RESEARCH PAPER

On the distortion and warping of cantilever beamswith hollow section

Timo Björk1 & Antti Ahola1 & Tuomas Skriko1

Received: 3 September 2019 /Accepted: 24 April 2020# The Author(s) 2020

AbstractThis paper deals with the stress analysis of a cantilever box beam subjected to static or fluctuating torsional moment loading.Such beams may have multiple critical locations from the strength point of view; one interesting detail is the cross section wherethe loading is imposed. However, such details can be typically designed to possess smooth shapes, resulting in moderate stressconcentrations, which mean that fatigue failures can be avoided. However, the distortional deformation of the cross sectioninduces transverse bending stresses, which may be detrimental, particularly in welded box beams. In addition, the fixing locationwhere the beam is typically welded to an end plate may become a critical point. This paper presents an analytical approach forcalculating the longitudinal stresses due to the warping of the cross section as well as the distortion-induced transverse andlongitudinal stresses in rectangular hollow sections. Finite element analyses (FEAs) are carried out to verify the analyticalapproach to shed light on the critical points in the end plate details with different degrees of weld penetration using the effectivenotch stress (ENS) concept and to suggest design proposals for an efficient structural detailing of diaphragm plates to decrease thewarping behavior.

Keywords Distortion .Warping . Secondarywarping . Hollow cross section .Welded joints . Ultra-high-strength steel

1 Introduction

Rectangular hollow sections (RHSs) usually offer good,or at least moderate, structural performance for beamssubjected to various loads, such as axial and shear forcesas well as bending and torsional moments. In warping, thecross section of the beam remains in its original shape,whereas in distortion, it will take on a parallelogramshape. Both these deformations cause axial displacementsand/or membrane stresses in the beam, but distortion alsocauses transverse bending stresses in the wall of the crosssection. These phenomena are independent degrees offreedom of the beam. Particularly from the fatigue designpoint of view, the bi-moment causing the warping of the

cross section and distortional loads of RHS beams is aninteresting, but lesser known load condition. The funda-mental basics of the primary warping of box beams can befound, e.g., in Ref. [1–4], while the secondary warping ofthe cross section is covered in Ref. [5–7]. Rubin [8] stud-ied the effect of the corner radius on the primary warpingof RHSs, showing that it also causes primary warping insquare hollow sections (SHSs). Meanwhile, the distortionof closed cross section has been studied in Ref. [9–12].Nevertheless, previous studies tend to focus on the theo-retical background of the distortion and warping of crosssections, and less attention has been paid to practical anal-ysis and design proposals to prevent fatigue failure due towarping and distortion. In weight-critical beams, the useof ultra-high-strength steels (UHSSs) with yield strengthsup to 1100 MPa becomes reasonable. With UHSS grades,hot- or cold-formed profiles are not typically available,and box beams made from welded sheet metals are usedinstead. In such structures, the fatigue-critical details arethe brackets, longitudinal welds, and transverse welds fix-ing the beam to the end plate.

The present paper introduces an analytical approach tocalculate the stresses that can be used in the fatigue

Recommended for publication by Commission XV-Design, Analysis,and Fabrication of Welded Structures

* Timo Björktimo.bjork@lut.fi

1 Laboratory of Steel Structures, LUT University, P.O. Box 20,53851 Lappeenranta, Finland

https://doi.org/10.1007/s40194-020-00911-5

/ Published online: 9 May 2020

Welding in the World (2020) 64:1269–1278

analysis of a cantilever beam. First, the bi-moment andwarping stress distributions along the beam length andover the cross section are obtained and, subsequently, thedistortional behavior is established based on the theory of abeam on an elastic foundation (BEF) [13]. Finite elementanalyses (FEAs) using 2D mid-plane element models areconducted to obtain numerical results for a comparison. Inaddition, a more detailed analysis of the secondary warpingof a square hollow section (SHS) with a round corner isconducted using a 3D solid element model. A fatigue anal-ysis is conducted on the end plate of a cantilever beamusing the effective notch stress (ENS) concept [14] andconsidering both weld toe and weld root fatigue failures.Furthermore, with FE modeling, the effect of a transversediaphragm plate and its location on the distortion of thecross section is investigated; this is followed by some de-sign recommendations. The work is focused on beamsloaded with torsional moment, and therefore, SHS profileis the main focus of this work. In addition, RHSs are alsoanalyzed to provide a comparison with an SHS.

2 Analytical approach

2.1 Geometry, load, and boundary conditionsof a cantilever beam

In this study, the force couple loading F is imposed on pair ofbrackets at the free end of a cantilever beam, as illustrated inFig. 1. The other end of the beam is fixed to a stiff end plate,thus representing a completely rigid boundary condition. Theloaded end of the beam is provided with a diaphragm plate,preventing the distortion of the cross section in the placewhere the brackets are fixed to the beam. However, as theplate is very thin, it does not prevent the free warping of thebeam in this section.

2.2 Warping due to torsional loading

Using the cross-section symbols denoted in Fig. 1, for a thin-walled closed section with constant plate thickness t, the tor-sional moment of inertia It is

I t ¼ 4A20

∮s

dst

¼ 4b2h2

∑i

siti

¼ 2b2h2tbþ h

ð1Þ

The warping constant of the cross section Iω consists ofprimary and secondary parts, whereby the primary part is cal-culated based on the sectorial coordinate ω of the middle lineof the cross section and the secondary part is based on thewarping over the plate thickness in direction rn for each crosssection part.

Iω ¼ ∮sω2tdsþ 2 ∮

s∫t=2

rn¼0r2ns

2drnds

¼ ∮s

rt−2Ao

t ∮s

dst

0BB@

1CCA

2

tdsþ ∮s

t3

48s2ds

¼ b2h2t24

h−bð Þ2hþ b

þ 2 b3 þ h3� �

t3

144ð2Þ

In the case of SHS, the dimension h = b and, consequently,primary warping is zero; however, except for a rotationallysymmetric cross section, there always exists secondarywarping, which for an SHS with sharp corners is:

Iω ¼ b3t3

36ð3Þ

In general loading cases, the beam can be subjectedto both axial and transverse (shear) loads and thus alsobending moments. Nevertheless, the main focus of this

Fig. 1 The studied and analyzedbox beam structure

1270 Weld World (2020) 64:1269–1278

study is on the torsional moment T caused by the cou-ple force F as follows:

T ¼ Fbb ð4Þ

The torsional moment T in the brackets generates the bi-moment Bω,0 at the loaded end of the beam:

Bω;0 ¼ Tc ¼ Fbbc ð5Þ

The bi-moment depends on the second derivate of thetwisting angle θ relative to the x-axis.

Bω xð Þ ¼ E0Iω

d2θ

dx2

¼ Eb2h2t24

h−bð Þ2hþ b

þ 2 b3 þ h3� �

t3

144 1−ν2ð Þ

" #d2θ

dx2; ð6Þ

where E and v are the Young’s modulus and the Poisson’s ratio ofmaterial, respectively. The bi-moment of the beam consists of Bω,0and the effect of the torsion moment T. Considering the presentboundary and load conditions, as shown in Fig. 1, the bi-momentdistribution Bω, (x) over the beam length can be defined:

Bω xð Þ ¼Bω;0coshk L−xð Þ þ T

ksinhkx

coshkL

¼ Fbbckcoshk L−xð Þ þ sinhkx

kcoshkL; ð7Þ

where the torsion constant k is

k ¼ffiffiffiffiffiffiffiffiGItEIω

r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI t

2 1þ vð ÞIω

sð8Þ

In Eq. (8),G is the shear modulus of material. At the loadedend of the beam, Bω,x = 0 = Fcbb, and at the fixed end:

Bω;x¼L ¼ Fbbck þ sinhkLkcoshkL

ð9Þ

The bi-moment induces axial stresses in the beam, whichconsists of membrane stresses due to the primary warping andshell bending stresses due to the secondary warping. For anSHS with sharp corners, the following equations can be de-rived:

σω sð Þ ¼ Bω xð ÞIω

ω ¼ 0

σω s; rnð Þ ¼ 36Bω xð Þb3t3

srn ¼ � 9Bω xð Þb2t2

; at

the outer or inner corner of the SHS‐profle

ð10Þ

This torsional moment T consists of the pure (or uniform)torsion TBredt and the warping torsion Tω.

T ¼ TBredt þ Tω ¼ GItdθdx

þ EIωd3θ

dx3ð11Þ

The shear stresses involved in warping can now be definedas

τω ¼ TωSωIωt

¼ dBω xð Þdx

SωIωt

¼ Fbbcoshkx−cksinhk L−xð Þ

coshkLSωIωt

; ð12Þ

where Sω is the sectorial cross section modulus. Based onBredt’s formula, the shear stress is

τBredt ¼ TBredt

2bht¼ T−Tω

2bht

¼ Fbb2bht

1þ cksinhk L−xð Þ−coshkxcoshkL

� �ð13Þ

The schematic distributions of internal forces over thebeam length and the warping stresses over the crosssection area are presented in Fig. 2. In the case ofSHS, the bi-moment will occur only at the loaded andfixed ends of the beam.

The secondary warping occurs very locally in the cor-ners of the box section, but it can induce detrimentalstresses if the local degree of freedom, involved in thesecondary warping, is prevented. This can occur whenthe wall of the cross section is locally prevented to rotate,e.g., due to joining the beam by welding to a stiff trans-verse plate component. It is worth noting, however, thatthe primary warping stresses depend on the global bound-ary conditions of the beam, while the secondary warpingstresses depend on the local boundary conditions of thecross-section wall. Consequently, primary and secondarywarping can occur independently on each other. The pri-mary warping of the cross sections with round corners hasbeen investigated analytically by Rubin [8]. In this study,this issue is addressed by means of the FEA in Sect. 3.

2.3 Distortion of the beam

In addition to the warping, the torsional load, causingthe local bi-moment at the loaded end of the beam, alsoestablishes a distortional loading. Unlike with thewarping, in the distortion, the cross section does notremain in its original form but deforms to a parallelo-gram shape. Because the corners of the box section arerigid, the distortion causes transverse bending stresses.Both warping and distortion includes longitudinal defor-mations and stresses, whereby the distributions alongthe profile perimeter are affine yet disassociated in thelongitudinal direction of the beam. Based on the theoryof BEF, Fig. 3 shows the longitudinal and transverse

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distributions of distortion, which are compared in somecases with the results of the numerical analyses de-scribed in Sect. 3. The distortional deformation v(x)(= d1 − d2, see Fig. 3) is the difference between the di-agonal dimensions of the cross section as defined by thedeformed geometry. The distortional deformation in the

longitudinal direction of the SHS beam with a semi-infinite length can be determined based on the theoryof BEF [15]:

v xð Þ ¼ 12ffiffiffi2

pe−βxsinβx

β2Etb3Fc; ð14Þ

Fig. 3 Distortion of the SHSbeam (L = 4000 mm) with a crosssection with the dimensions240 × 240 × 4 mm loaded with aforce couple of F = 150 kN (bb =240 mm, c = 100 mm); based onthe theory of BEF: a distortionaldeformation, b transversebending stress (σb) at the corner,and c longitudinal membranestress (σv) at the corner, obtainedfollowing the analytical approachand FEA. Further detailsregarding the FEAs can be foundin Sect. 3

Fig. 2 a Bi-moment and b tor-sional moment distributions overthe beam length, c primary andsecondary warping stresses at thecorner of an RHS cross section,and d Bredt’s and warping shearstresses in the cross section of anRHS

1272 Weld World (2020) 64:1269–1278

where

β ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12t2

b6 1−ν2ð Þ4

sð15Þ

The elastic foundation constant f for the BEF approach canbe defined by analyzing the cross section as a continuousframe and calculating the transverse stiffness of the framebased on the unit force-displacement-method.

f ¼ 4Et3

b3 1−ν2ð Þ ð16Þ

The maximum deformation vmax occurs when

dvdx

¼ 4ffiffiffi2

pβ2e−βx sinβx−cosβxð Þ

fFc ¼ 0→tanβx

¼ 1→xvmax ¼arctan1

β¼ π

4β¼ π

4

ffiffiffiffiffiffiffiffiffiffi1−ν2

12

4

r ffiffiffibt

rb ð17Þ

If the distortion occurs due to fluctuating torsion, itcan be a detrimental phenomenon leading to the fatigueof longitudinal welds if they are located in the cornersof the box beam. The longitudinal welds (all or at leastsome) are typically prepared using single-sided weldingand, thus, the weld root fatigue capacity may become acritical design criterion. The phenomenon can be dimin-ished through the use of diaphragm plates; however, thiseffect only occurs locally. Naturally, the first diaphragmplate is located at the fixing point on the brackets, atthe loaded end of the beam. However, this plate pre-vents the distortion of the cross section only at thatposition. Along the beam, the distortion v(x) increasesaccording to Eq. (14), reaching a maximum at the dis-tance of π/(4β) and, subsequently, fading gradually. Asimple engineering design concept could indicate plac-ing of a second diaphragm at precisely the place ofmaximum distortion. Theoretically, the optimal locationfor an additional diaphragm could be determined usingthe theory presented by Hetényi [13], but this requires

RHS h × b × t(mid-plane)

F = 150 kN

F = 150 kN

Location of diaphragm plate in 240 × 240 × 4

(l = 200; 400; 600 mm)Brackets with

nodal forces z

y

x

Fixed boundary

(all DOFs restrained)

End plate

l

Analyzed geometries

h [mm] b [mm] t [mm]

240 240 4

240 240 6

340 240 4

340 240 6

Corner radius r = 0

Fig. 4 Dimensions of the SHSbeam used in the FEA

Fig. 5 a General overview ofnormal stresses due to secondarywarping in a sharp corner of anSHS and b distributions of thesame stresses along the inner,middle, and outer surfaces of theprofile [17]

1273Weld World (2020) 64:1269–1278

very complicated analytical formulation and is, there-fore, ignored here. However, the optimal location ofthe second diaphragm plate is investigated numericallyby means of FEAs in Sect. 3.

The transverse bending stress σb(x) follows the distributionof distortional displacement v(x) and is

σb xð Þ ¼ 3Et

2ffiffiffi2

pb2 1−ν2ð Þ v xð Þ ð18Þ

The longitudinal membrane stress of the beam σv(x)follows the second derivative of the displacement and,consequently, can be calculated using the followingequation:

σv xð Þ ¼ 3e−βxcosβx

tb2Fc ð19Þ

3 Numerical analysis

3.1 FE model

The FE model for a beam with the load and boundary condi-tions is shown in Fig. 4. The beam is made of steel material

and, thus, E = 210 GPa and v = 0.3. Linear 4-noded plate ele-ment (CQUAD4) models with linear static analyses are car-ried out using NxNastran as a solver. More detailed descrip-tions of the models and analyses are given byAhlfors [16, 17].A beam length of L = 2000 mm was used as the basis in theanalyses. However, to validate the analytical approach (seeSect. 2), in which a semi-finite beam was assumed, an L =4000 mmmodel (240 × 240 × 4) was analyzed to also take thedamping effect (e−ßx) into account in the FEA. The results ofthose analyses are presented in Fig. 3. Furthermore, linear 8-noded solid element (CHEXA) models were used to investi-gate the secondary warping of the cross section and to analyzethe welded end plate detail using the ENS concept. The ele-ment mesh models used in the analyses can be found in Sect.3.2.

3.2 Warping of the cross section

In an SHS beam with sharp corners, the deformations andstresses due to primary warping are zero. However, secondarywarping occurs at the fixed end of the beam. A general over-view of the normal stress distribution in the fixed corner andthe same stress distribution in the inner, middle, and outersurfaces of the profile face derived using both plate and solidelements are shown in Fig. 5 a and b, respectively. The

Free end

(loaded)

Fixed end

War

pin

g

func

tion

4ω b2

0.04

0.03

0.02

0.01

0 0.1 0.2 0.3 0.4 0.5

r/h

(a) (b)

Fig. 6 a Axial stress distributionat the fixing point and b the effectof the corner radius to height ratioon the degree of warping of theSHS 150 × 150 × 8 [17]

s [mm] s [mm]

0 20 40 60 80 100 120 140 40 60 80 100 120 140

0.25

0.125

0

-0.125

-0.25

0

-0.25

-0.50

-0.75

Solid elem.

Plate elem.

Solid elem.

Plate elem.

ss

0.125

-0.125

-0.375

-0.625

centerline

of the corner

(a) (b)

Fig. 7 Membrane stress to shearstress ratio for a SHS 150 × 150 ×8 and b RHS 200 × 100 × 8beams subjected to pure torsionloading [17]. Parameter s is thedistance along the center line ofprofile perimeter and the stressdistribution is presented for onequarter of the whole cross section

1274 Weld World (2020) 64:1269–1278

adjacent normal stress components in the horizontal and ver-tical plates are opposite and, thus, induce high local shearstresses in the corner acting parallel to the longitudinal axisof the beam. The normal and shear stresses occur only locallyand have no impact on the static capacity of such a beamstructure, but they must be considered in fatigue analysis be-cause they can have a remarkable negative role on the struc-tural performance and durability.

If the box section has round corners, there also exist, inaddition to secondary (bending) stresses, primary(membrane) stresses due to the warping of the SHS.Figure 6 depicts the typical stress distribution of an SHS pro-file and the effect of the corner radius to beam height ratio onthe degree of primary warping in the corner area. If the cornerradius to height ratio of profile (r/h) approaches zero, theprimary warping will vanish, and if the ratio approaches avalue of 0.5, all warping will vanish. Figure 6 b shows thatthe maximum warping is reached when r/h = 0.18.

Figure 7 shows the ratio of normal stress to Bredt’s shearstress along the profile perimeter for the SHS and RHS pro-files. It can be seen that in the corner area, the axial membranestresses can be approximately half the shear stresses.

To consider the fatigue of the SHS with a round corner, theeffective notch stress (ENS) concept [14] was applied to a 3Dgeometry, with modeled welds and a reference radius of rref =1.0 mm at the weld toe and root, in the joint where the highestaxial stress occurs, as illustrated in Fig. 8a. The dimensions ofthe profile are given in the Fig. 8b.

The results of the analyses are presented in Fig. 9. Theweld throat thickness a ≈ 1.1 t was equal in all FE models,but the degree of penetration was varied so that themodels represented joints with fully and half penetratedas well as pure fillet welds. In all cases, the weld toe is thecritical part of the joint, not the root side. Consequently,the stress distributions over the plate thickness are pre-sented only at the weld toe. The presented distributions

r

B

H

t

rref = 1 mm

n

t

H = 150 mm

B = 150 mm

t = 8 mm

r = 16 mm

(a) (b)

Fig. 8 a SHS detail analyzedwiththe ENS method and b thedimensions of the analyzed SHSprofile (modified from [17])

Fig. 9 Stress ratio distributionsover the plate thickness at theweld toe with different degrees ofpenetration: a full penetration, bhalf penetration, and c pure filletweld [17]

1275Weld World (2020) 64:1269–1278

are normalized by dividing the normal stresses by Bredt’sshear stress.

3.3 Distortion of the cross section

In Fig. 10, the deformations for different profile sizes arepresented for a beam length of 2000 mm.

In Fig. 10, it can also be seen that some of the distortions ofthe beam are at the location of the first diaphragm plate (x = 0).This is due to the shear deformation of the diaphragm plate, ofwhich thickness is tp.

v0 ¼ γb ¼ τbG

¼ FbGhtp

¼ 150000⋅24080000⋅240⋅4

¼ 0:47mm ð20Þ

Figure 11 presents the effect of the plate thickness t on thedistortional deformation of the SHS profile with the dimen-sions 240 × 240 × t. A 50% increase in the plate thicknessdecreases the deformation nearly by approximately 50%.This cannot be concluded directly from Eq. (14) since thethickness t is involved in the β parameter.

Figure 11 also demonstrates the effect of the location of thesecond diaphragm plate in the SHS beam with the aforemen-tioned dimensions. The optimum location for the distance seems

to be roughly the dimension of the beam height. Figure 12 showsthe effect of diaphragm plates on the axial stresses of the SHSand RHS beams. The transverse plates slightly prevent thewarping deformation, acting as internal springs for the bi-mo-ment, but locally, they prevent the distortion efficiently.

4 Discussion

The normal and shear stresses due to warping and distortioncannot be neglected in structures subjected to torsional loading.Distortion should always be considered in the analysis of flexi-bility, stability, fatigue, and ultimate capacity of joints with hol-low sections. In the present paper, the distortion and warpingbehavior of SHS and RHS members subjected to torsional mo-ment were investigated analytically on the basis of theoreticalmodel for stress analysis and, numerically, conducting FEAs.

The length of a beam structure plays a crucial role in thewarping and distortion distributions along the beam length. InRHS beams, the primary warping is typically rather small (if h/b< 2) and the local peak value occurs only at the loading andfixing points of beam, as illustrated in Fig. 2. The distortion hasitsmaximum at distance of 0.412b√(b/t), after which it fades verysmoothly, as illustrated in Fig. 3. Even a 4000-mm-long beamwith the cross-section dimensions 240 × 240 × 4 mm is not longenough to completely eliminate the distortional deformations andstresses. This causes small differences between the theoreticalmodel and the FEA, see Fig. 3, but generally, the agreementbetween the analytical and FEA calculations is good.

Stresses due to warping and distortion can have a remark-able effect on the fatigue performance of a beam subjected tocyclic or fluctuating torsional loading. The primary warping,due to the round corner together with secondary warping, cancause fatigue failures even in the case of SHS, as shown inFig. 13a. The distortion-induced cyclic transverse bendingstresses, see Fig. 3b and Eq. (18), can also cause fatigue fail-ures longitudinally at the corner of SHS members, as demon-strated in Fig. 13b. Particularly, in the case of UHSSmaterials,fillet-welded hollow sections are fabricated for steel construc-tions, e.g., in boom components, due to the lack of tubularmembers made of high- and ultra-high-strength steel grades.In such welded details, lack of weld penetration and sharptransition at the weld root expose the longitudinal welds forweld root fatigue failures. Consequently, careful considerationof cyclic torsional loads in the stress analysis is particularlyrequired in welded box beams. In Ref. [18, 19], fatigue fail-ures in an SHS with an end plate, and subjected to the cyclictorsional moment, were found due to the warping and distor-tion. Nevertheless, these studies did not address the warpingbehavior in more detail.

The diaphragm plate at the loaded end of the beam does notprevent distortion, although it does curtail it locally. When con-sidering the optimal location for stiffeners and diaphragm plates,

0 400 800 1200 1600 2000

x [mm]

0

2

4

6

d1 – d2 [mm]

d1

d2

SHS 240 × 4 – No diaphragm plate

SHS 240 × 4 – Diaphragm plate (l = 200 mm)

SHS 240 × 4 – Diaphragm plate (l = 400 mm)

SHS 240 × 4 – Diaphragm plate (l = 600 mm)

SHS 240 × 6 – No diaphragm plate

Fig. 11 Effect of plate thickness and the location of the second diaphragmplate on the distortion of the cross section [16]

Profile: SHS 240 × 4

Profile: SHS 240 × 6

Profile: RHS 340 × 240 × 4

Profile: RHS 340 × 240 × 6

0 400 800 1200 1600 2000

x [mm]

0

2

4

6

d1 – d2 [mm]

d1

d2

Fig. 10 Effect of the profile cross section dimensions on the distortion ofthe beam [16]

1276 Weld World (2020) 64:1269–1278

these are usually located where the maximum deformation oc-curs without such stiffener, see Eq. (17). On the basis of theresults of this study, see Fig. 11, this approach is not applicablewhen considering the warping behavior. If diaphragm plates arelocated too far from each other, the distortion occurs between thediaphragm plates, and if they are too close, the effectiveness ofthe second plate is insufficient. The second plate after the endplate is important because it creates a transverse torsion bar insidethe beam, which connects the torsional couple moments. Thestiff torsion bar efficiently destroys the bi-moment induced bythe couple forces in the brackets. A transverse bar in the shape ofany closed hollow section, as illustrated in Fig. 14, is stiff andwill efficiently eliminate the bi-moment loading by causing

torsion shear stresses in the transverse box. This groundingspring effect is addressed in greater detail in Ref. [20]. The ef-fectiveness is based on the torsional stiffness St,p of the transversebeam created by these diaphragm plates and can be evaluatedusing the following equation:

St;p ¼ GItb

¼ 2Gh2l2

bhtpþ l

t

� � ð21Þ

The recommended distance for the second diaphragm platestill enables the welding, also in a case of commercially available

σv [MPa]

100

σv [MPa]

200

300

0

-1000 500 1000 1500 20000 500 1000 1500 2000

100

200

300

0

-100

No diaphragm plate (t = 4 mm)

Diaphragm plate (l = 200 mm)

Diaphragm plate (l = 400 mm)

Diaphragm plate (l = 600 mm)

No diaphragm plate (t = 6 mm)

x [mm]x [mm]

t = 4 mm

t = 6 mm

(a) (b)

Fig. 12 Longitudinal membranestresses (σv) for a SHS 240 ×240 × t and b RHS 340 × 240 × t[16]

(a) (b)

Fig. 13 Warping-induced fatiguefailures in a SHS beam with thecross section dimensions of100 × 100 × 5 mm subjected tocyclic torsional loading: a fatiguefailure at the weld toe of an endplate and b longitudinal fatiguecrack at the corner of a SHSmember [18]

(a) (b)

l

h

b

tp tpt

t

Fig. 14 The bi-moment isgrounded by a square or b circularshapes, which can be created bydiaphragm plates, separate tubes,or castings

1277Weld World (2020) 64:1269–1278

RHSbeams. If the box section is self-made design, the diaphragmcan be welded from three sides before closing the section, whichalso enables longer distance for the second diaphragm.

5 Conclusions

Based on the theoretical and numerical analyses along withexperimental findings from the literature, the following con-clusions can be drawn:

& Primary warping (causing either axial membrane stressesor deformations) occurs also in SHSs with round corners.

& Primary and secondary warping should be considered inthe fatigue analysis of SHSs subjected to fluctuating tor-sional loading, particularly at the rigid fixed end.

& Distortion can have effect on the flexibility, stability, andultimate capacity of joints, also in static loading.

& Transverse bending stresses due to distortion can have anegative impact on the fatigue performance of beams withhollow cross sections, especially if the fixing welds arelocated in the corners of the cross section.

& A diaphragm plate located only at the loaded end does notprevent distortion in a beam.

& A second diaphragm plate efficiently decreases the distor-tion if it is located at a distance of approximately theheight of the beam from the end plate.

& In the fillet-welded detail at the fixed end, higher stressesresulted at the weld toe than at the weld root, also in purefillet welds with no penetration.

Acknowledgments Open access funding provided by LUT University.The authors wish to thank Business Finland for funding the ISA-LUT-project and, hence, for enabling this research work to be completed.

Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing, adap-tation, distribution and reproduction in any medium or format, as long asyou give appropriate credit to the original author(s) and the source, pro-vide a link to the Creative Commons licence, and indicate if changes weremade. The images or other third party material in this article are includedin the article's Creative Commons licence, unless indicated otherwise in acredit line to the material. If material is not included in the article'sCreative Commons licence and your intended use is not permitted bystatutory regulation or exceeds the permitted use, you will need to obtainpermission directly from the copyright holder. To view a copy of thislicence, visit http://creativecommons.org/licenses/by/4.0/.

References

1. Vlasov VZ (1949) Строительная механика тонкостенныхпространственных систем (in Russian)

2. Vlasov VZ (1963) Thin-walled elastic beams. Israel Program forScientific Translations, Jerusalem

3. Murray NW (1984) Introduction to the theory of thin-walled struc-tures. Oxford University Press, New York

4. Roik K, Sedlacek G (1966) Theorie der Wölbkrafttorsion unterBerücksichtigung der sekundären Schubverformungen –Analogiebetrachtung zur Berechnung des querbelastetenZugstabes 2:43–52

5. Kollbrunner CF, Hajdin N (1975) Dünnwändige Stäbe, Band 2:Stäbe mit deformierbaren Querschnitten NichtelastischesVerhalten dünnwandiger Stäbe. doi: https://doi.org/10.1007/978-3-662-06782-6

6. Björk T (1989) Secondary warping constant of a beam subjected totorsional load (in Finnish). J Struct Mech 21:3–24

7. Rubin H (2005) Wölbkrafttorsion von Durchlaufträgern mitkonstantem Querschnitt unter Berücksichtigung sekundärerSchubverformungen. Stahlbau 74:826–842

8. Rubin H (2007) Torsions-Querschnittswerte für rechteckigeHohlprofile nach EN 10210–2 : 2006 und EN 10219–2 : 2006.Stahlbau 76:21–33. doi: https://doi.org/10.1002/stab.200710004

9. Křístek V (1979) Theory of box girders. John Wiley & Sons Ltd,Prague

10. Křístek V, Škaloud M (1991) Developments in civil engineering,32: advanced analysis and design of plated structures. ElsevierScience Publishing Company, New York

11. Kähönen A, Niemi E (1986) Distortion of a double symmetric boxsection subjected to eccentric loading - using the beam on elasticfounding approach. Research report. Lappeenranta University ofTechnology, Lappeenranta

12. Schardt R (1989) Verallgemeinerte Technische Biegetheore. doi:https://doi.org/10.1007/978-3-642-52330-4

13. Hetényi M (1946) Beams on elastic foundation - theory with appli-cations in the fields of civil and mechanical engineering. Universityof Michigan Press, Ann Arbor

14. Sonsino CM, Fricke W, De Bruyne F et al (2012) Notch stressconcepts for the fatigue assessment of welded joints - backgroundand applications. Int J Fatigue 34:2–16. https://doi.org/10.1016/j.ijfatigue.2010.04.011

15. Ylinen A (1976) Elasticity and strength of materials (in Finnish).Part I., 2nd edition. WSOY, Porvoo, Finland

16. Ahlfors M (2014) Design of a box girder againts distortion andtorsional load (in Finnish). Lappeenranta University ofTechnology, Bacherlor’s thesis

17. Ahlfors M (2015) Distortion and internal warping torsion of a dou-ble symmetric hollow section. Master’s thesis. LappeenrantaUniversity of Technology, Lappeenranta

18. Tong L (2007) The influence of variable amplitude loading on thefatigue strength of RHS corners. Master’s thesis. LappeenrantaUniversity of Technology, Lappeenranta

19. Bäckstrom M (2003) Multiaxial fatigue life assessment of weldsbased on nominal and hot spot stresses VTT Publications 502

20. Björk T (1991) On the spring constant in open section joints. In:Niemi E (ed) Proc. 4th FinnishMech. Days. Lappeenranta, pp 351–360

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