions they must satisfy. For the integrals in (2) to make sense,v and / must be square-integrable in the interval (0, l) and possesssquare-integrable first and second derivatives in that interval. Inaddition, v and / must satisfy the essential (i.e. kinematic) bound-ary conditions, which are necessarily homogeneous in this type ofproblem and will be addressed later. The real-valued functionsfulfilling these two requirements are termed ‘‘kinematicallyadmissible”.

Fig. 3. Comparison between the linearized warping-torsion behaviours of prismatic and web-tapered doubly symmetric I-beams.


Remarks:

3.1. In the energy functional (2), the effect of taper is dealt withthrough (i) additional, non-standard, mechanical propertiesEI�xw; EI�w and (ii) a modification of the minor axis bendingand warping rigidities by the use of a ‘‘reduced flange thick-ness” t�f ¼ tf cos3 u (this is indicated by the superscript *,which is dropped in the prismatic case). So that the reader

may grasp the physical significance of these two items, wecontrast in Fig. 3 the linearized warping-torsion behavioursof prismatic and web-tapered doubly symmetric I-beams.The following facts are worthy of notice:3.1.1. In both cases, the linearized version of the constraints

(H11)–(H12), together with symmetry considerations,imply qualitatively similar displacement fields for agiven cross-section mid-line (Fig. 3(a)): (i) the

Fig. 3 (continued)


mid-line rotates about the centroidal x-axis throughan angle / and (ii) the flanges warp out of the planeby rotating � h

2 /;x about the z-axis. To reach this con-clusion, one must not forget that in the web-taperedcase (i) the displacements along z are obviously notorthogonal to the flange mid-planes, as illustratedin the zooming of Fig. 3(a), and (ii) the centroidallines of the flanges are not parallel to the x-axis, sothat derivatives with respect to the arc-length ofthese lines are not identical to derivatives withrespect to x. In fact, if one denotes symbolically the

former operator by (�),s and the latter by (�),x, thenthe two are related through (�),s = ±cosu(�),x, wherethe choice of sign depends on the orientation of theflange centroidal lines. Incidentally, recall thath,x = �2tgu.

3.1.2. The displacement fields discussed above can now beused to find the membrane extensions in the flangesFig. 3(b). Comparing the ensuing strain–displace-ment relations for prismatic and web-tapered beams,one readily sees that the latter contain a correctivefactor (cos2u) and an additional term. The same is


true for the normal membrane forces in the flanges,as shown in Fig. 3(c).

3.1.3. The shear membrane forces shown in Fig. 3(d), whichhave a reactive character (in the sense that they stemfrom the stipulated kinematical constraints and arenot related to the strains via a constitutive equation),may be found by considering the equilibrium of a‘‘flange slice” acted by the previously obtained nor-mal membrane forces.

3.1.4. The normal (resp. shear) membrane forces arestatically equivalent to a bending moment (resp.shear force) in each flange (Fig. 3(e)–(f)). In theweb-tapered case, the modified warping stiffnessEI�x, calculated with the reduced flange thicknesst�f ¼ tf cos3 u, and the non-standard mechanicalproperty EI�xw arise naturally in this process.

3.1.5. Finally, in web-tapered beams the flange bendingmoments have an axial component, featuring EI�xw

and EI�w. Such an additional contribution to the totaltorque is obviously absent from prismatic beams(Fig. 3(e)).

Lee [29] and Kitipornchai and Trahair [30] have also provideda physical explanation for the warping-torsion behaviour ofdoubly symmetric web-tapered I-beams, by regarding eachflange as an individual Euler–Bernoulli beam in bending,with deflections � h

2 /. To put it in historical perspective, in-stead of starting from (a generalised form of) Vlassov’sassumptions, as done above, these authors tried to extendTimoshenko’s pioneering work of 1905 [31,32]. However,their analysis is marred by the fact that they fail to distin-guish between derivatives with respect to the arc-length ofthe flange centroidal lines and derivatives with respect to x,the longitudinal coordinate of the beam (see remark 3.1.1).Moreover, Kitipornchai and Trahair allow for (smooth) non-affine maps x ´ h(x) – that is, doubly symmetric I-beamswith curved flanges – in which case the equation used to re-late bending moments in the flanges to the rotation about thebeam axis – Eq. (14) in the cited paper – is incomplete. In par-ticular, it leads to the conclusion that a rigid infinitesimalrotation of the beam about its axis causes bending momentsin the flanges, which is obviously absurd. The term responsi-ble for this conclusion, which features h,xx/ and is ‘‘neglectedas being small ... for most structural beams”, would not havearisen in the first place if the complete equation for a curvedrectangular beam (e.g. equation (7.2.28) in [33, p. 400]) hadbeen used. The same kind of oversight is present in [34].

3.2. In the prismatic counterpart of Eq. (2), the St. Venant rigiditycould easily be adjusted to include the contribution of EI�w. Itwould be also a simple matter to use the reduced flangethickness to compute the minor axis bending and warpingrigidities. However, the term

Z l

0EI�xw/;xx/;xdx; ð11Þ

which couples the first and second derivatives of /, cannotbe directly accommodated in a prismatic beam formulation.It follows that it is conceptually incorrect to model aweb-tapered I-beam as an assembly of prismatic segments,regardless of the number of segments considered, even if theirproperties are modified according to the above suggestions.

3.3. The formula for J(x) given in Eq. (8) presupposes, inaddition to assumption (H2), that each plated component

(web and flanges) exhibits a sufficiently large width-to-thickness ratio throughout the whole length of thebeam (for more details, see [35, Sections 108–109]) – inparticular, it is assumed that the ratio (ah0)/tw meets thisrequirement.

We still have to consider the end restraints and, in particular,to distinguish between the elastic (non-rigid) and rigid cases. Inorder to account for the presence of elastic restraints (seeFig. 1), the energy functional (2) must be supplemented withthe terms

12

kRf v ;xð0Þ þh0

2/;xð0Þ

� �2

þ 12

kRf v ;xð0Þ �h0

2/;xð0Þ

� �2

þ 12

kRx/ðlÞ2 þ12

kTðvðlÞ � zT/ðlÞÞ2

¼ kRf v ;xð0Þ2 þh2

0

4/;xð0Þ

2

!þ 1

2kRx/ðlÞ2 þ

12

kTðvðlÞ � zT/ðlÞÞ2;

ð12Þ

which provide the strain energy stored in the restraints duringbuckling – it follows from the kinematical hypothesis underlyingthe one-dimensional model that the flange rotations about the z-axis are given by v ;x � h

2 /;x. Of course, when a restraint is taken asperfectly rigid, it does not store any strain energy and thus thereis no additional contribution to consider in the energy functional– however, relative to the unrestrained or elastically restrainedcases, the admissible functions must satisfy additional essentialboundary conditions (see remark 3.5 below).

Remark:

3.4. The first term on the right-hand side of (12) can be rewrittenin the form

12

kRzv ;xð0Þ2 þ12

kx/;xð0Þ2; ð13Þ

where

kRz ¼ 2kRf ; ð14Þ

kx ¼h2

0

42kRf : ð15Þ

This rearrangement clearly suggests the possibility of con-sidering independently the minor axis bending and warpingelastic restraints – simply replace the first term on the right-hand side of (12) by (13) and adopt two unrelated parame-ters, kRz and kx, instead of the single parameter kRf. In fact,such a generalization is needed to deal with several situa-tions of practical interest. For instance, if the warping re-straint is ‘‘internal” to the beam – provided by suchdevices as end plates [36–38], batten plates [36,38–40] orbox stiffeners [38,41–43] – then it is not coupled with aminor axis bending restraint (we would have kx > 0, butkRz = 0).

Suppose kRf, kRx, kT – +1 and let dv and d/ be arbitrary kinemat-ically admissible variations of v and / (i.e. differences between anytwo kinematically admissible functions v or /). Using (3)–(8) andassuming v and / to be sufficiently smooth to allow integrationby parts, the first variation of the energy functional F given by(2) + (12) reads

dF ¼Z l

0EI�z0v ;xxxx � ðl� xÞ/;xx � 2/;x

� Q ref k

� dv dxþ

Z l

01� ð1� aÞ x

l

h i2EI�x0/;xxxx �

4lð1� aÞ 1� ð1� aÞ x

l

h iEI�x0/;xxx

�

� 1� h0t3w

3J0ð1� aÞ x

l

� �GJ0/;xx þ

h0t3w

3J0

1� al

GJ0/;x � ðl� xÞQ ref kv ;xx

�d/dxþ EI�z0v ;xxxð0Þ � ðl/;xð0Þ � /ð0ÞÞQref k

� dvð0Þ

þ �EI�z0v ;xxð0Þ þ 2kRf v ;xð0Þ þ lQ ref k/ð0Þ �

dv ;xð0Þ þ EI�x0/;xxxð0Þ �2lð1� aÞEI�x0/;xxð0Þ �

2

l2 ð1� aÞ2EI�x0 þ GJ0

� �/;xð0Þ

� �d/ð0Þ

þ �EI�x0/;xxð0Þ þ2lð1� aÞEI�x0 þ

h20

42kRf

" #/;xð0Þ

( )d/;xð0Þ þ �EI�z0v ;xxxðlÞ þ kTðvðlÞ � zT/ðlÞÞ � Q ref k/ðlÞ

� dðvðlÞ � zT/ðlÞÞ þ EI�z0v ;xxðlÞdv ;xðlÞ

þ �zT EI�z0v ;xxxðlÞ � a2EI�x0/;xxxðlÞ þ2lað1� aÞEI�x0/;xxðlÞ þ

2

l2 ð1� aÞ2EI�x0 þ 1� h0t3w

3J0ð1� aÞ

� �GJ0

� �/;xðlÞ þ ðzQ � zTÞQ ref kþ kRx

� /ðlÞ

� �d/ðlÞ

þ a2EI�x0/;xxðlÞ �2lað1� aÞEI�x0/;xðlÞ

� �d/;xðlÞ: ð16Þ


The particular grouping criterion that led to the writing of theunderlined boundary terms in Eq. (16) calls for an explanation.By considering the variation d(v(l) � zT/(l)) of the lateral displace-ment at the level of the translational restraint (z = zT), it is ensuredthat the natural boundary condition corresponding to d/(l) in-volves only the torsional restraint stiffness kRx, instead of bothkRx and kT. Viewed from a complementary standpoint, this meansthat when kT = +1 (see remark 3.5 below), the values at x = l ofthe kinematically admissible variations d(v � zT/) and d/ are inde-pendent – on the contrary, if the variations dv and d/ were used,they would have to be related through dv(l) = zTd/(l).

Now, by virtue of the fundamental lemma of the calculus ofvariations [44, Theorem 1.24], the vanishing of the first variationof F for all kinematically admissible dv and d/ – often referred toin the literature as Trefftz’ criterion [45] – leads us to the classicalor strong form of the buckling problem, which may be phrased asfollows (recall that, for the moment, we are assuming kRf, kRx,kT – +1; moreover, kRf and kT are not simultaneously zero):

Problem 1. Find k 2 R and real-valued functions v, / 2 C4(0, l) \C2[0, l) \ C3(0, l], with / – 0, satisfying the differential equations

EI�z0v ;xxxx � ½ðl� xÞ/;xx � 2/;x�Q ref k ¼ 0; ð17Þ

1� ð1� aÞ xl

h i2EI�x0/;xxxx �

4lð1� aÞ 1� ð1� aÞ x

l

h iEI�x0/;xxx

� 1� h0t3w

3J0ð1� aÞ x

l

� �GJ0/;xx þ

h0t3w

3J0

1� al

GJ0/;x

� ðl� xÞQref kv ;xx ¼ 0; ð18Þ

in the open interval (0, l), together with the boundary conditions

vð0Þ ¼ 0; ð19Þ

EI�z0v ;xxð0Þ � 2kRf v ;xð0Þ � lQ ref k/ð0Þ ¼ 0; ð20Þ

/ð0Þ ¼ 0; ð21Þ

EI�x0/;xxð0Þ �2lð1� aÞEI�x0 þ

h20

42kRf

" #/;xð0Þ ¼ 0; ð22Þ

EI�z0v ;xxxðlÞ � kTðvðlÞ � zT/ðlÞÞ þ Qref k/ðlÞ ¼ 0; ð23Þ

v ;xxðlÞ ¼ 0; ð24Þ

zT EI�z0v ;xxxðlÞ þ a2EI�x0/;xxxðlÞ �2lað1� aÞEI�x0/;xxðlÞ

� 2

l2 ð1� aÞ2EI�x0 þ 1� h0t3w

3J0ð1� aÞ

� �GJ0

� �/;xðlÞ

� ðzQ � zTÞQ ref kþ kRx�

/ðlÞ ¼ 0; ð25Þ

aEI�x0/;xxðlÞ �2lð1� aÞEI�x0/;xðlÞ ¼ 0: ð26Þ

In the above list of boundary conditions, (19) and (21) are essentialand all the remaining ones are natural.

Remarks:

3.5. The modifications that must be incorporated into Problem 1to account for rigid restraints are now discussed. If kRf = +1,the cantilever is fully built-in at the support and the naturalboundary conditions (20) and (22) should be replaced by thecorresponding essential ones:

v ;xð0Þ ¼ 0; ð27Þ/;xð0Þ ¼ 0: ð28Þ

Similarly, if kT = +1, Eq. (23) should be replaced by

vðlÞ � zT/ðlÞ ¼ 0: ð29Þ

Finally, if kRx = +1, the boundary condition (25) should be re-placed by

/ðlÞ ¼ 0 ð30Þ

and, in this particular case, the buckling problem clearly be-comes independent of both zQ and zT.

3.6. The (not unusual) case of n > 1 translational restraints at onecross-section, placed at the levels z = zTi and with stiffnesseskTi, i = 1, . . ., n, possibly combined with a torsional restraintwith stiffness kRx, can always be reduced to a single transla-tional restraint with stiffness k�T , placed at the level z ¼ z�T ,together with a torsional restraint with stiffness k�Rx. If noneof the restraints is rigid, then

k�T ¼Xn

i¼1

kTi; ð31Þ

z�T ¼Pn

i¼1zTikTiPni¼1kTi

; ð32Þ

k�Rx ¼ kRx þXn

i¼1

zTi � z�T �

zTikTi: ð33Þ

If one (and only one) of the translational restraints is per-fectly rigid, say kTj = +1, then, with the customary algebraicconventions for the extended real number system (e.g., [46,Section 1.23]), Eq. (31) gives k�T ¼ þ1 and, after removingthe 11 indeterminacy, Eq. (32) yields z�T ¼ zTj. If two (or more)translational restraints are rigid, then k�T ¼ k�Rx ¼ þ1 and z�Tbecomes irrelevant.


3.7. Suppose that kT – +1. Upon integrating twice the differen-tial equation (17) and using the boundary conditions (23)and (24), together with the continuity properties of v, /and their derivatives, we get

EI�z0v ;xx � ðl� xÞ Q ref k/� kTðvðlÞ � zT/ðlÞÞ�

¼ 0; 0 6 x 6 l:

ð34Þ

In particular, EI�z0v ;xxð0Þ � lQref k/ð0Þ þ kT lðvðlÞ � zT/ðlÞÞ ¼ 0.Now, suppose in addition that kRf – +1. It follows at oncefrom (20) that

2kRf v ;xð0Þ þ kT lðvðlÞ � zT/ðlÞÞ ¼ 0: ð35Þ

The latter equation, which involves only boundary terms, hasa clear physical meaning – it is the global equilibrium equa-tion about the z-axis. We mention two consequences of Eq.(35):
3.7.1. If kT = 0, then kRf – 0 implies v,x(0) = 0 – in the
absence of a lateral translational restraint at the tip,every lateral bending restraint at the supportensures v,x(0) = 0, as observed in [8, Section 9.6.1.1].

3.7.2. If kRf = 0, then kT – 0 implies v(l) � zT/(l) = 0 – in theabsence of a lateral bending restraint at the support,every translational restraint at the tip precludes thelateral deflection at the level where it is located.

From a mathematical viewpoint, (17)–(26) (with the specifiedboundary condition replacements, whenever appropriate) define

an eigenvalue problem – the elastic buckling load factors kðnÞb and

the corresponding buckling modes v ðnÞb ;/ðnÞb

� �are its eigenvalues

and eigenfunctions. Given the self-adjoint character of the prob-lem, all eigenvalues are real and constitute an at most enumerableset with no finite cluster point; the eigenfunctions may be takento be real [47]. Since the analytical model constrains the cantileverto buckle in a pure lateral-torsional global mode, it is not possiblefor independent modes to coalesce, and so the eigenvalues are

necessarily simple. They are labelled so as to � � � < kð�2Þb < kð�1Þ

b <

0 < kð1Þb < kð2Þb < � � � and we put forward the conjecture (reminis-

cent of a classical result in Sturm–Liouville theory [47]) that /ðnÞb

has exactly jnj � 1 zeros in the open interval (0, l). In particular,the lowest positive eigenvalue is termed the (elastic) critical load

factor and we use the special notation kð1Þb ¼ kcr . Accordingly, thecorresponding buckling mode (vcr,/cr) is termed critical as well.

By an appropriate change of variable, the above problem may beconverted into an equivalent one, posed in a fixed reference do-main (i.e. independent of the cantilever length l) and written innon-dimensional form, which is a particularly adequate settingfor carrying out systematic parametric studies – within the specificcontext of the present paper, we cannot possibly agree with theassertion that it is ‘‘virtually impossible to non-dimensionalizethe many parameters in tapered sections [members]” found in[10, p. 83].

Choosing the closed unit interval [0,1] as the fixed reference do-main, the change of variable is defined by the map n: [0, l] ? [0,1],n(x) = x/l. Moreover, we set

�v : ½0;1� ! R; �vðnÞ ¼ 1l

ffiffiffiffiffiffiffiffiEI�z0

GJ0

svðnlÞ; ð36Þ

�/ : ½0;1� ! R; �/ðnÞ ¼ /ðnlÞ: ð37Þ

Finally, we introduce the family (fx)x2[0, l] of maps

z#fxðzÞ ¼2z

1� ð1� aÞ xl

� h0; ð38Þ

which defines, for each cross-section plane, a non-dimensionalordinate that is referred to (half) the cross-section’s height h(x).The non-dimensional version of Problem 1 is then stated asfollows:

Problem 2. Given the non-dimensional parameters

cQ ¼Qref l2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEI�z0GJ0

p ; ð39Þ

jx0 ¼pl

ffiffiffiffiffiffiffiffiffiEI�x0

GJ0

s; ð40Þ

jJ0 ¼h0t3

w

3J0; ð41Þ

rRf ¼2kRf lEI�z0

; ð42Þ

rRx ¼kRxlGJ0

; ð43Þ

rT ¼kT l3

EI�z0; ð44Þ

fT ¼ flðzTÞ ¼2zT

ah0; ð45Þ

fQ ¼ flðzQ Þ ¼2zQ

ah0; ð46Þ

where (i) rRf, rRx, rT – +1 and (ii) rRf and rT are not simultaneouslyzero, find cQ k 2 R and �v; �/ 2 C4ð0;1Þ \ C2½0;1Þ \ C3ð0;1�, with �/ – 0,satisfying the differential equations

�v ;nnnn � ½ð1� nÞ�/�;nncQk ¼ 0; ð47Þ

j2x0

p2 ½1� ð1� aÞn�2 �/;nnnn � 4j2

x0

p2 ð1� aÞ½1� ð1� aÞn��/;nnn

� ½1� jJ0ð1� aÞn��/;nn þ jJ0ð1� aÞ�/;n � ð1� nÞcQk�v ;nn ¼ 0 ð48Þ

in the open unit interval (0,1), together with the boundaryconditions

�vð0Þ ¼ 0; ð49Þ�v ;nnð0Þ � rRf �v ;nð0Þ � cQk�/ð0Þ ¼ 0; ð50Þ�/ð0Þ ¼ 0; ð51Þ�/;nnð0Þ � 2ð1� aÞ þ rRf

� �/;nð0Þ ¼ 0; ð52Þ

�v ;nnnð1Þ � rT �vð1Þ � jx0

pafT

�/ð1Þ� �

þ cQ k�/ð1Þ ¼ 0; ð53Þ

�v ;nnð1Þ ¼ 0; ð54Þ

jx0

pafT �v ;nnnð1Þ þ

j2x0

p2 a2 �/;nnnð1Þ � 2j2

x0

p2 að1� aÞ�/;nnð1Þ

� 1þ 2j2

x0

p2 ð1� aÞ2 � jJ0ð1� aÞ� �

�/;nð1Þ

� jx0

paðfQ � fTÞcQkþ rRx

h i�/ð1Þ ¼ 0; ð55Þ

a�/;nnð1Þ � 2ð1� aÞ�/;nð1Þ ¼ 0: ð56Þ

Remarks:

3.8. The following non-dimensional counterparts of Eqs. (27)–(30) should be used when rRf, rRx and/or rT are infinitelylarge:


�v ;nð0Þ ¼ 0; ð57Þ�/;nð0Þ ¼ 0; ð58Þ

�vð1Þ � jx0

pafT

�/ð1Þ ¼ 0; ð59Þ�/ð1Þ ¼ 0: ð60Þ

20

30

40

50

60

a=1

a=0.6

a=0.2

0Qζ = Rfσ = +∞ 0Rxσ = 0 0.1Jκ =

1.0α =

0.6α =

0.2α =

( )nQ bγ λ

nd2 mode

0 (free end)Tσ =

3.9. The set {a,cQk,jx0,jJ0,rRf,rRx,rT,fT,fQ} constitutes a com-plete set of independent non-dimensional parameters char-acterizing the lateral-torsional buckling behaviour of thestructural system depicted in Fig. 1. This conclusion couldalso have been reached through Vaschy–Buckingham’s theo-rem [48], the fundamental result upon which all of dimen-sional analysis rests.

3.10. In common design practice, jx0 ranges from 0.1 to 2.5 [49].Low (resp. high) values of this parameter correspond to long(resp. short) cantilevers and/or compact (resp. slender)cross-sections at the support. As for jJ0, which measuresthe relative contribution of the web to the St. Venant tor-sional rigidity of the support cross-section, it lies in the openinterval (0,1) – the extreme values jJ0 = 0 and jJ0 = 1, notaddressed here, are associated with two degenerate cases:webless and narrow rectangular beams, respectively. Previ-ous parametric studies have shown that the effect of jJ0 onkcr is always small [50,51].

3.11. In view of (36), it would have been more natural or, at anyrate, more immediate to define the map

0

10

0.0 0.5 1.0 1.5 2.0 2.5

20

25

30

35

a=1

a=0.6

a=0.2

0ωκ

st1 (critical)mode

1.0Tζ = −

Q crγ λ

1.0α =

0.6α =

0.2α =

Tσ = +∞

z#eðzÞ ¼ zl

ffiffiffiffiffiffiffiffiEI�z0

GJ0

sð61Þ

instead of the family (fx)x2[0,l] given by (38). Our choice wasdictated by practical reasons, since (38) is easily visualisedand conveys a clear perception of position (in relation tothe cross-section height) – for instance, fx = �1 (resp. fx = 1)indicates the level of the top flange (resp. bottom flange)mid-line of the cross-section defined by the abscissa x. Onthe contrary, (61) is far from being intuitive. However, weshall use (61) in Section 5. Note the relationship

eðzÞ ¼ 1� ð1� aÞ xl

h ijx0

pfxðzÞ: ð62Þ

10

15 1.0Tζ =0Tζ =
3.12. The differential equations (47) and (48) and boundary condi-
tions (50), (52)–(56) are the Euler–Lagrange equations andnatural boundary conditions associated with the vanishingof the first variation of the functional

0

5

0.0 0.5 1.0 1.5 2.0 2.5

0

20

40

60

80

100

120

140

160

0.0 0.5 1.0 1.5 2.0 2.5

a=1

a=0.6

a=0.2

0ωκ

0ωκ

100%λ λ

λ−

Δ = ×rigidly restrained free endcr cr

free endcr

1.0α =

0.6α =

0.2α = 1.0Tζ = −

1.0Tζ =

0Tζ =

Fig. 4. Effect of a rigid translational restraint at the tip.

F½�v; �/; k� ¼ 12

Z 1

0�v2;nndnþ 1

2j2

x0

p2

Z 1

0½1� ð1� aÞn�2 �/2

;nndn

� 2ð1� aÞj2x0

p2

Z 1

0½1� ð1� aÞn��/;nn

�/;ndn

þ 2ð1� aÞ2 j2x0

p2

Z 1

0

�/2;ndn

þ 12

Z 1

0½1� jJ0ð1� aÞn��/2

;ndn

þ rRf

2�v ;nð0Þ2 þ

j2x0

p2�/;nð0Þ2

� �þ rRx

2�/ð1Þ2

þ rT

2�vð1Þ � afT

jx0

p�/ð1Þ

� �2

� cQ kZ 1

0ð1� nÞ�v ;nn

�/dnþ 12afQ

jx0

pcQ k�/ð1Þ2

¼ lGJ0

F½v;/; k�: ð63Þ

For later convenience, we write (63) in abbreviated form as

F½�v; �/; k� ¼ A½�v ; �/� � cQkB½�v ; �/�; ð64Þ

that is, A½�v ; �/� stands for the sum of the material terms in (63)(independent of k), while B½�v ; �/� is the sum of the geometricalterms per unit value of cQk.

4. Numerical solution

Among the structural engineering community, the finite ele-ment method (FEM) is unquestionably the most popular choicefor solving the eigenproblem 2 (or, more frequently, its dimen-sional counterpart 1) – we mention in particular (i) the works of


Krajcinovic [52], Barsoum and Gallagher [53], Powell and Klingner[54] and Tebedge and Tall [55], concerning the lateral-torsionalbuckling analysis of doubly symmetric prismatic beams, (ii) theirgeneralization to asymmetric members by Yoo [56], and (iii) theformulations developed for tapered members by Wekezer [57],Yang and Yau [58], Bradford and Cuk [14], Bradford [13] – the onlyone on this list that accounts for the presence of elastic restraints –,Rajasekaran [59], Ronagh et al. [60] and Boissonnade and Muzeau[61] (a critical appraisal of the one-dimensional models underlyingthese tapered beam finite element formulations is given in [2]).The FEM is based on a variational (or weak) form of the problem,which involves derivatives of lower order than those appearingin the classical (or strong) form, and therefore poses less stringentregularity requirements. Moreover, the FEM procedures are verysimple and flexible when it comes to (i) describe irregular-shaped(multi-dimensional) domains and (ii) specify boundary conditions.

20

40

60

80

0.64 0.66 0.68 0.70 0.72

a=1.0

a=0.6

a=0.2

0

20

40

60

80

100

120

0.20 0.40 0.60 0.80 1.00

a=1.0

a=0.6

a=0.2

( )crv l

(φcr l

RC

( )ε −free endRC Tz z

1.0α =

0.6α =

0.2α =

λref crQ

0Qζ = Rfσ = +∞

%)Δ ( 0 0.1ωκ =


1.0α =

0.6α =

0.2α =

%)Δ ( 0 1.0ωκ =

Fig. 5. Influence of the distance between the rotation centre of the tip cross-section

However, these two features, responsible for the key role played bythe FEM in solving boundary value problems for partial differentialequations, are not essential in the case of ordinary differentialequations: the construction of a high-order B-spline basis, for in-stance, is more or less straightforward [62] and no particular geo-metrical flexibility is required. In fact, the method of choiceadvocated for abstract one-dimensional eigenproblems in thenumerical analysis literature, particularly when only a small num-ber of eigenvalues and eigenfunctions is sought, consists in theirreformulation so that they can be solved using a general-purposecode for boundary value problems (BVPs) in ordinary differentialequations (ODEs) [63–66]. As Seydel puts it, we ‘‘stay in the‘‘infinite-dimensional space” of ODEs and let standard softwaretake care of the transition to the finite-dimensional world ofnumerical approximation” [67, p. 263]. This approach has beensuccessfully applied to beam vibration problems [68] and to the

0

20

40

60

80

100

0.40 0.50 0.60 0.70 0.80 0.90

a=1.0

a=0.6

a=0.2

0

20

40

60

80

100

120

140

0.00 0.50 1.00 1.50

a=1.0

a=0.6

a=0.2

=

free end

free endRCz

)

λref crQ

0σ =Rx 0 0.1Jκ =


1.0α =

0.6α =

0.2α =

%)Δ ( 0 0.5ωκ =


1.0α =

0.6α =

0.2α =

%)Δ ( 0 2.0ωκ =

in the free-end case and the location where the translational restraint is placed.


lateral-torsional buckling analysis of prismatic and tapered cantile-ver beams and beam-columns [69–71].

The basic idea is to convert the original eigenproblem into aninhomogeneous two-point BVP in the ‘‘standard” form requiredby the software, for which the desired eigenpair corresponds toan isolated solution – recall that it has been argued, on physicalgrounds, that in the present case – i.e. the eigenproblem definedby (47)–(56), with the specified boundary condition replacementswhenever appropriate – the eigenvalues are simple. The conver-sion can be achieved in a variety of ways. Our choice was to supple-ment (47)–(56) with the first-order ODEs

ðcQkÞ;n ¼ 0; ð65Þ

#;n ¼12

�v2;nn þ

12

j2x0

p2 ½1� ð1� aÞn�2 �/2;nn

� 2ð1� aÞj2x0

p2 ½1� ð1� aÞn��/;nn�/;n

þ 12

1þ 4ð1� aÞ2 j2x0

p2 � jJ0ð1� aÞn� �

�/2;n ð66Þ

4

9

14

19

24

29

1.0E-03 1.0E-01 1.0E+01 1.0E+03

a=1

a=0.6

a=0.2

0

20

40

60

80

100

1.0E-03 1.0E-01 1.0E+01 1.0E+03

0Qζ = 1.0Tζ = − Rfσ = +∞ 0Rxσ = 0 0.1Jκ =

Q crγ λ

Tσ

0 2.0ωκ =

1.0α =

0.6α =

0.2α =

0 1.0ωκ =

0 0.5ωκ =

0 0.1ωκ =

100%elastically restrained free end

rcrcrigidly restrained free endcr cr

λ λρ

λ λ−

×=−

Tσ

0 0.1; 0.5;1.0; 2.0ωκ =

1.0; 0.6; 0.2α =

Fig. 6. Effect of a linearly elastic top-flange translational restraint at the tip.

and the boundary conditions

#ð0Þ ¼ rRf

2�v ;nð0Þ2 þ

j2x0

p2�/;nð0Þ2

� �; ð67Þ

#ð1Þ ¼ 1� rRx

2�/ð1Þ2 � rT

2�vð1Þ � afT

jx0

p�/ð1Þ

� �2: ð68Þ

The first additional ODE is just a statement of the fact that an eigen-value, when formally regarded as a function of n, is a constant. Thesecond ODE, together with the added boundary conditions, is equiv-alent to the normalization

A½�v ; �/� ¼ 1 ð69Þ

and makes the eigenfunctions unique, up to sign. Note that this nor-malization also allows one to drop the �/ – 0 requirement. (Ofcourse, when writing the boundary conditions (67) and (68) itwas tacitly assumed that rRf, rRx, rT – +1; the modifications re-quired to deal with an infinitely large r are obvious.) It should benoticed that (i) the augmented BVP is nonlinear, even though theoriginal eigenproblem is linear, and (ii) the boundary conditions re-main separated.

The problem can now be solved using standard software devel-oped for two-point BVPs. In this work, we employ the collocationcode COLNEW [72,73], designed to solve nonlinear multi-point BVPsfor mixed-order systems of ordinary differential equations andseparated boundary conditions. It is based on spline collocationat Gaussian points and incorporates rather sophisticated and ro-bust mesh selection and nonlinear iteration strategies. Unlike mostother general-purpose BVP codes, such as MATLAB’s bvp4c solver[74], COLNEW tackles (47)–(56) + (65)–(68) directly, without

0

10

20

30

40

50

60

0.0 0.5 1.0 1.5 2.0 2.5

a=1

a=0.6

a=0.2

0

20

40

60

80

100

120

0.0 0.5 1.0 1.5 2.0 2.5

a=1

a=0.6

a=0.2

Rfσ = +∞ 0Tσ = 0 0.1Jκ =

1.0α =0.6α =

0.2α =

( )nQ bγ λ

0ωκ

nd2 mode(free end; = 0)Qζ

st1 mode(free end ; = 0)Qζ

0ωκ

100%rigid restrained free endcr cr

free endcr

λ λλ

−Δ = ×

1.0α =

0.6α =

0.2α =

st1 mode( = + )Rxσ ∞

Fig. 7. Effect of a rigid torsional restraint at the tip.

4

6

8

10

12

14

16

18

20

22

24

1.0E-03 1.0E-01 1.0E+01 1.0E+03 1.0E+05

a=1

a=0.6

a=0.2

0Qζ = Rfσ = +∞ 0Tσ = 0 0.1Jκ =

Q crγ λ

Rxσ

0 2.0ωκ =

1.0α =

0.6α =

0.2α =

0 1.0ωκ =

0 0.5ωκ =

0 0.1ωκ =

100%elastically restrained free end

rcrcrigidly restrained free end

λ λρ

λ λ−

×=−


requiring its prior conversion into an equivalent first-order system– a definite advantage from the user’s viewpoint. An overview ofthe numerical techniques used by COLNEW and also by its immediatepredecessor COLSYS, together with references to the relevant mathe-matical literature, can be found in [71].

Since the parametric study reported in the next section requiresthe solution of chains of ‘‘nearby problems” (known as homotopychains [75, p. 65]), a relatively crude, but very effective, continua-tion strategy was implemented: the solution of a previous problemis used as the initial guess for the next one, having the same dataexcept for a small increment in a single parameter, a possibilitythat is already encoded in COLNEW.

5. Parametric study

Richard Hamming summed up his views on scientific comput-ing with the motto ‘‘the purpose of scientific computing is insight,not numbers” [76]. In the parametric study presented next, we willtry to live up to Hamming’s famous words and shed some light onthe behaviour of restrained tapered beams. Each type of restraint isaddressed separately – more specifically, we consider in successionthe cases (i) rT > 0, with rRx = 0 and rRf = +1, (ii) rRx > 0, withrT = 0 and rRf = +1, and (iii) rRf > 0, with rT = rRx = 0. In order to fo-cus on the effect of the restraints and its interplay with the degreeof web tapering, the parametric study is restricted to the centroidalloading case (fQ = 0). Moreover, having previously remarked thatthe parameter jJ0 is relatively unimportant, we always setjJ0 = 0.1 – as an indication, this corresponds roughly to memberswith h0 ffi 2b and tf ffi 2tw.

5.1. Built-in cantilevers (rRf = +1) with a translational restraint at thetip (rT > 0,rRx = 0)

For selected values of the web-taper ratio a ¼ hðlÞh0

and for

0:1 6 jx0 ¼ pl

ffiffiffiffiffiffiffiEI�x0GJ0

q6 2:5, Fig. 4 shows:

(i) The non-dimensional buckling loads cQ kcr ¼Qref l2ffiffiffiffiffiffiffiffiffiffiffiEI�z0GJ0

p kcr and

cQ kð2Þb , corresponding to the first (critical) and second buck-ling modes of cantilevers that are entirely free at the tip

rT ¼ kT l3

EI�z0¼ 0

� �– these results are shown for reference

purposes;(ii) The non-dimensional critical loads of cantilevers with a rigid

translational tip restraint (rT = +1), located at the top flange

fT ¼ 2zTah0¼ �1

� �, mid-height (fT = 0) and bottom flange

(fT = 1);(iii) The percentage increase of the critical load factor, relative to

the free-end case, due to the rigid restraint,

80

100 cr cr

D ¼ krigidly restrainedcr � kfree end

cr

kfree endcr

� 100%: ð70Þ

0

20

40

60

1.0E-03 1.0E-01 1.0E+01 1.0E+03 1.0E+05

Rxσ

0 2.0ωκ =

1.0α =

0 0.1ωκ =

0.2α =

Fig. 8. Effect of a linearly elastic torsional restraint at the tip.

The results presented in this figure unveil the importance of therestraint location. Indeed, the effectiveness of the restraint inincreasing the elastic buckling strength relative to the free-endcase is greatest when it is located at the top flange (fT = �1). More-over, the effectiveness of a top-flange restraint increases with theweb-taper ratio a. It is also visible in Fig. 4 that the benefit of atop-flange restraint generally increases with jx0 for a sufficientlylarge a. On the other hand, for moderate-to-high jx0 values, a bot-tom flange restraint is almost completely ineffective, especiallywhen the taper is not very pronounced. The main issue in explain-ing these behavioural features is the distance, when buckling setsoff, between the rotation centre of the tip cross-section in the

free-end case (i.e. without any restraint) and the location wherethe translational restraint is placed – the greatest this distance,the more effective the restraint will be. In order to back this asser-tion, we plot in Fig. 5, for selected values of a and jx0, the percent-

age increase D defined by Eq. (70) versus e zfree endRC � zT

� �¼

1l

ffiffiffiffiffiffiEI�z0GJ0

qzfree end

RC � zT

� �, where zfree end

RC denotes the z-coordinate of the

rotation centre in the free-end case and is given by

zfree endRC ¼ vcrðlÞ

/crðlÞ

� �free end

: ð71Þ

(In the present circumstances, it would not make sense to expresszfree end

RC � zT in relation to the cross-section height ah0, that is, touse fl instead of e.) Observe that:

(i) D always increases with e zfree endRC � zT

� �.

(ii) For a given jx0, the variation of D with e zfree endRC � zT

� �is

only mildly sensitive to the web-taper ratio a. For a givena, the dependence of these plots on jx0 is more pronounced,although this is not immediately perceived in Fig. 5.


Finally, notice that the critical load of a cantilever with arigid translational restraint (even if located at the top flange)is always well below the second-mode buckling load of thesame cantilever without the restraint – this is easily under-stood if the conjecture concerning the zeros of /ðnÞb proves tobe true.

Let us now turn to the influence of the restraint stiffness param-eter rT ¼ kT l3

EI�z0. We restrict ourselves to the case of a top-flange re-

straint (fT = �1). As shown in Fig. 6, there is a well-definedthreshold beyond which a further increase in rT causes a negligiblebuckling strength raise. Moreover, the ratio

q ¼ kelastically restrainedcr � kfree end

cr

krigidly restrainedcr � kfree end

cr

� 100% ð72Þ

is practically independent of a and jx0, and this is perhaps thedominant feature in this figure. It is therefore possible to state, withgenerality, that rT = 10 and rT = 50 guarantee q = 60% and q = 90%,respectively. These figures provide useful quantitative guidelinesfor designing the tip translational restraint as far as its stiffness isconcerned.

4

5

6

7

8

9

10

11

12

1.0E-03 1.0E-01 1.0E+01

a=1

a=0.6

a=0.2

4

5

6

7

8

0.2 0.4 0.6 0.8 1.0

0Qζ = Rx Tσ σ=

Q crγ λ

1.0α =

0.6α =

0.2α =

0 0.5ωκ =

0 1.0ωκ =

0 2.0ωκ =

Q crγ λ

210σ −=Rf

α

0 2.0

piecewise prismatic modelωκ =

Fig. 9. Effect of linearly elastic warping and mi

5.2. Built-in cantilevers (rRf = +1) with a torsional restraint at the tip(rRx > 0,rT = 0)

We now investigate the effect of a torsional restraint located atthe tip of a built-in cantilever. The examination of Fig. 7, concern-

ing a perfectly rigid restraint rRx ¼ kRxlGJ0¼ þ1

� �, prompts the fol-

lowing observations:

(i) The critical load of a cantilever with a rigid torsionalrestraint is always well below the second-mode bucklingload of the same cantilever with the tip entirely free – recallthat the same was found in the case of a rigid translationalrestraint.

(ii) The critical load factor percentage increase D ¼krigidly restrained

cr � kfree endcr

� �=kfree end

cr due to the rigid torsional

restraint generally grows with jx0. However, the rate ofgrowth gradually diminishes as the web-taper ratio adecreases. For a = 0.2, D remains comparatively constant.

The influence of the restraint stiffness parameter rRx isdepicted in Fig. 8. Once again, there is a limit value beyond which

1.0E+03 1.0E+05 1.0E+07

4

6

8

10

12

0.2 0.4 0.6 0.8 1.0

0= 0 0.1Jκ =

Rfσ

0 2.0ωκ =

0 1.0ωκ =

0 0.5ωκ =

0 0.1ωκ =

610σ =Rf

Q crγ λ

α

0 0.5ωκ =

0 1.0ωκ =

0 2.0ωκ =

0 2.0

piecewise prismatic modelωκ =

nor axis bending restraints at the support.


further increases in rRx lead to imperceptible gains in buckling

strength. As for the ratio q ¼ kelastically restrainedcr � kfree end

cr

� �=

krigidly restrainedcr � kfree end

cr

� �, it now exhibits some sensitivity with

respect to both a and jx0. Nevertheless, it can be safely said thatrRx = 10 and rRx = 60 guarantee at least q = 60% and q = 90%,respectively.

5.3. Overhanging beam segments free at the tip (rRx = rT = 0), withlinearly elastic warping and minor axis bending restraints at thesupport (0 < rRf < +1)

Fig. 9 shows the influence of the stiffness parameter rRf ¼2kRf lEI�z0

in

overhanging beam segments. For large jx0 (segments with shortspan and/or slender cross-section at the support) and fixed rRf,the non-dimensional critical load cQkcr does not increase monoton-

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

α=1α=0.6α=0.2

0.0

0.2

0.4

0.6

0.8

1.0

-0.4

0.0

0.4

0.8

1.2

0.0 0.2 0.4 0.6 0.8 1.0

α=1α=0.6α=0.2

-0.

0.

0.

0.

1.

0 2.0ωκ = 0Qζ = Rσ

1.0α =0.6α =0.2α =

210σ −=Rf

ξ

crv

φ cr

ξ

1.0α =0.6α =0.2α =

( )*

00

00

21 ωκφ φπ

− = −zcr cr cr cr

EI zv z v

l GJ h

topflange

bottomflange

( )crv x

( )φcr x

12( ) ( ) (φ+cr crv x h x x

12( ) ( )φ−cr crv x h x

Fig. 10. Critical buckling modes ð�vcr ; �/crÞ, normalized so as to have A½�vcr ; �/cr � ¼ 1, andflange centroidal lines.

ically with the web-taper ratio a. As a matter of fact, for low valuesof rRf, the trend is the opposite: cQkcr decreases as a increases. Forlarge values of rRf, cQkcr reaches a minimum at an intermediate va-lue of a. At first glance, these findings seem to defy our intuition,even more so because the piecewise prismatic model yields a stea-dy increase of cQkcr with a (see the dashed lines in the bottomgraphs of Fig. 9). However, this seeming paradox has an explana-tion, which is addressed next. The buckling strength of cantile-vers/segments with large jx0 (for illustration, we take jx0 = 2.0as reference throughout the discussion) depends to a large extenton the warping torsion (i.e. torsion due to non-uniform warping)generated when buckling occurs. Non-uniform warping reflects it-self in the membrane strains shown in Fig. 3(b) – in web-tapered I-beams, these strains comprise two terms, one associated with h /,xx

(also present in prismatic members) and the other depending ontgu/,x (peculiar to tapered members). For warping to be uniform,it is required that

0.0 0.2 0.4 0.6 0.8 1.0

α=1

α=0.6α=0.2

4

0

4

8

2

0.0 0.2 0.4 0.6 0.8 1.0

α=1α=0.6α=0.2

0x Tσ= = 0 0.1Jκ =

610σ =Rf

ξ

1.0α =0.6α =0.2α =

crv

φ cr

ξ

1.0α =0.6α =0.2α =

( )*

00

00

21 ωκφ φπ

− = −zcr cr cr cr

EI zv z v

l GJ h

topflange

bottomflange

12 ( )h x

)

12 ( )h x

( )x

corresponding non-dimensional lateral deflections �vcr � ½1� ð1� aÞn� jx0p

�/cr of the


h4

/;xx � tgu/;x ¼ 0; 8x 2 ð0; lÞ; ð73Þ

or, in non-dimensional terms,

½1� ð1� aÞn��/;nn � 2ð1� aÞ�/;n ¼ 0; 8n 2 ð0;1Þ: ð74Þ

In prismatic beams, (73) and (74) become simply

/;xx ¼ 0; 8x 2 ð0; lÞ; ð75Þ�/;nn ¼ 0; 8n 2 ð0;1Þ: ð76Þ

Let us examine first the low rRf case (say rRf = 10�2, for illustra-tion), corresponding to a practically non-existent warping restraintat the support, and see how the preceding interpretative keyapplies. The graphs on the left-hand side of Fig. 10, concerning can-tilevers with jx0 = 2.0, provide the basis for the ensuing discussion– they display (i) the critical buckling modes ð�vcr; �/crÞ associatedwith selected values of a, normalized so as to have A½�vcr ; �/cr � ¼ 1,and (ii) the corresponding non-dimensional lateral deflections ofthe flange centroidal lines. The careful consideration of thesegraphs prompts the following observations:

(i) One need not be concerned with the lateral bending of thecantilevers, since �vcr is practically independent of a.

(ii) In a prismatic member (a = 1), the �/-component of thecritical buckling mode is almost linear (the L2 norm of its

second derivative, k�/cr;nnkL2ð0;1Þ ¼R 1

0�/2

cr;nndn� �1

2, is very small,

as shown in Fig. 11), indicating that warping is nearly uni-form – this was already noted in [19] and leads to a lowbuckling strength.

(iii) Now, decrease a, starting from a = 1. Up to a certain point(roughly a = 0.4 0.5), the �/-components of the criticalbuckling modes remain almost linear, and their (practicallyconstant) slope gradually decreases (e.g., see the casea = 0.6 in Fig. 10). However, in view of (74), a linear �/cr nolonger means that warping is uniform, since the second termon the left-hand side of this equation, intrinsically associatedwith the web-height variation, does not vanish – notice thedifference in lateral bending curvature between the top andbottom flange centroidal lines for a = 0.6, which is clearlyhigher than in the prismatic case. There is thus some amountof warping torsion generated during buckling, responsiblefor the gradual increase in cQkcr with decreasing a observedin Fig. 9 (bottom left-hand corner).

Fig. 11. L2 norm squared of �/cr;nn per unit value of A½�vcr ; �/cr �.

(iv) When a drops below 0.4 (a situation typified by a = 0.2 inFig. 10), the critical buckling mode shapes �/cr exhibit a morenoticeable, even if still mild, curvature (as reflected in ahigher L2 norm of �/cr;nn – see Fig. 11), so that both termson the left-hand side of (74) are clearly non-zero. In the cen-tral portion of the span, these two terms have the same sign(�/cr;n and �/cr;nn have opposite signs), thus reinforcing eachother and leading to larger warping strains. Near the ends,on the contrary, they tend to cancel each other out (�/cr;n

and �/cr;nn have the same sign). The net effect, in this case,is to further increase cQkcr as a decreases.

A closer look at the material part of the non-dimensional energyfunctional (63) – i.e. at A½�vcr ; �/cr� – corroborates the above analysis.This functional is disassembled in its individual terms and, for easeof reference, the following notation is adopted:

Az½�v � ¼12

Z 1

0

�v2;nndn; ð77Þ

Ax½�/� ¼12

j2x0

p2

Z 1

0½1� ð1� aÞn�2 �/2

;nndn; ð78Þ

Axw½�/� ¼ �2ð1� aÞj2x0

p2

Z 1

0½1� ð1� aÞn��/;nn

�/;ndn; ð79Þ

Aw½�/� ¼ 2ð1� aÞ2 j2x0

p2

Z 1

0

�/2;ndn; ð80Þ

ASV ½�/� ¼12

Z 1

0½1� jJ0ð1� aÞn��/2

;ndn; ð81Þ

ARf ½�v; �/� ¼ rRf

2�v ;nð0Þ2 þ

j2x0

p2�/;nð0Þ2

� �: ð82Þ

In Fig. 12, the percentage contribution of these individual terms tothe aggregate total A½�vcr ; �/cr� is plotted as a function of a. The obser-vation of the top graph shows that:

(i) The ratio Az½�vcr �=A½�vcr ; �/cr� is independent of a and equal to50%. In fact, this result can be easily proven. Since rT = 0,one has

�v ;nn ¼ ð1� nÞcQ k�/; 0 6 n 6 1; ð83Þ

which is just the non-dimensional version of Eq. (34). There-fore (recall that we are taking fQ = 0),

cQkB½�v ; �/� ¼ cQ kZ 1

0ð1� nÞ�v ;nn

�/dn ¼Z 1

0�v2;nndn ¼ 2Az½�v �:

ð84ÞThe desired result now follows at once from the identity [77,Section 6.4]

A �vcr; �/cr�

¼ cQkcrB �vcr ; �/cr�

: ð85Þ

Note that this reasoning does not rest on any particular valueor order of magnitude assumed for rRf.�

(ii) The ratio Ax½�/cr �=A �vcr ; �/cr is always small, never exceeding5%, and its dependence on a is very mild.

(iii) As a decreases, one notices a steady decay in the ratioASV ½�/cr �=A �vcr; �/cr

� associated with St. Venant torsion, which

is compensated by an increase in Axw½�/cr � þ Aw½�/cr��

=

A �vcr; �/cr�

(i.e. an increase in the contribution of the termsinvolving the non-standard mechanical properties). The lat-ter is mainly due to Aw½�/cr�, whilst Axw½�/cr �=A �vcr ; �/cr

� remains always very small, oscillating between 0 and 3%.

Fig. 12. Contribution of the individual material terms (77)–(82), with �v ¼ �vcr and �/ ¼ �/cr , to the aggregate total A½�vcr ; �/cr �.


Suppose now that rRf is large (say rRf = 106, for illustration), sothat warping at the support is effectively prevented. Again, we mayfocus exclusively on the torsional behaviour at buckling, since lat-eral bending is unaffected by the web-taper ratio a. Near the sup-port, and for the entire range of variation of a, the �/-component ofthe critical buckling modes exhibits a substantial curvature and theflanges bend in opposite directions, as shown on the right-handside of Fig. 10. Further away from the support, the shape of �/cr de-pends on a: for moderate-to-high values of a (including the pris-matic case), it is almost a straight line; for smaller a, however,the �/cr display some curvature (this trend is reflected in Fig. 11).We therefore conclude that:

(i) In a prismatic member, warping is clearly non-uniform nearthe support, but nearly uniform throughout the remainingportion of the beam.

(ii) In a tapered member, both terms on the left-hand side of Eq.(74) are non-zero near the support, but since �/cr;n and �/cr;nn

have the same sign, they tend to cancel each other out, lead-ing to the generation of a smaller amount of warping torsionin this beam segment, when compared with the prismaticcase.

(iii) On the remaining portion of the beam, away from the sup-port, the first term on the left-hand side of (74) is practi-cally zero when a takes on moderate-to-high values; theeffect of the second term (the one associated with theheight variation), which is non-zero, turns out to be insuffi-cient to compensate for item (ii), and the net result is agradual decrease in cQkcr with decreasing a, at a slightly fas-ter rate than predicted by the piecewise prismatic model(see Fig. 9).

(iv) For sufficiently low a, both terms on the left-hand side of(74) are non-zero away from the support, with �/cr;n and�/cr;nn sharing the sign near the tip and having opposite signsin the central portion of the span. The overall effect is toraise the non-dimensional buckling load cQkcr as a decreasesin the range a < 0.6, as shown in Fig. 9.

The plot at the bottom of Fig. 12 corroborates the above analysis– indeed:

(i) The ratio Ax½�/cr �=A �vcr ; �/cr�

is practically independent of a,but its contribution is now substantial, in stark contrast withthe low rRf case.


(ii) The ratio Axw½�/cr �=A �vcr ; �/cr�

is always negative for 0.2 6 a <

1.0. The sum Axw½�/cr � þ Aw½�/cr ��

=A �vcr ; �/cr�

reaches a mini-

mum at a = 0.6 (which virtually coincides with the mini-mum in the graph cQkcr vs. a for jx0 = 2.0 shown in Fig. 9)and is positive for 0.2 6 a < 0.4 (roughly agreeing with thedomain where the values of cQkcr for jx0 = 2.0, yielded bythe tapered model, are above those predicted by the piece-wise prismatic model).

6. Conclusion

The paper presented an investigation on the elastic lateral-tor-sional buckling behaviour of discretely restrained tapered beamsthrough the analysis of a specific model problem that ‘‘containsall the germs of generality”.

The one-dimensional model previously developed and validatedby the authors was briefly described and physical interpretationswere given for its non-classical features – i.e. the introduction ofnovel mechanical properties in addition to those appearing in Vlas-sov’s theory for prismatic beams. This one-dimensional model wasthen extended to include the presence of linearly elastic or rigiddiscrete restraints, which may have a translational, torsional, min-or axis bending and/or warping nature. The resulting self-adjointeigenproblem was given in both variational and strong forms,being subsequently cast in non-dimensional form over a fixed ref-erence domain. A complete set of independent non-dimensionalparameters was thereby defined, providing an adequate settingfor carrying out parametric studies. For the numerical solution ofthe non-dimensional eigenproblem, we departed from the tradi-tional finite element approach. Instead, the problem was first con-verted into a ‘‘standard” inhomogeneous two-point boundaryvalue problem and then solved numerically with the collocationpackage COLNEW.

It was shown, both theoretically and through numerical results,that it is conceptually incorrect to model a web-tapered I-beam asan assembly of prismatic segments, regardless of the number ofsegments considered. In fact, a piecewise prismatic modellingmay even fail to capture the trend of variation of the non-dimen-

sional critical load cQ kcr ¼Qref l2ffiffiffiffiffiffiffiffiffiffiffiEI�z0GJ0

p kcr with the web-taper ration

a ¼ hðlÞh0

. The differences are particularly visible when the warping

torsion plays an important role in resisting buckling. Indeed, inthe analysis of overhanging segments free at the tip and with lin-early elastic warping and minor axis bending restraints at the sup-

port, it was found that for a large jx0 ¼ pl

ffiffiffiffiffiffiffiEI�x0GJ0

q(short span and/or

slender cross-section at the support) and a fixed restraint stiffness

parameter rRf ¼2kRf lEI�z0

, the non-dimensional critical load cQkcr does

not increase monotonically with the web-taper ratio a. For low val-ues of rRf, cQkcr decreases as a increases, while for large values ofrRf, cQkcr reaches a minimum at an intermediate value of a.

From the analysis of built-in cantilevers with a translational or arotational restraint at the tip, the following conclusions can bedrawn:

(i) The critical load of a cantilever with a rigid restraint at thetip (either translational or torsional) is well below the sec-ond-mode buckling load of the same cantilever withoutthe restraint.

(ii) The main factor in explaining the effectiveness of a rigidtranslational restraint in increasing the buckling strength isthe distance between the rotation centre of the cross-sectionin the free-end case and the location of the translationalrestraint – the greatest the distance is, the more effectivethe restraint will be.

(iii) There is a well-defined threshold beyond which the outcome,in terms of buckling strength, of further increases in the stiff-ness of the restraints is negligible. In the case of a translationalrestraint at the tip, the ratio q ¼ kelastically restrained

cr ��

kfree endcr Þ= krigidly restrained

cr � kfree endcr

� �was found to be practically

independent of a and jx0. On the contrary, in the case of arotational restraint, q exhibits some sensitivity with respectto both a and jx0. Simple quantitative guidelines for the stiff-ness of the restraints were given in both cases.

Acknowledgements

The first author gratefully acknowledges the financial supportof FCT (Portuguese Foundation for Science and Technology)through the Doctoral Grant SFRH/BD/39115/2007.

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elastic lateral-torsional buckling of restrained web-tapered i-beams

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