SimplifiedLateralTorsionalBuckling(LTB)AnalysisofGlass

Research ArticleSimplified Lateral Torsional Buckling (LTB) Analysis of GlassFins with Continuous Lateral Restraints at the Tensioned Edge

Chiara Bedon

University of Trieste Department of Engineering and Architecture Piazzale Europa 1 Trieste Italy

Correspondence should be addressed to Chiara Bedon chiarabedondiaunitsit

Received 31 December 2020 Accepted 10 March 2021 Published 25 March 2021

Academic Editor Jose Renato de Sousa

Copyright copy 2021 Chiara Bedon ampis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Within multiple design challenges the lateral torsional buckling (LTB) analysis and stability check of structural glass members is awell-known issue for design Typical examples can be found not only in glass slabs with slender bracing members but also infacades and walls where glass fins are used to brace the vertical panels against input pressures Design loads such as wind suctiongive place to possible LTB of fins with LR at the tensioned edge and thus require dedicated tools In the present investigation theLTB analysis of structural glass fins that are intended to act as bracers for facade panels and restrained via continuous flexiblejoints acting as lateral restraints (LRs) is addressed Geometrically simplified but refined numerical models developed in Abaqusare used to perform a wide parametric study and validate the proposed analytical formulations Special care is spent for theprediction of the elastic critical buckling moment with LRs given that it represents the first fundamental parameter for bucklingdesign However the LR stiffness and resistance on the one side and the geometricalmechanical features of the LR glass memberson the other side are mutually affected in the final LTB prediction In the case of laminated glass (LG) members composed of twoor more glass panels moreover further design challenges arise from the bonding level of the constituent layers A simplified butrational analytical procedure is thus presented in this paper to support the development of a conservative and standardized LTBstability check for glass fins with LR at the tensioned edge

1 Introduction

It is recognized that glass represents a structural materialwith an increasing number of applications in buildingsHarmonized standards in support of safe structural designexist but are still under preparation [1ndash3] or do not includerecommendations for the whole multitude of practicalconditions ampe last few years in this regard showed a hugespread of technical guidelines codes of practice and doc-uments in support of designers [4ndash6] and research effortsare continuously in evolution

Among others the stability analysis of glass members isknown to require dedicated calculation methods and veri-fication procedures in order to achieve safe structural per-formances (Figure 1) Literature efforts can be found forseveral loading and boundary configurations of technicalinterest including investigations on glass columns [7ndash11]beams [12ndash15] members under combined design loads

[16 17] plates under compression or shear [18ndash20] andproposals of generalized design buckling procedures[21 22] ampe use of equivalent thickness formulations forbuckling purposes has also been addressed [21 23 24] ampebuckling response of glass columns affected by variable inputparameters is described in [25] with the support of theMonte Carlo simulation method with nominal materialproperties [26] randomly modified

ampe present study investigates the sensitivity of theelastic critical buckling moment for structural glass beams inlateral torsional buckling (LTB) as a major effect derivingfrom the presence of continuous lateral restraints (LRs) atthe tensioned edge ampe problem is typical of slender glassfins that are used to brace facade panels under wind suctionampe presence of LRs however makes the LTB responsemarkedly different from laterally unrestrained (LU) mem-bers ampe current analytical study is first developed for fullymonolithic LR members in LTB ampe effects of geometrical

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 6667373 21 pageshttpsdoiorg10115520216667373

parameters and LR shear stiffness are explored in terms ofelastic critical moment giving evidence of the expectedmagnification factor as a function of the input properties ofthe system to verify Later on the analysis is extended tolaminated glass (LG) elements of typical use in practicewhere two or more glass layers are bonded together but theviscoelastic behaviour of interlayers makes the LTB analysisof the so-assembled composite sections even more complexAs shown the solution of the governing mathematicalproblem for LR members in LTB is rather complex due to amutual influence of multiple parameters Besides simpleformulations of practical use are developed to capture theLTB trends of a given LR member ampe proposed analyticalapproach is validated with the support of finite element (FE)numerical analyses [27]ampanks to a linearized procedure asshown the proposed empirical equations can support anefficient and reliable estimation of the theoretical LTB ca-pacity with LRs

2 State of the Art

Design applications for glass fins and bracings are charac-terized by the presence of adhesive mechanical or com-bined joints that provide mechanical interaction to thelinked components In the case of fins and stiffeners such asin Figure 1 this results in the LTB analysis of glass membersthat are characterized by the presence of continuous ordiscrete lateral restraints (LRs) that can be variably designed

Compared to LU members in LTB [21] the use ofspecific calculation methods for LR members is a basic needto account for the severe modifications in the LU structuralbehaviour However several input parameters are respon-sible for these behaviour modifications and make the defi-nition of generalized approaches complex

Studies that emphasize the beneficial effect of LRs forglass members in LTB can be found in [28ndash30] An extendedinvestigation and design proposal is presented in [31 32] forLR members in LTB characterized by the presence ofcontinuous LRs at the compressed edge (ie as in the case ofa roof fin under sustained vertical loads) ampe same loadingcondition is further explored in [33] by taking into accountthe presence of discrete mechanical LRs Experiments arefinally reported in [34] for glass members with fully rigiddiscrete LR in LTB In accordance with [31 32] and Figure 2

the proposed research outcomes and design procedure forthe LTB analysis of glass fins with flexible (or discrete [33])LRs at the compressed edge show that the actual criticalbuckling moment M(E)

crR of a given LR member progressivelyincreases with the LR shear stiffness ky

M(E)crR gtM

(E)cr

πL

EJzGJt

1113968 (1a)

or

M(E)crR RMM

(E)cr RM ge 1 (1b)

with RM in Figure 2(b) an unknown magnification factorFor a btimes tmonolithic member E and G are Youngrsquos and

shear moduli of glass while Jz is the flexural bending mo-ment of the resisting section and Jt is the torsional momentof inertia

Jz bt

3

12 (2)

Jt bt

3

3 (3)

21 Mathematical Model for Monolithic Members ampe LTBanalysis of LR glass fins with continuous flexible connectionsat the tensile edge is typical of practical applications in whicha facade wall and the bracing fins are subjected to suction(Figure 1) Figure 3 schematizes the typical cross section of aLR glass fin with a continuous flexible joint As far as thestiffness of the braced glass wall is disregarded the joint canbe efficiently represented in the form of distributed shearand rotational springs [31] By increasing ky (kθ 0) the LTBresponse of the LR member progressively tends to increasecompared to the base LU condition (ky⟶ 0)

ampe monolithic glass fin has a rectangular btimes t crosssection and is simply supported at the ends of the bucklinglength L Fork-end restraints for the fin enable possible out-of-plane displacements due to the applied positive constantbending moment My (ldquobottomrdquo edge in tension) while thetensioned edge is variably restrained depending on ky As astarting reference for the current analysis the LTB issue ofglass fins with LR at the tensioned edge is explored first withthe adaptation of past efforts from steel constructions and

(a) (b)

Figure 1 Structural glass used to brace roofs and facade panels

2 Mathematical Problems in Engineering

L0

My My

ZM

ky

θx

kθ

x

z

t

yb b

G y

z z

t

(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RM

RMC infin

RMT RMlim

T

(b)

Figure 2 LR glass members in LTB [31] (a) LU or LR member sections (compressive edge) with (b) magnification factor RM for the criticalbuckling moment as a function of ky ldquoCrdquo and ldquoTrdquo denote respectively the compressed or tensioned edge with the LR

qz

My

MyZ

L

Cont

inuo

us L

R

qy

MyMM Z

Cont

inuo

us L

R

(a)

b

Z

y

ZM

ky

kθt

b

Z

y

Z

ky

kt

(b)

Figure 3 Reference mathematical model of a LR glass member in LTB with tensioned edge restrained (a) side view and (b) cross section

Mathematical Problems in Engineering 3

girders [35ndash38] especially [39ndash41] Following [31 32] andFigure 3 the LTB behaviour of the monolithic LR membercan be derived from the solution of

EJz

z4v

zx4 + kyv +

z2 θxMy1113872 1113873

zx2 minus kyzMθx

⎡⎣ ⎤⎦ 0 (4a)

EJωz4θx

zx4 minus GJt

z2θx

zx2 + θx zqqz + kθ1113872 1113873 +

z2v

zx2My

minus kyzM v minus zMθx( 1113857 0

(4b)

where

Jω is the warping constantky represents the translational (shear) rigidity of thecontinuous elastic restraint per unit of length alongthe y-axiskθ is the rotational rigidity of the continuous elasticrestraint per unit of length around the x-axisMy is the applied bending moment (Mygt 0 when thebottom flange is in tension)qz represents a possible transversal distributed loadzM is the distance between the continuous LR and the x-axiszq is the distance between the point of introduction ofqz load and the x-axisv represents the vertical deflection of the member in thez-directionθx is the rotation of the cross section around thelongitudinal x-axis of the beam

In this paper the closed-form solution of the differentialsystem in equations (4a) and (4b) is further discussed inSections 3 and 4 It is first applied to a wide set of glassmembers with both monolithic or LG cross section andvariable LR stiffness Successively empirical expressions ofpractical use are proposed to facilitate the prediction of theelastic critical buckling moment ampe effect of input pa-rameters is explored with the support of parametriccalculations

22 Laminated Glass Members From a practical point ofview the use of LG sections that are obtained by bondingtwo or more glass panels represents the reference technicalsolution for safe structural glass design applications (Fig-ure 4) In this regard the literature offers a number offormulations and proposals for design that are based on theuse of the simplified equivalent thickness method and allowthe analysis of approximated glass sectionsmembers with a(flexural) monolithic thickness Otherwise the most im-portant aspect for LTB considerations is represented by theneed of dedicated calculations in terms of torsional stiffnessof the so-defined equivalent monolithic glass sections so asto include the viscoelastic properties of common interlayerson the final LTB performance indicators also [42] ampetypical shear stiffness that is offered by the interlayers in useis in fact generally weak or anyway (for most of the design

actions of technical interest) far away from the ldquofully rigidrdquobonding condition It is thus required to first calculate thetorsional contribution of the independent glass panels andthen the equivalent term given by the interposed flexibletorsional bonding

According to the literature the governing equations forLG members in LTB that are composed of two or three glasslayers are given in Tables 1 and 2 (with ng being the numberof glass layers) with evidence of the equivalent flexuralstiffness terms (Table 1) and of the equivalent torsionalstiffness contribution due to partial bonding for the glasslayers (Table 2) While the equivalent thickness concept cancorrectly capture the flexural rigidity of the composite LGsection as far as Gint modifies the trouble arises from theequivalent torsional term Jt that is largely affected by Gintmodifications and directly involves some variations in thetheoretical critical buckling moment of the member in LTB

ampe limit conditions for the equivalent expressions inTables 1 and 2 are represented by the presence of a weakinterlayer (ldquolayeredrdquo limit Gint ⟶ 0) or a rigid bond(ldquomonolithicrdquo limit Gint⟶infin) ampe effect of these limitconfigurations manifests in flexural and torsional parame-ters for a given LG composite section that progressivelymodify between the two reference values

Jz

⟶ JzL 1113944

ng

i1

bt3i

12 layered limit

⟶ JzM b

121113944

ng

i1ti

⎛⎝ ⎞⎠

3

monolithic limit

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

Jt

⟶ JtL 1113944

ng

i1

bt3i

3 layered limit

⟶ JtM b

31113944

ng

i1ti

⎛⎝ ⎞⎠

3

monolithic limit

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(6)

For a practical comparison let us consider a LG fincomposed of three glass layers (t 10mm each) and bondingfoils with tint 076mm thickness ampe member has totaldimensions L 3000mm and b 300mm ampe first step toaddress for the LTB analysis of the LG member is the cal-culation of the equivalent flexural and torsional contribu-tions under LU conditions as a function of the actualinterlayer stiffness Gint ampe composite section character-ization follows Tables 1 and 2 Figure 5 shows the variation ofboth flexural and torsional stiffness terms as compared tothe ldquolayeredrdquo conditions (RFLEX and RTOR respectively) as afunction of Gint

In Figure 5(a) it is possible to notice that the monolithicthickness teq in Table 1 well captures the flexural propertiesof the multilayer member However compared to thecomposite model for torsion (Table 2) it disregards theactual capacity of the same section as far as Gint modifies(Figure 5(b)) It turns out that the real torsional contributionof the LG member in LTB based on teq could be highly


by

Ztint

t1 t2

(a)

b

Z

y

tint

t1 t2 t1

(b)

Figure 4 Example of LG sections with (a) two (ng 2) or (b) three (ng 3) layers

Table 1 Equivalent thickness definition for LG members in LTB [21] with ng being the number of glass layers

ng 2 ng 3

teq

t31 + t32 + 12ΓbJs3

1113969

2t31 + t32 + 12ΓbJs3

1113969

Γb 1(1 + π2(E t1 t2 tint(t1 + t2) GintL2)) 1(1 + π2(E t1tint4GintL

2))

Js t1 (05t1 + 05tint)2 + t2(05t2 + 05tint)

2 t1 (05t1 + 05t2 + tint)2

Table 2 Equivalent torsional stiffness definition for LG members in LTB [21] with ng being the number of glass layers

ng 2 ng 3

Jt Jt1 + Jt2 + Jtint 2Jt1 + Jt2 + JtintJti bt3i 3 i 1 2Jtint JsLT (1 minus (tanh (05 λLT b)05 λLT b))

JsLT 4b (t1t2(t1 + t2))(05t1 + 05t2 + tint)2 4b (t1t2(t1 + t2))(t1 + t2 + 2tint)

2

λLT(GintG)((t1 + t2)t1t2tint)

1113968 (GintG)((2t1 + t2)4t1t2tint)

1113968

10ndash2 100 102 10410ndash4

Gint (Nmm2)

1

3

5

7

9

11

13

R FLE

X

LayeredGint 0

MonolithicGint infin

PVB

(30deg

3s)

teq

Composite

H = 3000mm b = 300mmng = 3t = 10mm tint = 076mm

(a)

10ndash2 100 102 10410ndash4

Gint (Nmm2)

1

3

5

7

9

11

13

R TO

R


LayeredGint 0 PV

B (3

0deg 3

s)

teq

Composite


(b)

Figure 5 Variation of (a) flexural and (b) torsional stiffness parameters for a LU laminated glass member based on the ldquocompositerdquo sectionapproach (Tables 1 and 2) or on the flexural equivalent thickness (Table 1)


overestimated depending on the input parameters FinallyFigure 5 extends to the Gint range 10minus 4ndash104MPa ampis choiceis derived from calculation purposes only and does notreflect real design applications for structural glass where thecommon interlayers are characterized (for ordinary designconditions) by Gint values that hardly exceed the order ofasymp100MPa [3 4] As an example the typicalGint effect of PVBbonds under conventional wind loads is also emphasized(with 30degC reference operating temperature and 3 s loadduration [3 4])

3 Elastic Critical Buckling Moment of GlassMembers with LR at the Tensioned Edge

31 Monolithic Members As also discussed in [31 32] theproblem in Figure 3 requires the implementation of closed-form solutions of suitable use for designers However the

presence of a continuous LR is unavoidably associated to thedefinition of rather complex analytical models in whichmultiple parameters affect each other Often additionallythe derived closed-form solutions (when available) areobtained for limited loadingboundary conditions only ampealternative can take the form of more generalized and ac-curate (but computationally expensive) numerical models todevelop case by case [33]

For the LTB analysis of glass members with flexiblecontinuous LRs at the tensile edge the first calculationapproach should be focused on the definition of the criticalbuckling moment and shapeampe structural benefit providedby continuous LRs specifically could be rationally quan-tified in the form of a magnification coefficient RM

T as inequation (1b) with 0lt kyltinfin and kθ 0 which could beextrapolated from the critical buckling moment definition

M(E)crR zMky

L

π1113874 1113875

2minus

EJz

πL

1113874 11138752

+ ky

L

π1113874 1113875

21113890 1113891 EJω

πL

1113874 11138752

+ GJt + z2Mky + kθ1113872 1113873

L

π1113874 1113875

21113890 1113891

1113971

(7)

where all the input parameters in equation (7) are definedfrom the differential system in equations (4a) and (4b)

It follows from equations (1a) (1b) and (7) mainlybased on the input kygt 0 that the estimated M

(E)crR could be

significantly higher than Eulerrsquos critical moment of the sameLU geometry At the same time however the presence ofLRs at the tensioned edge is characterized by well-knownperformance limit conditions in LTB as also summarized inFigure 2(b) ampe design challenge of the present studycompared to past literature efforts on compressed edges withLR [31 32] is thus to quantify these tensioned edge LReffects on LTB and develop a practical calculation approachfor design considerations

ampe magnification factor RTM unavoidably includes the

effects derived from various parameters such as the shearrigidity ky of the continuous joint and also the beam aspectratio span slenderness of the glass member (both mono-lithic or LG) and the position of the elastic restraints (zM)When zM minus b2 as in the schematic drawing of Figure 4(b)and the elastic LR is applied at the tensioned edge equations(1a) (1b) and (7) can be rearranged as

RTM

minus 6 bL3ky +

3b2L2ky + 16Ebt

31113872 1113873 12L

4ky + 974Ebt

31113872 1113873

1113969

395 bt3E

(8)

where E 70GPa and G 041 E for glass and1leR

TM leR

TMlim (9)

As far as ky modifies for a given geometry RTM pro-

gressively changes as in equations (8) and (9) and furtheraffects the overall LTB response ampe critical buckling shapemoreover is limited in lateral displacements of the LRtensioned edge and can follow the qualitative trends inFigure 6

ampe top limit condition associated to a ldquofully rigidrdquocontinuous LR is in fact given by

RTMlim R

TM ky⟶infin1113872 1113873 (10)

and thus allows to calculate

M(E)crR R

TMlimM

(E)cr ⟶Mcrinfin asymp

GJt

b (11)

For a monolithic member equation (11) can also beexpressed as

RTMlim asymp

L

24532b(12)

and allows to extrapolate the threshold limit ky that is as-sociated to a ldquofully rigidrdquo restraint (also see Section 34) Asfar as ky increases in equation (8) the RT

M factor progres-sively tends to the limit value in equation (11) At the sametime it is also recognized that a given geometry to verifycould receive a minimum benefit from the LR in use Studiesreported in [29ndash32] proved that the typical LR adhesivejoints for structural glass applications are characterized bymoderate shear stiffness thus suggesting the need of ageneralized calculation for RT

M

32 Discussion of Analytical Results It is clear that the shearstiffness ky alone is not representative of a constant am-plification factor for a given glass member object of study butcan be further exploited by the flexural and torsional stiffnessparameters of the member itself Under the above condi-tions the design calculation could first take advantage of thetop limit value that the member object of study could ideallyachieve According to Figure 7 such a limit value can beeasily expressed as a function of the Lb ratio while the effectof thickness t can be disregarded ampe collected analytical


U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(a)

U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(b)

U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(c)

Figure 6 Qualitative critical buckling shapes and magnification factor RTM for a monolithic fin with continuous LR at the tensioned edge

(AbaqusStandard) as a function of ky (with kθ 0) (a) ky⟶ 0 and RTM 1 (b) 0lt kyltinfin and 1ltRT

M ltRTMlim and (c) ky⟶infin and

RTM RT

Mlim

0

3

6

9

RMT

ky ge kymin

ky ge kymax

5 10 15 200Lb

y = 0406x

y = 0396x + 0178(y = x2453)

RMlim (analytical)

RMinf (analytical)

Empirical RMlim (ky infin)

Empirical RMinfin (ky infin)

(ky 0)

T

T

T

T

Figure 7 Analysis of limit RTM conditions for LR monolithic glass members in LTB as a function of ky


predictions for the top limit more in detail are derived fromequation (11) or (12)

Regarding all the possible intermediate conditions oftechnical interest a concise calculation example is proposedin Figure 8 where the RT

M variation for glass fins withcontinuous flexible restraints are presented as a function ofky Key variations take the form of modifications for span L(the examined range 1000ndash5000mm) section height b(100ndash400mm) and fin thickness t (the examined range5ndash50mm)

While the grouped members are characterized by aunivocal limit RT

M value that represents the effects of a rigidcontinuous LR at the tensioned edge severe modificationsfor the intermediate RT

M values can be observed when tchanges

Based on Figure 8 a concise definition of RTM trends for

general LR glass members with variable geometries andstiffness ky cannot be directly expressed In this regardFigure 9 shows how the input parameters can affect theoverall prediction of critical buckling moment More pre-cisely the analytical estimates are shown in Figure 9(a) interms of percentage of RT

Mlim as a function of ky for a3000times 300mm member with variable thickness A given value and thickness t give evidence of the variability inrequired ky to achieve a similar LR effect Similar consid-erations can be drawn for Figure 9(a) To achieve a certain of LR the minimum required ky increases with the increaseof stiffness (based on t) in the examined 3000times 300mmmember Furthermore modification in the span or height ofthe glass member also affects the overall LTB performanceas shown in Figure 9(c) for a selection of values in termsof RT

Mlim ky values and geometrical parameters In otherwords equation (8) can be used case by case for single-LTBcalculations or applied iteratively so as to describe the limitconditions and explore the effect of input design parameters

33 Finite Element Numerical Assessment of AnalyticalResults In order to further assess the analytical approachpresented in Section 32 a set of FE models was developed inAbaqusStandard ampe numerical study included linear bi-furcation analyses that were carried out on various geo-metrical properties of glass members and also various shearstiffness parameters for the LRs ampe critical buckling mo-ment of each beam subjected to a constant and negativebending momentMy was predicted numerically by changingky As a further attempt of validation the analysis was alsofocused on the numerically predicted buckling shapes of thesame members (ie Figure 6) to qualitatively explore theeffects of weak or rigid LRs Finally the additional advantageof the developed FE models was represented by the para-metric analysis of the selected LR glass members underdifferent load (moment) distributions so as to includeadditional configurations that are typical of design appli-cations in the study

To this aim the reference FE model schematized inFigure 10 was developed in agreement with earlier studiesreported in [31 32] ampe typical FE assembly consisted ofS4R 4-node quadrilateral stressdisplacement shell

elements with reduced integration and large-strain formu-lation For the purpose of the parametric analysis a linearelastic constitutive law was taken into account for glass(E 70 GPa and v 023) while the continuous LR at thetensile edge was described in the form of a set of equivalentshear springs ampese springs were placed at each edge node(tensioned edge) of the meshed glass members with thecorresponding shear stiffness given by the unit of lengthvalue and the mesh edge ampis edge size was derived from aregular mesh pattern of quadrilateral shell elements whosetotal number was set in a fixed number for each examinedspan L Loads and boundaries for the simply supportedfork-end restrained beams in LTB were finally introducedin each FE model by means of bending moments Myapplied at the barycentric node of the end sections as wellas nodal translational and rotational restraints for the samecross-sectional nodes

Selected comparative results from the parametric FEstudy are reported in Figure 11(a) in the form of predictedRM

T factor for LR glass beams in LTB as a function of kyWorth of interest is the close correlation of ABAQUS nu-merical predictions and analytical estimates from equation(8) as far as ky modifies

In the same Figure 11(a) additional numerical dots arepresented for members under triangular or parabolic mo-ment distribution

For LU members in LTB it is in fact reasonable toestimate the maximum effects derived from distributed q ormidspan concentrated F loads by accounting for anequivalent bending moment that is given by

My equivMyeq Mymax

k1 (13)

with

k1 a correction factor depending on the distribution ofthe applied loadsMymax the maximum bending moment due to theapplied q or F loads respectively that is Mymax

qL28 (with k1 113) or Mymax FL4 (withk1 132)

While computationally efficient for design equation (13)is not valid for LR members due to the presence of partiallyrigid continuous lateral restraints which are able to modifythe global LTB behaviour It was shown for example in[31 32] that a compressed edge with continuous LR canseverely influence the LTB performance of a given glassmember under different loading distributions

ampe FE parametric study herein summarized for glassmembers with LR at the tensioned edge further confirmedsome important effects due to the imposed moment dis-tribution for a given member ampe extended analysis ofvarious geometries and LR stiffnesses for instance gaveevidence or a rather stable LTB behaviour with someinfluencing effects due to moderately stiff LRs in slendermembers only Figures 11(b) and 11(c) in this regardpresent the numerically derived correction factor k1 (fromequations (14a) to (14e)) for a multitude of LR glass members


under distributed load q or midspan concentrated load F Asshown the average numerical predictions for k1 asymp 113andasymp 132 are still in the order of the conventional values in

the simplified model of equation (13) Besides the samenumerical derivations show some deviations as a function ofboth the input ky and the geometrical parameters At the first

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 100mm

RMT RMlim

T

(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 300mm

RMT RMlim

T

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 100mm RM

T RMlimT

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 300mm

RMT RMlim

T

(d)

Figure 8 Analysis of critical load variation (RTM) for selected geometries of monolithic glass beams as a function of the restraint stiffness ky

(a) Lb 10 (b) Lb 333 (c) Lb 20 and (d) Lb 666


calculation stage in this regard the conventional coefficientsk1 in equation (13) for the parabolic or triangular momentdistributions can be used for preliminary estimates only

34 Linearized Calculation Procedure

341 Magnification Factor RTM From a practical point of

view based on previous results the overall discussion on theeffects of input features for a given LR glass member in LTBmanifests in a set of well-defined configurations It is in factexpected that the possible LR conditions that can be

theoretically achieved while changing ky for a generalmember geometry are in any case bounded as in equation(9) ampe comparative analysis as in Figure 8 otherwiseproves that the geometry and thus the flexuraltorsionalparameters are also influencing parameters for RT

MTo develop a practical approach for design a simplified

expression of equation (8) could be for example written as

RTM minus A + B

CD

radic (14a)

with

10ndash3 10ndash1 101 103

ky (Nmm2)

0

20

40

60

80

100

R

Mli

m

L = 3000mmb = 300m

102040

t (mm)

T

(a)


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mmb = 300m

52550

75100

RMlimT

(b)


ky (Nmm2)

10

20

30

40

t (m

m)

RMlim25

L = 3000mm20001000

Lb = 10

T RMlim50

T RMlim75

T


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10

(c)

Figure 9 Analysis of shear stiffness effects ky on the expected magnification factor RTM for monolithic glass members in LTB


A minus 6bL

3ky

395bt3E

(14b)

B 1

395bt3E

(14c)

C 3b2L2ky + 16Ebt

3 (14d)

D 12L4ky + 974Ebt

3 (14e)

but the AndashD parameters are still sensitive to a large numberof input properties that make the derivation of generalempirical expressions of practical use hard

Alternatively equation (8) could be simplified andrearranged as

RTM

minus CR1L3

t3 +

CR2

bt3

b bL2

+ CR3t3

1113872 1113873 CR4L4

+ bt3

1113872 1113873

1113969

(15a)

so as to put in evidence the input (b t L) geometricalproperties with E a known parameter for glass In this casethe advantage of equations (15a)ndash(15d) is that the values forCR1 CR2 CR3 and CR4 are implicitly inclusive of ky featuresand effects for the LR in use with

CR1 CR4 f ky1113872 1113873 (15b)

CR2 fky

11139691113874 1113875 (15c)

CR3 f1ky

1113888 1113889 (15d)

In Figure 12 their general trend is shown with ky andfitting equations are proposed for their calculation It ispossible to see thatCR1 CR2 andCR4 havemostly a null valuefor a wide range of ky values and thus the RT

M definition inequation (15a) finds simplification in some terms On thecontrary CR3 in Figure 12 has an opposite trend with ky dueto the basic assumption of equation (15a) ampe goal of such akind of elaboration is that for a given geometry (b t L) thetheoretical effectiveness of ky on the actual LTB response canbe more easily quantified and addressed with a graphicalapproach or with the empirical equations in Figure 12

ampe approach is able to fit equation (8) and thus providesaccurate estimations for LR members in LTB Besides thesolution of equations (14a)ndash(15d) itself still involves thecalculation of multiple coefficients to account for theirvariation with the geometry of glass members and shearstiffness of LRs

342 Limit LR Stiffness Values In order to develop apractical tool the current study proposes a linearizedprocedure that is based on the derivation of simple butgeneralized and reliable empirical formulations Differingfrom Section 341 these equations are useful to directlyexpress the required limit values kymax and kymin (for ageneral LR glass member in LTB) that can be associated tothe limit conditions of ldquorigidrdquo or ldquoweakrdquo LR at the ten-sioned edge

Let us assume a general glass member in LTB with (b t L)geometrical properties and a continuous LR ampe proposedprocedure first requires the estimation of kymax and kyminvalues Moreover a linearized trend is then presented for allthe intermediate conditions in which kyminlt kylt kymax

(1)Maximum LR Configurationampe LR can be considered asldquofully rigidrdquo when its shear stiffness ky is at least equal to agiven top limit value kymax Given that kymax modifies withthe geometrical and mechanical features of a given LRmember in LTB this limit condition can be generally de-tected as

kymax ≔ D1

D2k2y + D3ky + D4

1113969

minus D5ky 095RTMlim

(16)

ampe D1ndashD5 parameters are implicitly inclusive of themember features (b t L) given that equation (16) is ex-trapolated from equations (8) and (12) with the target limitvalue of 095 RT

Mlim ampe advantage of the solution given byequation (16) is the direct expression of the ky valuereproducing a mostly complete bracing system for the glassmember to analyze Furthermore assumed that the thick-ness t in equation (16) has minimum effects on the top limitvalue RT

Mlim in the same way as the Lb ratio (as in Figures 7and 8 and equation (12)) the combination of equations (8)(12) and (16) makes possible to elaborate a further sim-plification of empirical formulations for the so-definedminimum top threshold kymax value

Spring local axis

b

My

lMesh

L0

Ky

t

xz y

xprime zprimeyprime

Figure 10 Detail of the reference FE numerical model (ABAQUSStandard) adapted from [31 32]


(2) Minimum LR Configuration ampe minimum LTBcontribution for a given LR and a general member is arelevant issue in the same way as the ldquorigidrdquo top conditionIn the present study the LR flexibility is assumed asldquoefficientrdquo as far as the ky value in use is able to offer at leasta minimum increase of LTB resistance for the glassmember to verify compared to its LU configuration Assuch the knowledge of the LU member performance inLTB (that is min(RT

M) 1 for ky⟶ 0) is used as a ref-erence to capture a minimum increase of the base

theoretical value ampe so-defined minimum configurationcan be expressed as

kymin ≔ D6

D2k2y + D3ky + D4

1113969

minus D7ky 105 (17)

where the D2ndashD7 parameters are again defined as in the caseof equation (16) Worth of interest is that in both equations(16) and (17) for a given set of (b t L) features coefficientsD2ndashD4 are identical for the kymax and kymin calculationsFrom a practical point of view however all the Dn

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

L = 3000mm b = 300mmt = 10mm

AnalyticalFE (M = cost)

FE (M = triangular)FE (M = parabolic)

(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1

L = var b = var t = varFE (M = parabolic)

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1

L = var b = var t = varFE (M = triangular)

(c)

Figure 11 Analytical (equation (9)) and numerical (ABAQUSStandard) analysis of LR effects on the critical buckling moment of glassbeams with the restrained tensioned edge (a) magnification factor RT

M with numerical derivation of the correction factor k1 for (b) parabolicor (c) triangular moment distribution


coefficients are still characterized by intrinsic complexitythat is derived from the mutual effect of all the inputproperties Some typical values are reported in Table 3 for aselection of geometries It is anyway worth of interest thatthe final result of equations (16) and (17) can support thedefinition of the generalized expression allowing to accountfor variations in the member geometry

(3) Empirical Model for Maximum and Minimum LRConfigurations To facilitate and further simplify the LTBanalysis of a general LR member some typical limit valueskymax and kymin are reported in Figure 13 with practicalexpressions that are based on geometrical features only ampekymin value for a given monolithic member in particular isfound to follow equation (17) and the trends in Figure 13(a)where it can be seen that

kymin asympJt

L

cmin

Cmin1113888 1113889 middot max RLb RL( 1113857 (18)

In equation (18) Jt is given by equation (3) L is the span(in mm) cmin 2 is a calibrated correction factor and

Cmin 185 times 10minus 8L37

1113872 1113873 (19)

Moreover RLb and RL in equation (18) denote thevariation ratio in terms of (Lb) and span L respectively forthe member object of study compared to the base

ldquoreferencerdquo conditions that are set respectively in (RL

b)0 10 and (RL)0 1000mmampe same analytical study from equation (16) and

summarized in Figures 13(b)ndash13(d) shows that the top limitvalue kymax corresponding to a ldquofully rigidrdquo LR at thetensioned edge is given by

kymax asympJt

L

1Cmax

1113888 11138891

cmax middot max RLb RL( 11138571113888 1113889 (20)

with Jt from equation (3) L being the span (in mm) and

Cmax DmaxL

b

minus 351113888 1113889 (21)

where

Dmax 0026L

2minus 597L + 45500 (Lle 3000mm)

432L minus 40250 (Lgt 3000mm)

⎧⎨

⎩

(22)

while cmax 085 is a calibrated correction factor (forLge 1500mm) In case of short LR members (Llt 1500mm)the second term of equation (20) vanishes

(4) Intermediate LR Configurations Once the maximum andminimum LR conditions and parameters are known fromequations (18) and (20) the intermediate configurations can be

10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025

C R1

y = 21679ndash6x +11123ndash10

(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

000

005

010

015

020

025

C R2

y = 0001634x05

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0

109

2 times 109

3 times 109

4 times 109

C R3

y = 374456xndash1

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025C R

4y = 1176ndash6x +

1087ndash11

(d)

Figure 12 Derivation of coefficients CR1 CR2 CR3 and CR4 for the LTB critical moment analysis of monolithic glass members as a functionof the restraint stiffness ky


roughly estimated as a linear interpolation of the above definedlimit values A practical example is shown in Figure 14 for a3000times 300times10mm member with variable ky input

In conclusion the developed empirical procedure allowsto develop the overall LTB assessment of a given monolithicLR member in LTB based on three simple steps namely (i)the prediction of LU critical buckling moment fromequation (1a) (ii) the definition of ky limit values fromequations (18) and (20) and (iii) the linear interpolation ofintermediate ky conditions as in Figure 14

4 Analysis of Laminated Glass Elements

41 Magnification Factor RTM Following Section 3 it is

expected that the proposed closed-form formulations couldbe underconservative for the LTB analysis of LGmembers inwhich the bonding foils can provide only partial connectionto the glass panels thus resulting in a minimum shearbonding and torsional stiffness for the layered cross section(Figure 5) ampis design issue can be efficiently and conser-vatively addressed as far as the LTB calculation is carried outin the assumption of weak interlayers (Gint⟶ 0) as inequation (6)

When zM minus b2 and the elastic LR is applied at thetensioned edge of the LG member to verify the magnifi-cation factor can still be expressed as a function of the (b tL) geometrical properties (with t being the thickness of eachglass layer) and the shear stiffness ky From equation (6) therearranged equation (8) for ldquolayeredrdquo LG sections with ng 2or 3 respectively (Gint⟶ 0) takes the form of

RTM2

minus 6 bL3ky +

6b2L2ky + 64Ebt

31113872 1113873 6L

4ky + 974Ebt

31113872 1113873

1113969

79bt3E

(23)

RTM3

minus 2 bL3ky minus

b2L2ky + 16Ebt

31113872 1113873 4L

4ky + 974Ebt

31113872 1113873

1113969

395bt3E

(24)

In both ng 2 or 3 conditions from equations (11) and(12) it is also expected that the LR critical buckling moment

could ideally reach the top limit value due to the presence ofa ldquorigidrdquo restraint that is

M(E)crR R

TMlimM


GJtL

b (25)

where the ldquolayeredrdquo torsional term JtL for the compositesection is still given by equation (6) ampe same equation (25)could be theoretically used with JtM in place of JtL fromequation (6) ampe achievement of monolithic torsionalbonding however is a difficult condition to realize inpractical LG solutions as also previously noted

For a general LG section disregarding the shear rigidityGint the magnification factor RT

M due to a variable LRflexibility ky can at maximum achieve the top limit valueRT

Mlim that is quantified in

RTMlim

RTM2lim asymp

L

17957b

RTM3lim asymp

L

2035b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)

with ng 2 or 3 respectively ampe assumption of a ldquolayeredrdquoconfiguration is on the conservative side and can represent aminimum value that the LG member object of study will beable to offer in LTB conditions with flexible LRs

Figure 15 in this regard shows the correlation of thelimit RT

M values defined above It is worth of interest that ng

directly reflects in a possible limitation of the LTB perfor-mance capacity for the member to verify In any case theoverall LTB performance with a rigid LR (kyge kymax) is notable to equal the maximum performance of a fully mono-lithic glass member and this is a major effect of compositeldquolayeredrdquo properties ampe use of monolithic equivalentthickness formulations as in Tables 1 and 2 with the em-pirical equations presented for fully monolithic LR glassmembers in LTB can represent an alternative solution as faras the intrinsic effects on composite flexural and torsionalstiffnesses are properly considered

From the interpolation and iterative calculation ofequations (23) and (24) finally limit values ky for LGsections are reported in Tables 4 and 5

Table 3 Analytical derivation of limit LR stiffness parameters kymin and kymax for fully monolithic glass members in LTB

Lb L(mm)

t(mm) RT

Mlimkymin (RT

M 105)(Nmm2)

kymax (RTM 095RT

Mlim)(Nmm2)

10 1000 10 4138 7699E minus 03 1302E+ 0110 1000 20 4138 6159E minus 02 1041E+ 0210 1000 30 4138 2079E minus 01 3515E+ 0210 1000 40 4138 4927E minus 01 8331E+ 0210 2000 10 4138 9624E minus 04 1627E+ 0010 2000 20 4138 7699E minus 03 1302E+ 0110 2000 30 4138 2598E minus 02 4393E+ 0110 2000 40 4138 6159E minus 02 1041E+ 0210 3000 10 4138 2852E minus 04 4821E minus 0110 3000 20 4138 2281E minus 03 3857E+ 0010 3000 30 4138 7699E minus 03 1302E+ 0110 3000 40 4138 1825E minus 02 3086E+ 01


411 Layered Members with ng 2 (Gint 0) ampe extendedanalysis of parametric analytical data further allows tohighlight that the kymin value has a typical trend as inFigure 16 and can be estimated as

kymin asympJtL

L

1C2min

(27)

with

C2min max(1 967L minus 105473) (28)

Furthermore the upper limit value kymax can be roughlyderived as

kymax asympJtL

L

1C2max

(29)

with

C2max D2maxeminus 02(Lb)

D2max max 1 045L

bminus 3151113874 1113875

(30)

10ndash5 10ndash3 10ndash1 101

kymin (Nmm2)

0

2000

4000

6000

8000

J tL

Monolithicb = var t = var

L = 3000mm20001000

(a)

10ndash3 10ndash1 101 10310ndash5

kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 1000mm

Monolithict = var

Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 2000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

2000

4000

6000

8000J t

LL

L = 3000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(d)

Figure 13 Analytical derivation of limit LR parameters (a) kymin and (b)ndash(d) kymax for monolithic glass members in LTB


412 Layered Members with ng 3 (Gint 0) ampe limitstiffness values for LRs are found to follow the trends inFigure 17 ampe analysis of collected analytical data furtherallows to express the limit values as

kymin asympJtL

L

1C3min

(31)

where

C3min asymp 145C2min

kymax asympJtL

L

1C3max

(32)

with


D3max max 1 145L

bminus 13521113874 1113875

(33)

5 LTB Example for a Laminated GlassFin (ng = 2)

In conclusion a calculation example is presented to addressthe LTB stability check of LGmembers that are used to bracea glass wall against wind suction based on the simplifiedapproach herein developed ampe case study system takesinspiration and input data from [4]

51 Geometry and Loads ampe wall has dimensionsH 4mtimesB 8m and is sustained (for dead loads only) byanother supporting system (Figure 18)ampis means that eachfin comes into play with the action of wind EachH L 4mlongtimes b 045m fin is i 2m spaced and connected to thewall panels by means of a continuous silicone joint able toprovide a certain structural interaction between them ampecross section of each fin consists of LG obtained fromheat-strengthened glass and amiddle PVB layer (ng 2)ampenominal thickness of each glass layer is set int t1 t2 15mm with tint 152mm for PVB foils

ampe gust of wind (peak speed averaged over 3 seconds)has a pressure equal to pwind 12 kNm2 that corresponds toa q pwind times i 24 kNm distributed load on each fin ForLTB purposes moreover the self-weight of the glass fin isdisregarded A conventional load time duration of 3 secondsand a reference temperature T 30degC are considered for thedesign action In addition the strengthening and stiffeningcontribution of LRs is conservatively neglected and LU finsin LTB are preliminary verified

ampe maximum bending moment at the midspan sectionof each fin is

Mqmax

cqqL2

8 72 kNm (34)

with cq 15 the partial safety factor for live loads

52 Design Buckling Strength ampe design tensile strength fgdand the characteristic strength for buckling fgkst are twomandatory parameters that must be first calculated for glassampe first one is given by

10ndash2 100 10210ndash4

kymin (Nmm2)

1

10

RMT

Equation (8)Linearized with equations (18) and (20)

L = 3000mm b = 300mm t = 10mm

105

kymin

kymax

095RMlim

RMT RMlim

T

T

Figure 14 Analysis of critical load variation (magnification factorRT

M) for monolithic glass beams as a function of ky based onequation (8) or the simplified linearized procedure (from equations(18) and (20))

0

3

6

9

12

15

18

RTM

10 20 30 400Lb

ky le kymin

ky ge kymaxky ge kymax

y = x2035

y = x178

ng = 2 (Gint = 0)ng = 3 (Gint = 0)

RMlimT

Figure 15 Analysis of limit RTM conditions for ldquolayeredrdquo LG

members in LTB (Gint 0) as a function of ky


fgd kmodke dksfλgAλgAfgk

RMcM

+ke dprimekv fbk minus fgk1113872 1113873

RMvcMv

2449MPa

(35)

with all the input coefficients derived from [4] while

fgd kmodke dksfλgAλgAfgk + ke dprimekv fbk minus fgk1113872 1113873 4419MPa

(36)

ampe shear stiffness value Gq

int 08MPa should also betaken into account in PVB for the estimation of the designbuckling strength of each fin (3 s T 30degC) For safety issuesit is assumed that T 50degC and thus G

qint 044MPa ampe

equivalent glass thickness for each fin is first calculated fromTables 1 and 2 where

teq 1856mm

Jzeq 4252 times 105 mm4

Jttot Jt1 + Jt2 + Jtint 1095 times 106 mm4

(37)

where it is possible to notice the limited torsional contri-bution of the interlayer (ldquolayeredrdquo limit condition fortorsion)

Disregarding the presence of continuous LR at thetensioned edge the normalized torsional slenderness ratio ofeach fin is preliminary calculated

λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)

Eulerrsquos critical buckling moment of the reference LU finampe torsional buckling reduction factor for the LU fin

would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where

Φ 05 1 + αimp λLT minus α01113872 1113873 + λ2LT1113876 1113877 1239 αimp 026 α0 021113872 1113873

(41)

and thus

Mq

bR d χLTMR χLTWxfgd 1042 middot 107 Nmm (42)

ampe LTB verification of the examined fins (LU prelim-inary hypothesis) under wind suction is positively satisfiedgiven that

Table 4 Analytical derivation of limit LR values kymin and kymax for LG members in LTB (ng 2 and Gint 0)

Lb L (mm) t (mm) kymin (RTM 105) (Nmm2) kymax (RT

M 095 RTMlim) (Nmm2)

10 1000 8 3797E minus 03 1263E+ 0110 1000 10 7288E minus 03 2179E+ 0110 1000 12 1245E minus 02 3534E+ 0110 1000 15 2416E minus 02 6550E+ 0110 2000 8 4706E minus 04 1581E+ 0010 2000 10 9109E minus 04 2724E+ 0010 2000 12 1556E minus 03 4417E+ 0010 2000 15 3019E minus 03 8187E+ 0010 3000 8 1401E minus 04 4689E minus 0110 3000 10 2699E minus 04 8070E minus 0110 3000 12 4611E minus 04 1309E+ 0010 3000 15 8946E minus 04 2426E+ 00




10 1000 8 7586E minus 03 1564E+ 0110 1000 10 1480E minus 02 2882E+ 0110 1000 12 2532E minus 02 4840E+ 0110 1000 15 4950E minus 02 9209E+ 0110 2000 8 9448E minus 04 1958E+ 0010 2000 10 1850E minus 03 3602E+ 0010 2000 12 3164E minus 03 6050E+ 0010 2000 15 6188E minus 03 1151E+ 0110 3000 8 2807E minus 04 5793E minus 0110 3000 10 5483E minus 04 1069E+ 0010 3000 12 9376E minus 04 1792E+ 0010 3000 15 1833E minus 03 3411E+ 00


Mq

E d

Mq

bR d

7201042

069le 1 (43)

However it is worth to note that the LR fins can takeadvantage of the beneficial contribution of the continuousjoint at the tensioned edge as discussed in Section 4 for LGmembers Disregarding the exact ky values the top ldquolayeredrdquocondition is associated to

RTM2lim asymp

L

17957b 495 (44)

ampe knowledge of allowable theoretical performances forthe examined fin geometry can thus be used for the LTBverification check of the LR members and for the detaileddesign of the joint

ampe iterative LTB design check based onkyminle kyle kymax and the developed linearization approachhas in fact a direct effect on RT

M thus on the correspondingM

(E)Rcr and the LTB buckling parameters λLT and χLT that

govern the above LTB verification condition From RTM 2lim

the top condition gives χLT 0908 and

10ndash3 10ndash110ndash5

kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000

Layered (Gint = 0)ng = 2b = var t = var

(a)

10ndash3 10ndash1 10110ndash5

kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm

Layered (Gint = 0)ng = 2t = var

Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)

Figure 16 Analytical derivation of limit LR parameters (a) kymin and (b)ndash(d) kymax for laminated glass members in LTB (ng 2 layeredlimit)


Mq

E d

Mq

bR d

043le 1 (45)

At the same time it is possible to notice that thecalculated kymin value is rather small Considering sometypical bonds and LR joint sections in use for similarapplications (see for example [30ndash32]) this suggests thateven a minor LR joint can offer a certain beneficial

contribution in terms of stiffness and resistance for theinvestigated geometry and thus enforce the proposedanalytical procedure for LR members in LTB Besides ithas to be noted that the current approach does not ex-clude the need of separate design checks that the samefins should satisfy in terms of ultimate resistance andserviceability deformation under the imposed designloads


kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions

ampis paper investigated the lateral torsional buckling (LTB)analysis of glass members with continuous flexible lateralrestraint (LR) at the tensioned edge ampe condition is typicalof fins and stiffeners that are used to brace roofs and facadepanels under negative pressure

To provide empirical formulations of practical use insupport of design analytical models validated toward finiteelement numerical simulation have been discussed givingevidence of the effects of geometrical and mechanical inputparameters on the overall performance of LR glass membersin LTB As shown major challenges are represented by thepresence of multiple input parameters that mutually affectthe final LTB performance for a given memberampis suggeststhe need of computationally practical but reliable tools thatcould account for the main parameters of design

Empirical formulations have thus been presented forglass members with tensioned edge in LR and characterizedby a fully monolithic or laminated (with ng 2 or 3 glasslayers) resisting section Practical expressions have beenproposed to estimate the threshold shear stiffness parame-ters of continuous LRs in use for a given member geometryso as to facilitate the critical buckling moment calculationampe major advantage is taken from a linearization procedurethat interpolates the effects of variable LR stiffnesses for ageneral member geometry starting from the top and bottomlimit conditions of rigid or weak restraints ampe reportedstudy also showed that for members under different mo-ment distributions (such as parabolic or triangular) thepresence of LRs can still be efficiently accounted in the formof traditional correction factors for the critical buckling loadpredictions thus facilitating the overall LTB design process

Data Availability

ampe data used to support the findings of this study areavailable upon request to the author

Conflicts of Interest

ampe author declares that there are no conflicts of interest

References

[1] CENTC 250 prCENTS xxxx-1 2019mdashin-plane loaded glasscomponents CEN-European Committee for StandardizationBrussels Belgium 2019

[2] CENTC 250 prCENTS xxxx-2 2019mdashout of-plane loadedglass components CEN-European Committee for Standardi-zation Brussels Belgium 2019

[3] K Langosch S Dimova A V Pinto et al ldquoGuidance forEuropean structural design of glass components-support tothe implementation harmonization and further developmentof the eurocodesrdquo in Report EUR 26439-Joint Research Centre-Institute for the Protection and Security of the CitizenP Dimova and D Feldmann Eds European Union Lux-embourg City Luxembourg 2014

[4] CNR-DT 2102013 Istruzioni per la Progettazione lrsquoEsecu-zione ed il Controllo di Costruzioni con Elementi Strutturali inVetro (Guideline for Design Execution and Control of Con-structions Made of Structural Glass Elements (in Italian))National Research Council (CNR) Rome Italy 2013

[5] Buildings Department ldquoCode of practice for the structural useof glassrdquo 2018 httpwwwbdgovhk

[6] A I S Glass ldquoCode of practice for use of glass in buildingsrdquoDecember 2020 httpswwwaisglasscomsitesdefaultfilespdfstechnical20papersAIS-59pdf

[7] J Blaauwendraad ldquoBuckling of laminated glass columnsrdquoHeron-English Edition vol 52 no 1-2 p 147 2007

[8] M Feldmann and K Langosch ldquoBuckling resistance andbuckling curves of pane-like glass columns with monolithicsections of heat strengthened and tempered glassrdquo Chal-lenging Glass Conference Proceedings vol 2 pp 319ndash3302010

[9] P Foraboschi ldquoBuckling of a laminated glass column undertestrdquo Structural Engineer vol 87 no 1 pp 2ndash8 2009

[10] S Aiello G Campione G Minafo and N Scibilia ldquoCom-pressive behaviour of laminated structural glass membersrdquoEngineering Structures vol 33 no 12 pp 3402ndash3408 2011

[11] C Amadio and C Bedon ldquoBuckling of laminated glass ele-ments in compressionrdquo Journal of Structural Engineeringvol 137 no 8 2011

[12] O Pesek and J Melcher ldquoLateral-torsional buckling oflaminated structural glass beamsrdquo Procedia Engineeringvol 190 pp 70ndash77 2017

[13] L Valarihno J R Correia M Machado-e-CostaF A Branco and N Silvestre ldquoLateral-torsional bucklingbehaviour of long-span laminated glass beams analyticalexperimental and numerical studyrdquo Materials and Designvol 102 pp 264ndash275 2016

[14] C Amadio and C Bedon ldquoBuckling of laminated glass ele-ments in out-of-plane bendingrdquo Engineering Structuresvol 32 no 11 pp 3780ndash3788 2010

[15] J Belis C Bedon C Louter C Amadio and R Van ImpeldquoExperimental and analytical assessment of lateral torsionalbuckling of laminated glass beamsrdquo Engineering Structuresvol 51 pp 295ndash305 2013

[16] C Amadio and C Bedon ldquoA buckling verification approachfor monolithic and laminated glass elements under combinedin-plane compression and bendingrdquo Engineering Structuresvol 52 pp 220ndash229 2013

[17] C Bedon and C Amadio ldquoFlexural-torsional buckling ex-perimental analysis of laminated glass elementsrdquo EngineeringStructures vol 73 no 8 pp 85ndash99 2014

200m

400

mFigure 18 Reference geometry for the LG fins in LTB


[18] A Luible and M Crisinel ldquoBuckling strength of glass ele-ments in compressionrdquo Structural Engineering Internationalvol 14 no 2 pp 120ndash125 2004

[19] D Mocibob ldquoGlass panels under shear loadingmdashuse of glassenvelopes in building stabilizationrdquo PhD thesis no 4185EPFL Lausanne Lausanne Switzerland 2008

[20] C Bedon and C Amadio ldquoBuckling analysis of simplysupported flat glass panels subjected to combined in-planeuniaxial compressive and edgewise shear loadsrdquo EngineeringStructures vol 59 pp 127ndash140 2014

[21] C Bedon and C Amadio ldquoDesign buckling curves for glasscolumns and beamsrdquo Proceedings of the Institution of CivilEngineersmdashStructures and Buildings vol 168 no 7pp 514ndash526 2015

[22] C Bedon and C Amadio ldquoA unified approach for the shearbuckling design of structural glass walls with non-ideal re-straintsrdquo American Journal of Engineering and Applied Sci-ences vol 9 no 1 pp 64ndash78 2016

[23] G DrsquoAmbrosio and L Galuppi ldquoEnhanced effective thicknessmodel for buckling of LG beams with different boundaryconditionsrdquo Glass Struct Engvol 5 pp 205ndash210 2020

[24] L Galuppi and G Royer-Carfagni ldquoEnhanced Effectiveampickness for laminated glass beams and plates under tor-sionrdquo Engineering Structures vol 206 p 110077 2020

[25] M Momeni and C Bedon ldquoUncertainty assessment for thebuckling analysis of glass columns with random parametersrdquoInternational Journal of Structural Glass and Advanced Ma-terials Research vol 4 no 1 pp 254ndash275 2020

[26] EN 572-22004 Glass in BuildingsmdashBasic Soda Lime SilicateGlass Products CEN Brussels Belgium 2004

[27] Simulia ABAQUS v614 Computer Software and OnlineDocumentation Dassault Systemes Velizy-VillacoublayFrance 2020

[28] J Belis R V Impe G Lagae andW Vanlaere ldquoEnhancementof the buckling strength of glass beams by means of lateralrestraintsrdquo Structural Engineering and Mechanics vol 15no 5 pp 495ndash511 2003

[29] E Verhoeven ldquoEffect van constructieve kitvoegen op destabiliteit van glazen liggers (in Dutch)rdquo Master thesis LMO-Laboratory for Research on Structural Models Ghent Uni-versity Ghent Belgium 2008

[30] D Sonck and J Belis ldquoElastic lateral-torsional buckling ofglass beams with continuous lateral restraintsrdquo Glass Struc-tures amp Engineering vol 1 no 1 pp 173ndash194 2016

[31] C Bedon J Belis and C Amadio ldquoStructural assessment andlateral-torsional buckling design of glass beams restrained bycontinuous sealant jointsrdquo Engineering Structures vol 102pp 214ndash229 2015

[32] C Bedon and C Amadio ldquoAnalytical and numerical as-sessment of the strengthening effect of structural sealant jointsfor the prediction of the LTB critical moment in laterallyrestrained glass beamsrdquo Materials and Structures vol 49no 6 pp 2471ndash2492 2015

[33] D Santo S Mattei and C Bedon ldquoElastic critical moment forthe lateral-torsional buckling (LTB) analysis of structural glassbeams with discrete mechanical lateral restraintsrdquo Materialvol 13 no 11 p 2492 2020

[34] A Luible and D Scharer ldquoLateral torsional buckling of glassbeams with continuous lateral supportrdquo Glass Structures ampEngineering vol 1 no 1 pp 153ndash171 2016

[35] X-T Chu R Kettle and L-Y Li ldquoLateral-torsion bucklinganalysis of partial-laterally restrained thin-walled channel-section beamsrdquo Journal of Constructional Steel Researchvol 60 no 8 pp 1159ndash1175 2004

[36] R H J Bruins RHJ ldquoLateral-torsional buckling of laterallyrestrained steel beamsrdquo Report nr A-2007-7 TechnischeUniversiteit Eindhoven Eindhoven ampe Netherlands 2007

[37] X-T Chu J Rickard and L-Y Li ldquoInfluence of lateral re-straint on lateral-torsional buckling of cold-formed steelpurlinsrdquo Jin-Walled Structures vol 43 no 5 pp 800ndash8102005

[38] A Taras and R Greiner ldquoTorsional and flexural torsionalbuckling - a study on laterally restrained I-sectionsrdquo Journal ofConstructional Steel Research vol 64 no 8 pp 725ndash7312008

[39] A Khelil and B Larue ldquoSimple solutions for the flexural-torsional buckling of laterally restrained I-beamsrdquo Engi-neering Structures vol 30 no 10 pp 2923ndash2934 2008

[40] B Larue A Khelil and M Gueury ldquoElastic flexural-torsionalbuckling of steel beams with rigid and continuous lateralrestraintsrdquo Journal of Constructional Steel Research vol 63pp 692ndash708 2006

[41] B Larue A Khelil and M Gueury ldquoEvaluation of the lateral-torsional buckling of an I beam section continuously re-strained along a flange by studying the buckling of an isolatedequivalent profilerdquo Jin-Walled Structures vol 45 no 1pp 77ndash95 2007

[42] C Bedon J Belis and A Luible ldquoAssessment of existinganalytical models for the lateral torsional buckling analysis ofPVB and SG laminated glass beams via viscoelastic simula-tions and experimentsrdquo Engineering Structures vol 60 no 2pp 52ndash67 2014


parameters and LR shear stiffness are explored in terms ofelastic critical moment giving evidence of the expectedmagnification factor as a function of the input properties ofthe system to verify Later on the analysis is extended tolaminated glass (LG) elements of typical use in practicewhere two or more glass layers are bonded together but theviscoelastic behaviour of interlayers makes the LTB analysisof the so-assembled composite sections even more complexAs shown the solution of the governing mathematicalproblem for LR members in LTB is rather complex due to amutual influence of multiple parameters Besides simpleformulations of practical use are developed to capture theLTB trends of a given LR member ampe proposed analyticalapproach is validated with the support of finite element (FE)numerical analyses [27]ampanks to a linearized procedure asshown the proposed empirical equations can support anefficient and reliable estimation of the theoretical LTB ca-pacity with LRs

2 State of the Art

Design applications for glass fins and bracings are charac-terized by the presence of adhesive mechanical or com-bined joints that provide mechanical interaction to thelinked components In the case of fins and stiffeners such asin Figure 1 this results in the LTB analysis of glass membersthat are characterized by the presence of continuous ordiscrete lateral restraints (LRs) that can be variably designed

Compared to LU members in LTB [21] the use ofspecific calculation methods for LR members is a basic needto account for the severe modifications in the LU structuralbehaviour However several input parameters are respon-sible for these behaviour modifications and make the defi-nition of generalized approaches complex

Studies that emphasize the beneficial effect of LRs forglass members in LTB can be found in [28ndash30] An extendedinvestigation and design proposal is presented in [31 32] forLR members in LTB characterized by the presence ofcontinuous LRs at the compressed edge (ie as in the case ofa roof fin under sustained vertical loads) ampe same loadingcondition is further explored in [33] by taking into accountthe presence of discrete mechanical LRs Experiments arefinally reported in [34] for glass members with fully rigiddiscrete LR in LTB In accordance with [31 32] and Figure 2

the proposed research outcomes and design procedure forthe LTB analysis of glass fins with flexible (or discrete [33])LRs at the compressed edge show that the actual criticalbuckling moment M(E)

crR of a given LR member progressivelyincreases with the LR shear stiffness ky

M(E)crR gtM

(E)cr

πL

EJzGJt

1113968 (1a)

or

M(E)crR RMM

(E)cr RM ge 1 (1b)

with RM in Figure 2(b) an unknown magnification factorFor a btimes tmonolithic member E and G are Youngrsquos and

shear moduli of glass while Jz is the flexural bending mo-ment of the resisting section and Jt is the torsional momentof inertia

Jz bt

3

12 (2)

Jt bt

3

3 (3)

21 Mathematical Model for Monolithic Members ampe LTBanalysis of LR glass fins with continuous flexible connectionsat the tensile edge is typical of practical applications in whicha facade wall and the bracing fins are subjected to suction(Figure 1) Figure 3 schematizes the typical cross section of aLR glass fin with a continuous flexible joint As far as thestiffness of the braced glass wall is disregarded the joint canbe efficiently represented in the form of distributed shearand rotational springs [31] By increasing ky (kθ 0) the LTBresponse of the LR member progressively tends to increasecompared to the base LU condition (ky⟶ 0)

ampe monolithic glass fin has a rectangular btimes t crosssection and is simply supported at the ends of the bucklinglength L Fork-end restraints for the fin enable possible out-of-plane displacements due to the applied positive constantbending moment My (ldquobottomrdquo edge in tension) while thetensioned edge is variably restrained depending on ky As astarting reference for the current analysis the LTB issue ofglass fins with LR at the tensioned edge is explored first withthe adaptation of past efforts from steel constructions and

(a) (b)

Figure 1 Structural glass used to brace roofs and facade panels


L0

My My

ZM

ky

θx

kθ

x

z

t

yb b

G y

z z

t

(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RM

RMC infin

RMT RMlim

T

(b)


qz

My

MyZ

L

Cont

inuo

us L

R

qy

MyMM Z

Cont

inuo

us L

R

(a)

b

Z

y

ZM

ky

kθt

b

Z

y

Z

ky

kt

(b)




EJz

z4v

zx4 + kyv +

z2 θxMy1113872 1113873

zx2 minus kyzMθx

⎡⎣ ⎤⎦ 0 (4a)

EJωz4θx

zx4 minus GJt

z2θx

zx2 + θx zqqz + kθ1113872 1113873 +

z2v

zx2My


(4b)

where







Jz

⟶ JzL 1113944

ng

i1

bt3i

12 layered limit

⟶ JzM b

121113944

ng

i1ti

⎛⎝ ⎞⎠

3

monolithic limit

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

Jt

⟶ JtL 1113944

ng

i1

bt3i

3 layered limit

⟶ JtM b

31113944

ng

i1ti

⎛⎝ ⎞⎠

3

monolithic limit

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(6)




by

Ztint

t1 t2

(a)

b

Z

y

tint

t1 t2 t1

(b)



ng 2 ng 3

teq

t31 + t32 + 12ΓbJs3

1113969

2t31 + t32 + 12ΓbJs3

1113969


2))

Js t1 (05t1 + 05tint)2 + t2(05t2 + 05tint)

2 t1 (05t1 + 05t2 + tint)2


ng 2 ng 3



2


1113968 (GintG)((2t1 + t2)4t1t2tint)

1113968

10ndash2 100 102 10410ndash4

Gint (Nmm2)

1

3

5

7

9

11

13

R FLE

X

LayeredGint 0


PVB

(30deg

3s)

teq

Composite


(a)

10ndash2 100 102 10410ndash4

Gint (Nmm2)

1

3

5

7

9

11

13

R TO

R


LayeredGint 0 PV

B (3

0deg 3

s)

teq

Composite


(b)









M(E)crR zMky

L

π1113874 1113875

2minus

EJz

πL

1113874 11138752

+ ky

L

π1113874 1113875

21113890 1113891 EJω

πL

1113874 11138752

+ GJt + z2Mky + kθ1113872 1113873

L

π1113874 1113875

21113890 1113891

1113971

(7)



(E)crR could be




RTM

minus 6 bL3ky +

3b2L2ky + 16Ebt

31113872 1113873 12L

4ky + 974Ebt

31113872 1113873

1113969

395 bt3E

(8)


TM leR

TMlim (9)




RTMlim R

TM ky⟶infin1113872 1113873 (10)


M(E)crR R

TMlimM


GJt

b (11)


RTMlim asymp

L

24532b(12)



M



U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(a)

U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(b)

U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(c)




RTM RT

Mlim

0

3

6

9

RMT

ky ge kymin

ky ge kymax

5 10 15 200Lb

y = 0406x

y = 0396x + 0178(y = x2453)

RMlim (analytical)

RMinf (analytical)



(ky 0)

T

T

T

T




















My equivMyeq Mymax

k1 (13)

with








10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 100mm

RMT RMlim

T

(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 300mm

RMT RMlim

T

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 100mm RM

T RMlimT

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 300mm

RMT RMlim

T

(d)











RTM minus A + B

CD

radic (14a)

with


ky (Nmm2)

0

20

40

60

80

100

R

Mli

m

L = 3000mmb = 300m

102040

t (mm)

T

(a)


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mmb = 300m

52550

75100

RMlimT

(b)


ky (Nmm2)

10

20

30

40

t (m

m)

RMlim25

L = 3000mm20001000

Lb = 10

T RMlim50

T RMlim75

T


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10

(c)



A minus 6bL

3ky

395bt3E

(14b)

B 1

395bt3E

(14c)

C 3b2L2ky + 16Ebt

3 (14d)

D 12L4ky + 974Ebt

3 (14e)



RTM

minus CR1L3

t3 +

CR2

bt3

b bL2

+ CR3t3

1113872 1113873 CR4L4

+ bt3

1113872 1113873

1113969

(15a)


CR1 CR4 f ky1113872 1113873 (15b)

CR2 fky

11139691113874 1113875 (15c)

CR3 f1ky

1113888 1113889 (15d)







kymax ≔ D1

D2k2y + D3ky + D4

1113969


(16)




Spring local axis

b

My

lMesh

L0

Ky

t

xz y

xprime zprimeyprime






kymin ≔ D6

D2k2y + D3ky + D4

1113969

minus D7ky 105 (17)


10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

L = 3000mm b = 300mmt = 10mm



(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(c)






kymin asympJt

L

cmin




1113872 1113873 (19)





kymax asympJt

L

1Cmax

1113888 11138891



Cmax DmaxL

b

minus 351113888 1113889 (21)

where

Dmax 0026L

2minus 597L + 45500 (Lle 3000mm)

432L minus 40250 (Lgt 3000mm)

⎧⎨

⎩

(22)



10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025

C R1


(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

000

005

010

015

020

025

C R2

y = 0001634x05

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0

109

2 times 109

3 times 109

4 times 109

C R3

y = 374456xndash1

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025C R

4y = 1176ndash6x +

1087ndash11

(d)









RTM2

minus 6 bL3ky +

6b2L2ky + 64Ebt

31113872 1113873 6L

4ky + 974Ebt

31113872 1113873

1113969

79bt3E

(23)

RTM3

minus 2 bL3ky minus

b2L2ky + 16Ebt

31113872 1113873 4L

4ky + 974Ebt

31113872 1113873

1113969

395bt3E

(24)



M(E)crR R

TMlimM


GJtL

b (25)





RTMlim

RTM2lim asymp

L

17957b

RTM3lim asymp

L

2035b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)







Lb L(mm)

t(mm) RT

Mlimkymin (RT

M 105)(Nmm2)

kymax (RTM 095RT

Mlim)(Nmm2)




kymin asympJtL

L

1C2min

(27)

with

C2min max(1 967L minus 105473) (28)


kymax asympJtL

L

1C2max

(29)

with


D2max max 1 045L

bminus 3151113874 1113875

(30)


kymin (Nmm2)

0

2000

4000

6000

8000

J tL


L = 3000mm20001000

(a)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 1000mm

Monolithict = var

Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 2000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

2000

4000

6000

8000J t

LL

L = 3000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(d)




kymin asympJtL

L

1C3min

(31)

where


kymax asympJtL

L

1C3max

(32)

with


D3max max 1 145L

bminus 13521113874 1113875

(33)






Mqmax

cqqL2

8 72 kNm (34)




kymin (Nmm2)

1

10

RMT


L = 3000mm b = 300mm t = 10mm

105

kymin

kymax

095RMlim

RMT RMlim

T

T



0

3

6

9

12

15

18

RTM

10 20 30 400Lb

ky le kymin


y = x2035

y = x178

ng = 2 (Gint = 0)ng = 3 (Gint = 0)

RMlimT





RMcM


RMvcMv

2449MPa

(35)



(36)



qint 044MPa ampe


teq 1856mm



(37)



λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)


would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where


(41)

and thus

Mq












Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400





























L0

My My

ZM

ky

θx

kθ

x

z

t

yb b

G y

z z

t

(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RM

RMC infin

RMT RMlim

T

(b)


qz

My

MyZ

L

Cont

inuo

us L

R

qy

MyMM Z

Cont

inuo

us L

R

(a)

b

Z

y

ZM

ky

kθt

b

Z

y

Z

ky

kt

(b)




EJz

z4v

zx4 + kyv +

z2 θxMy1113872 1113873

zx2 minus kyzMθx

⎡⎣ ⎤⎦ 0 (4a)

EJωz4θx

zx4 minus GJt

z2θx

zx2 + θx zqqz + kθ1113872 1113873 +

z2v

zx2My


(4b)

where







Jz

⟶ JzL 1113944

ng

i1

bt3i

12 layered limit

⟶ JzM b

121113944

ng

i1ti

⎛⎝ ⎞⎠

3

monolithic limit

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

Jt

⟶ JtL 1113944

ng

i1

bt3i

3 layered limit

⟶ JtM b

31113944

ng

i1ti

⎛⎝ ⎞⎠

3

monolithic limit

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(6)




by

Ztint

t1 t2

(a)

b

Z

y

tint

t1 t2 t1

(b)



ng 2 ng 3

teq

t31 + t32 + 12ΓbJs3

1113969

2t31 + t32 + 12ΓbJs3

1113969


2))

Js t1 (05t1 + 05tint)2 + t2(05t2 + 05tint)

2 t1 (05t1 + 05t2 + tint)2


ng 2 ng 3



2


1113968 (GintG)((2t1 + t2)4t1t2tint)

1113968

10ndash2 100 102 10410ndash4

Gint (Nmm2)

1

3

5

7

9

11

13

R FLE

X

LayeredGint 0


PVB

(30deg

3s)

teq

Composite


(a)

10ndash2 100 102 10410ndash4

Gint (Nmm2)

1

3

5

7

9

11

13

R TO

R


LayeredGint 0 PV

B (3

0deg 3

s)

teq

Composite


(b)









M(E)crR zMky

L

π1113874 1113875

2minus

EJz

πL

1113874 11138752

+ ky

L

π1113874 1113875

21113890 1113891 EJω

πL

1113874 11138752

+ GJt + z2Mky + kθ1113872 1113873

L

π1113874 1113875

21113890 1113891

1113971

(7)



(E)crR could be




RTM

minus 6 bL3ky +

3b2L2ky + 16Ebt

31113872 1113873 12L

4ky + 974Ebt

31113872 1113873

1113969

395 bt3E

(8)


TM leR

TMlim (9)




RTMlim R

TM ky⟶infin1113872 1113873 (10)


M(E)crR R

TMlimM


GJt

b (11)


RTMlim asymp

L

24532b(12)



M



U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(a)

U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(b)

U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(c)




RTM RT

Mlim

0

3

6

9

RMT

ky ge kymin

ky ge kymax

5 10 15 200Lb

y = 0406x

y = 0396x + 0178(y = x2453)

RMlim (analytical)

RMinf (analytical)



(ky 0)

T

T

T

T




















My equivMyeq Mymax

k1 (13)

with








10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 100mm

RMT RMlim

T

(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 300mm

RMT RMlim

T

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 100mm RM

T RMlimT

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 300mm

RMT RMlim

T

(d)











RTM minus A + B

CD

radic (14a)

with


ky (Nmm2)

0

20

40

60

80

100

R

Mli

m

L = 3000mmb = 300m

102040

t (mm)

T

(a)


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mmb = 300m

52550

75100

RMlimT

(b)


ky (Nmm2)

10

20

30

40

t (m

m)

RMlim25

L = 3000mm20001000

Lb = 10

T RMlim50

T RMlim75

T


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10

(c)



A minus 6bL

3ky

395bt3E

(14b)

B 1

395bt3E

(14c)

C 3b2L2ky + 16Ebt

3 (14d)

D 12L4ky + 974Ebt

3 (14e)



RTM

minus CR1L3

t3 +

CR2

bt3

b bL2

+ CR3t3

1113872 1113873 CR4L4

+ bt3

1113872 1113873

1113969

(15a)


CR1 CR4 f ky1113872 1113873 (15b)

CR2 fky

11139691113874 1113875 (15c)

CR3 f1ky

1113888 1113889 (15d)







kymax ≔ D1

D2k2y + D3ky + D4

1113969


(16)




Spring local axis

b

My

lMesh

L0

Ky

t

xz y

xprime zprimeyprime






kymin ≔ D6

D2k2y + D3ky + D4

1113969

minus D7ky 105 (17)


10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

L = 3000mm b = 300mmt = 10mm



(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(c)






kymin asympJt

L

cmin




1113872 1113873 (19)





kymax asympJt

L

1Cmax

1113888 11138891



Cmax DmaxL

b

minus 351113888 1113889 (21)

where

Dmax 0026L

2minus 597L + 45500 (Lle 3000mm)

432L minus 40250 (Lgt 3000mm)

⎧⎨

⎩

(22)



10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025

C R1


(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

000

005

010

015

020

025

C R2

y = 0001634x05

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0

109

2 times 109

3 times 109

4 times 109

C R3

y = 374456xndash1

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025C R

4y = 1176ndash6x +

1087ndash11

(d)









RTM2

minus 6 bL3ky +

6b2L2ky + 64Ebt

31113872 1113873 6L

4ky + 974Ebt

31113872 1113873

1113969

79bt3E

(23)

RTM3

minus 2 bL3ky minus

b2L2ky + 16Ebt

31113872 1113873 4L

4ky + 974Ebt

31113872 1113873

1113969

395bt3E

(24)



M(E)crR R

TMlimM


GJtL

b (25)





RTMlim

RTM2lim asymp

L

17957b

RTM3lim asymp

L

2035b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)







Lb L(mm)

t(mm) RT

Mlimkymin (RT

M 105)(Nmm2)

kymax (RTM 095RT

Mlim)(Nmm2)




kymin asympJtL

L

1C2min

(27)

with

C2min max(1 967L minus 105473) (28)


kymax asympJtL

L

1C2max

(29)

with


D2max max 1 045L

bminus 3151113874 1113875

(30)


kymin (Nmm2)

0

2000

4000

6000

8000

J tL


L = 3000mm20001000

(a)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 1000mm

Monolithict = var

Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 2000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

2000

4000

6000

8000J t

LL

L = 3000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(d)




kymin asympJtL

L

1C3min

(31)

where


kymax asympJtL

L

1C3max

(32)

with


D3max max 1 145L

bminus 13521113874 1113875

(33)






Mqmax

cqqL2

8 72 kNm (34)




kymin (Nmm2)

1

10

RMT


L = 3000mm b = 300mm t = 10mm

105

kymin

kymax

095RMlim

RMT RMlim

T

T



0

3

6

9

12

15

18

RTM

10 20 30 400Lb

ky le kymin


y = x2035

y = x178

ng = 2 (Gint = 0)ng = 3 (Gint = 0)

RMlimT





RMcM


RMvcMv

2449MPa

(35)



(36)



qint 044MPa ampe


teq 1856mm



(37)



λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)


would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where


(41)

and thus

Mq












Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400






























EJz

z4v

zx4 + kyv +

z2 θxMy1113872 1113873

zx2 minus kyzMθx

⎡⎣ ⎤⎦ 0 (4a)

EJωz4θx

zx4 minus GJt

z2θx

zx2 + θx zqqz + kθ1113872 1113873 +

z2v

zx2My


(4b)

where







Jz

⟶ JzL 1113944

ng

i1

bt3i

12 layered limit

⟶ JzM b

121113944

ng

i1ti

⎛⎝ ⎞⎠

3

monolithic limit

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

Jt

⟶ JtL 1113944

ng

i1

bt3i

3 layered limit

⟶ JtM b

31113944

ng

i1ti

⎛⎝ ⎞⎠

3

monolithic limit

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(6)




by

Ztint

t1 t2

(a)

b

Z

y

tint

t1 t2 t1

(b)



ng 2 ng 3

teq

t31 + t32 + 12ΓbJs3

1113969

2t31 + t32 + 12ΓbJs3

1113969


2))

Js t1 (05t1 + 05tint)2 + t2(05t2 + 05tint)

2 t1 (05t1 + 05t2 + tint)2


ng 2 ng 3



2


1113968 (GintG)((2t1 + t2)4t1t2tint)

1113968

10ndash2 100 102 10410ndash4

Gint (Nmm2)

1

3

5

7

9

11

13

R FLE

X

LayeredGint 0


PVB

(30deg

3s)

teq

Composite


(a)

10ndash2 100 102 10410ndash4

Gint (Nmm2)

1

3

5

7

9

11

13

R TO

R


LayeredGint 0 PV

B (3

0deg 3

s)

teq

Composite


(b)









M(E)crR zMky

L

π1113874 1113875

2minus

EJz

πL

1113874 11138752

+ ky

L

π1113874 1113875

21113890 1113891 EJω

πL

1113874 11138752

+ GJt + z2Mky + kθ1113872 1113873

L

π1113874 1113875

21113890 1113891

1113971

(7)



(E)crR could be




RTM

minus 6 bL3ky +

3b2L2ky + 16Ebt

31113872 1113873 12L

4ky + 974Ebt

31113872 1113873

1113969

395 bt3E

(8)


TM leR

TMlim (9)




RTMlim R

TM ky⟶infin1113872 1113873 (10)


M(E)crR R

TMlimM


GJt

b (11)


RTMlim asymp

L

24532b(12)



M



U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(a)

U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(b)

U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(c)




RTM RT

Mlim

0

3

6

9

RMT

ky ge kymin

ky ge kymax

5 10 15 200Lb

y = 0406x

y = 0396x + 0178(y = x2453)

RMlim (analytical)

RMinf (analytical)



(ky 0)

T

T

T

T




















My equivMyeq Mymax

k1 (13)

with








10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 100mm

RMT RMlim

T

(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 300mm

RMT RMlim

T

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 100mm RM

T RMlimT

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 300mm

RMT RMlim

T

(d)











RTM minus A + B

CD

radic (14a)

with


ky (Nmm2)

0

20

40

60

80

100

R

Mli

m

L = 3000mmb = 300m

102040

t (mm)

T

(a)


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mmb = 300m

52550

75100

RMlimT

(b)


ky (Nmm2)

10

20

30

40

t (m

m)

RMlim25

L = 3000mm20001000

Lb = 10

T RMlim50

T RMlim75

T


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10

(c)



A minus 6bL

3ky

395bt3E

(14b)

B 1

395bt3E

(14c)

C 3b2L2ky + 16Ebt

3 (14d)

D 12L4ky + 974Ebt

3 (14e)



RTM

minus CR1L3

t3 +

CR2

bt3

b bL2

+ CR3t3

1113872 1113873 CR4L4

+ bt3

1113872 1113873

1113969

(15a)


CR1 CR4 f ky1113872 1113873 (15b)

CR2 fky

11139691113874 1113875 (15c)

CR3 f1ky

1113888 1113889 (15d)







kymax ≔ D1

D2k2y + D3ky + D4

1113969


(16)




Spring local axis

b

My

lMesh

L0

Ky

t

xz y

xprime zprimeyprime






kymin ≔ D6

D2k2y + D3ky + D4

1113969

minus D7ky 105 (17)


10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

L = 3000mm b = 300mmt = 10mm



(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(c)






kymin asympJt

L

cmin




1113872 1113873 (19)





kymax asympJt

L

1Cmax

1113888 11138891



Cmax DmaxL

b

minus 351113888 1113889 (21)

where

Dmax 0026L

2minus 597L + 45500 (Lle 3000mm)

432L minus 40250 (Lgt 3000mm)

⎧⎨

⎩

(22)



10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025

C R1


(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

000

005

010

015

020

025

C R2

y = 0001634x05

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0

109

2 times 109

3 times 109

4 times 109

C R3

y = 374456xndash1

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025C R

4y = 1176ndash6x +

1087ndash11

(d)









RTM2

minus 6 bL3ky +

6b2L2ky + 64Ebt

31113872 1113873 6L

4ky + 974Ebt

31113872 1113873

1113969

79bt3E

(23)

RTM3

minus 2 bL3ky minus

b2L2ky + 16Ebt

31113872 1113873 4L

4ky + 974Ebt

31113872 1113873

1113969

395bt3E

(24)



M(E)crR R

TMlimM


GJtL

b (25)





RTMlim

RTM2lim asymp

L

17957b

RTM3lim asymp

L

2035b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)







Lb L(mm)

t(mm) RT

Mlimkymin (RT

M 105)(Nmm2)

kymax (RTM 095RT

Mlim)(Nmm2)




kymin asympJtL

L

1C2min

(27)

with

C2min max(1 967L minus 105473) (28)


kymax asympJtL

L

1C2max

(29)

with


D2max max 1 045L

bminus 3151113874 1113875

(30)


kymin (Nmm2)

0

2000

4000

6000

8000

J tL


L = 3000mm20001000

(a)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 1000mm

Monolithict = var

Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 2000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

2000

4000

6000

8000J t

LL

L = 3000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(d)




kymin asympJtL

L

1C3min

(31)

where


kymax asympJtL

L

1C3max

(32)

with


D3max max 1 145L

bminus 13521113874 1113875

(33)






Mqmax

cqqL2

8 72 kNm (34)




kymin (Nmm2)

1

10

RMT


L = 3000mm b = 300mm t = 10mm

105

kymin

kymax

095RMlim

RMT RMlim

T

T



0

3

6

9

12

15

18

RTM

10 20 30 400Lb

ky le kymin


y = x2035

y = x178

ng = 2 (Gint = 0)ng = 3 (Gint = 0)

RMlimT





RMcM


RMvcMv

2449MPa

(35)



(36)



qint 044MPa ampe


teq 1856mm



(37)



λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)


would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where


(41)

and thus

Mq












Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400





























by

Ztint

t1 t2

(a)

b

Z

y

tint

t1 t2 t1

(b)



ng 2 ng 3

teq

t31 + t32 + 12ΓbJs3

1113969

2t31 + t32 + 12ΓbJs3

1113969


2))

Js t1 (05t1 + 05tint)2 + t2(05t2 + 05tint)

2 t1 (05t1 + 05t2 + tint)2


ng 2 ng 3



2


1113968 (GintG)((2t1 + t2)4t1t2tint)

1113968

10ndash2 100 102 10410ndash4

Gint (Nmm2)

1

3

5

7

9

11

13

R FLE

X

LayeredGint 0


PVB

(30deg

3s)

teq

Composite


(a)

10ndash2 100 102 10410ndash4

Gint (Nmm2)

1

3

5

7

9

11

13

R TO

R


LayeredGint 0 PV

B (3

0deg 3

s)

teq

Composite


(b)









M(E)crR zMky

L

π1113874 1113875

2minus

EJz

πL

1113874 11138752

+ ky

L

π1113874 1113875

21113890 1113891 EJω

πL

1113874 11138752

+ GJt + z2Mky + kθ1113872 1113873

L

π1113874 1113875

21113890 1113891

1113971

(7)



(E)crR could be




RTM

minus 6 bL3ky +

3b2L2ky + 16Ebt

31113872 1113873 12L

4ky + 974Ebt

31113872 1113873

1113969

395 bt3E

(8)


TM leR

TMlim (9)




RTMlim R

TM ky⟶infin1113872 1113873 (10)


M(E)crR R

TMlimM


GJt

b (11)


RTMlim asymp

L

24532b(12)



M



U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(a)

U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(b)

U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(c)




RTM RT

Mlim

0

3

6

9

RMT

ky ge kymin

ky ge kymax

5 10 15 200Lb

y = 0406x

y = 0396x + 0178(y = x2453)

RMlim (analytical)

RMinf (analytical)



(ky 0)

T

T

T

T




















My equivMyeq Mymax

k1 (13)

with








10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 100mm

RMT RMlim

T

(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 300mm

RMT RMlim

T

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 100mm RM

T RMlimT

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 300mm

RMT RMlim

T

(d)











RTM minus A + B

CD

radic (14a)

with


ky (Nmm2)

0

20

40

60

80

100

R

Mli

m

L = 3000mmb = 300m

102040

t (mm)

T

(a)


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mmb = 300m

52550

75100

RMlimT

(b)


ky (Nmm2)

10

20

30

40

t (m

m)

RMlim25

L = 3000mm20001000

Lb = 10

T RMlim50

T RMlim75

T


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10

(c)



A minus 6bL

3ky

395bt3E

(14b)

B 1

395bt3E

(14c)

C 3b2L2ky + 16Ebt

3 (14d)

D 12L4ky + 974Ebt

3 (14e)



RTM

minus CR1L3

t3 +

CR2

bt3

b bL2

+ CR3t3

1113872 1113873 CR4L4

+ bt3

1113872 1113873

1113969

(15a)


CR1 CR4 f ky1113872 1113873 (15b)

CR2 fky

11139691113874 1113875 (15c)

CR3 f1ky

1113888 1113889 (15d)







kymax ≔ D1

D2k2y + D3ky + D4

1113969


(16)




Spring local axis

b

My

lMesh

L0

Ky

t

xz y

xprime zprimeyprime






kymin ≔ D6

D2k2y + D3ky + D4

1113969

minus D7ky 105 (17)


10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

L = 3000mm b = 300mmt = 10mm



(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(c)






kymin asympJt

L

cmin




1113872 1113873 (19)





kymax asympJt

L

1Cmax

1113888 11138891



Cmax DmaxL

b

minus 351113888 1113889 (21)

where

Dmax 0026L

2minus 597L + 45500 (Lle 3000mm)

432L minus 40250 (Lgt 3000mm)

⎧⎨

⎩

(22)



10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025

C R1


(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

000

005

010

015

020

025

C R2

y = 0001634x05

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0

109

2 times 109

3 times 109

4 times 109

C R3

y = 374456xndash1

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025C R

4y = 1176ndash6x +

1087ndash11

(d)









RTM2

minus 6 bL3ky +

6b2L2ky + 64Ebt

31113872 1113873 6L

4ky + 974Ebt

31113872 1113873

1113969

79bt3E

(23)

RTM3

minus 2 bL3ky minus

b2L2ky + 16Ebt

31113872 1113873 4L

4ky + 974Ebt

31113872 1113873

1113969

395bt3E

(24)



M(E)crR R

TMlimM


GJtL

b (25)





RTMlim

RTM2lim asymp

L

17957b

RTM3lim asymp

L

2035b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)







Lb L(mm)

t(mm) RT

Mlimkymin (RT

M 105)(Nmm2)

kymax (RTM 095RT

Mlim)(Nmm2)




kymin asympJtL

L

1C2min

(27)

with

C2min max(1 967L minus 105473) (28)


kymax asympJtL

L

1C2max

(29)

with


D2max max 1 045L

bminus 3151113874 1113875

(30)


kymin (Nmm2)

0

2000

4000

6000

8000

J tL


L = 3000mm20001000

(a)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 1000mm

Monolithict = var

Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 2000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

2000

4000

6000

8000J t

LL

L = 3000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(d)




kymin asympJtL

L

1C3min

(31)

where


kymax asympJtL

L

1C3max

(32)

with


D3max max 1 145L

bminus 13521113874 1113875

(33)






Mqmax

cqqL2

8 72 kNm (34)




kymin (Nmm2)

1

10

RMT


L = 3000mm b = 300mm t = 10mm

105

kymin

kymax

095RMlim

RMT RMlim

T

T



0

3

6

9

12

15

18

RTM

10 20 30 400Lb

ky le kymin


y = x2035

y = x178

ng = 2 (Gint = 0)ng = 3 (Gint = 0)

RMlimT





RMcM


RMvcMv

2449MPa

(35)



(36)



qint 044MPa ampe


teq 1856mm



(37)



λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)


would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where


(41)

and thus

Mq












Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400



































M(E)crR zMky

L

π1113874 1113875

2minus

EJz

πL

1113874 11138752

+ ky

L

π1113874 1113875

21113890 1113891 EJω

πL

1113874 11138752

+ GJt + z2Mky + kθ1113872 1113873

L

π1113874 1113875

21113890 1113891

1113971

(7)



(E)crR could be




RTM

minus 6 bL3ky +

3b2L2ky + 16Ebt

31113872 1113873 12L

4ky + 974Ebt

31113872 1113873

1113969

395 bt3E

(8)


TM leR

TMlim (9)




RTMlim R

TM ky⟶infin1113872 1113873 (10)


M(E)crR R

TMlimM


GJt

b (11)


RTMlim asymp

L

24532b(12)



M



U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(a)

U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(b)

U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(c)




RTM RT

Mlim

0

3

6

9

RMT

ky ge kymin

ky ge kymax

5 10 15 200Lb

y = 0406x

y = 0396x + 0178(y = x2453)

RMlim (analytical)

RMinf (analytical)



(ky 0)

T

T

T

T




















My equivMyeq Mymax

k1 (13)

with








10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 100mm

RMT RMlim

T

(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 300mm

RMT RMlim

T

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 100mm RM

T RMlimT

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 300mm

RMT RMlim

T

(d)











RTM minus A + B

CD

radic (14a)

with


ky (Nmm2)

0

20

40

60

80

100

R

Mli

m

L = 3000mmb = 300m

102040

t (mm)

T

(a)


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mmb = 300m

52550

75100

RMlimT

(b)


ky (Nmm2)

10

20

30

40

t (m

m)

RMlim25

L = 3000mm20001000

Lb = 10

T RMlim50

T RMlim75

T


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10

(c)



A minus 6bL

3ky

395bt3E

(14b)

B 1

395bt3E

(14c)

C 3b2L2ky + 16Ebt

3 (14d)

D 12L4ky + 974Ebt

3 (14e)



RTM

minus CR1L3

t3 +

CR2

bt3

b bL2

+ CR3t3

1113872 1113873 CR4L4

+ bt3

1113872 1113873

1113969

(15a)


CR1 CR4 f ky1113872 1113873 (15b)

CR2 fky

11139691113874 1113875 (15c)

CR3 f1ky

1113888 1113889 (15d)







kymax ≔ D1

D2k2y + D3ky + D4

1113969


(16)




Spring local axis

b

My

lMesh

L0

Ky

t

xz y

xprime zprimeyprime






kymin ≔ D6

D2k2y + D3ky + D4

1113969

minus D7ky 105 (17)


10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

L = 3000mm b = 300mmt = 10mm



(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(c)






kymin asympJt

L

cmin




1113872 1113873 (19)





kymax asympJt

L

1Cmax

1113888 11138891



Cmax DmaxL

b

minus 351113888 1113889 (21)

where

Dmax 0026L

2minus 597L + 45500 (Lle 3000mm)

432L minus 40250 (Lgt 3000mm)

⎧⎨

⎩

(22)



10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025

C R1


(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

000

005

010

015

020

025

C R2

y = 0001634x05

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0

109

2 times 109

3 times 109

4 times 109

C R3

y = 374456xndash1

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025C R

4y = 1176ndash6x +

1087ndash11

(d)









RTM2

minus 6 bL3ky +

6b2L2ky + 64Ebt

31113872 1113873 6L

4ky + 974Ebt

31113872 1113873

1113969

79bt3E

(23)

RTM3

minus 2 bL3ky minus

b2L2ky + 16Ebt

31113872 1113873 4L

4ky + 974Ebt

31113872 1113873

1113969

395bt3E

(24)



M(E)crR R

TMlimM


GJtL

b (25)





RTMlim

RTM2lim asymp

L

17957b

RTM3lim asymp

L

2035b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)







Lb L(mm)

t(mm) RT

Mlimkymin (RT

M 105)(Nmm2)

kymax (RTM 095RT

Mlim)(Nmm2)




kymin asympJtL

L

1C2min

(27)

with

C2min max(1 967L minus 105473) (28)


kymax asympJtL

L

1C2max

(29)

with


D2max max 1 045L

bminus 3151113874 1113875

(30)


kymin (Nmm2)

0

2000

4000

6000

8000

J tL


L = 3000mm20001000

(a)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 1000mm

Monolithict = var

Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 2000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

2000

4000

6000

8000J t

LL

L = 3000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(d)




kymin asympJtL

L

1C3min

(31)

where


kymax asympJtL

L

1C3max

(32)

with


D3max max 1 145L

bminus 13521113874 1113875

(33)






Mqmax

cqqL2

8 72 kNm (34)




kymin (Nmm2)

1

10

RMT


L = 3000mm b = 300mm t = 10mm

105

kymin

kymax

095RMlim

RMT RMlim

T

T



0

3

6

9

12

15

18

RTM

10 20 30 400Lb

ky le kymin


y = x2035

y = x178

ng = 2 (Gint = 0)ng = 3 (Gint = 0)

RMlimT





RMcM


RMvcMv

2449MPa

(35)



(36)



qint 044MPa ampe


teq 1856mm



(37)



λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)


would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where


(41)

and thus

Mq












Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400





























U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(a)

U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(b)

U magnitude10

09

08

07

07

06

05

04

03

03

02

01

00

(c)




RTM RT

Mlim

0

3

6

9

RMT

ky ge kymin

ky ge kymax

5 10 15 200Lb

y = 0406x

y = 0396x + 0178(y = x2453)

RMlim (analytical)

RMinf (analytical)



(ky 0)

T

T

T

T




















My equivMyeq Mymax

k1 (13)

with








10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 100mm

RMT RMlim

T

(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 300mm

RMT RMlim

T

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 100mm RM

T RMlimT

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 300mm

RMT RMlim

T

(d)











RTM minus A + B

CD

radic (14a)

with


ky (Nmm2)

0

20

40

60

80

100

R

Mli

m

L = 3000mmb = 300m

102040

t (mm)

T

(a)


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mmb = 300m

52550

75100

RMlimT

(b)


ky (Nmm2)

10

20

30

40

t (m

m)

RMlim25

L = 3000mm20001000

Lb = 10

T RMlim50

T RMlim75

T


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10

(c)



A minus 6bL

3ky

395bt3E

(14b)

B 1

395bt3E

(14c)

C 3b2L2ky + 16Ebt

3 (14d)

D 12L4ky + 974Ebt

3 (14e)



RTM

minus CR1L3

t3 +

CR2

bt3

b bL2

+ CR3t3

1113872 1113873 CR4L4

+ bt3

1113872 1113873

1113969

(15a)


CR1 CR4 f ky1113872 1113873 (15b)

CR2 fky

11139691113874 1113875 (15c)

CR3 f1ky

1113888 1113889 (15d)







kymax ≔ D1

D2k2y + D3ky + D4

1113969


(16)




Spring local axis

b

My

lMesh

L0

Ky

t

xz y

xprime zprimeyprime






kymin ≔ D6

D2k2y + D3ky + D4

1113969

minus D7ky 105 (17)


10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

L = 3000mm b = 300mmt = 10mm



(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(c)






kymin asympJt

L

cmin




1113872 1113873 (19)





kymax asympJt

L

1Cmax

1113888 11138891



Cmax DmaxL

b

minus 351113888 1113889 (21)

where

Dmax 0026L

2minus 597L + 45500 (Lle 3000mm)

432L minus 40250 (Lgt 3000mm)

⎧⎨

⎩

(22)



10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025

C R1


(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

000

005

010

015

020

025

C R2

y = 0001634x05

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0

109

2 times 109

3 times 109

4 times 109

C R3

y = 374456xndash1

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025C R

4y = 1176ndash6x +

1087ndash11

(d)









RTM2

minus 6 bL3ky +

6b2L2ky + 64Ebt

31113872 1113873 6L

4ky + 974Ebt

31113872 1113873

1113969

79bt3E

(23)

RTM3

minus 2 bL3ky minus

b2L2ky + 16Ebt

31113872 1113873 4L

4ky + 974Ebt

31113872 1113873

1113969

395bt3E

(24)



M(E)crR R

TMlimM


GJtL

b (25)





RTMlim

RTM2lim asymp

L

17957b

RTM3lim asymp

L

2035b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)







Lb L(mm)

t(mm) RT

Mlimkymin (RT

M 105)(Nmm2)

kymax (RTM 095RT

Mlim)(Nmm2)




kymin asympJtL

L

1C2min

(27)

with

C2min max(1 967L minus 105473) (28)


kymax asympJtL

L

1C2max

(29)

with


D2max max 1 045L

bminus 3151113874 1113875

(30)


kymin (Nmm2)

0

2000

4000

6000

8000

J tL


L = 3000mm20001000

(a)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 1000mm

Monolithict = var

Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 2000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

2000

4000

6000

8000J t

LL

L = 3000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(d)




kymin asympJtL

L

1C3min

(31)

where


kymax asympJtL

L

1C3max

(32)

with


D3max max 1 145L

bminus 13521113874 1113875

(33)






Mqmax

cqqL2

8 72 kNm (34)




kymin (Nmm2)

1

10

RMT


L = 3000mm b = 300mm t = 10mm

105

kymin

kymax

095RMlim

RMT RMlim

T

T



0

3

6

9

12

15

18

RTM

10 20 30 400Lb

ky le kymin


y = x2035

y = x178

ng = 2 (Gint = 0)ng = 3 (Gint = 0)

RMlimT





RMcM


RMvcMv

2449MPa

(35)



(36)



qint 044MPa ampe


teq 1856mm



(37)



λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)


would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where


(41)

and thus

Mq












Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400














































My equivMyeq Mymax

k1 (13)

with








10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 100mm

RMT RMlim

T

(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 300mm

RMT RMlim

T

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 100mm RM

T RMlimT

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 300mm

RMT RMlim

T

(d)











RTM minus A + B

CD

radic (14a)

with


ky (Nmm2)

0

20

40

60

80

100

R

Mli

m

L = 3000mmb = 300m

102040

t (mm)

T

(a)


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mmb = 300m

52550

75100

RMlimT

(b)


ky (Nmm2)

10

20

30

40

t (m

m)

RMlim25

L = 3000mm20001000

Lb = 10

T RMlim50

T RMlim75

T


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10

(c)



A minus 6bL

3ky

395bt3E

(14b)

B 1

395bt3E

(14c)

C 3b2L2ky + 16Ebt

3 (14d)

D 12L4ky + 974Ebt

3 (14e)



RTM

minus CR1L3

t3 +

CR2

bt3

b bL2

+ CR3t3

1113872 1113873 CR4L4

+ bt3

1113872 1113873

1113969

(15a)


CR1 CR4 f ky1113872 1113873 (15b)

CR2 fky

11139691113874 1113875 (15c)

CR3 f1ky

1113888 1113889 (15d)







kymax ≔ D1

D2k2y + D3ky + D4

1113969


(16)




Spring local axis

b

My

lMesh

L0

Ky

t

xz y

xprime zprimeyprime






kymin ≔ D6

D2k2y + D3ky + D4

1113969

minus D7ky 105 (17)


10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

L = 3000mm b = 300mmt = 10mm



(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(c)






kymin asympJt

L

cmin




1113872 1113873 (19)





kymax asympJt

L

1Cmax

1113888 11138891



Cmax DmaxL

b

minus 351113888 1113889 (21)

where

Dmax 0026L

2minus 597L + 45500 (Lle 3000mm)

432L minus 40250 (Lgt 3000mm)

⎧⎨

⎩

(22)



10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025

C R1


(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

000

005

010

015

020

025

C R2

y = 0001634x05

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0

109

2 times 109

3 times 109

4 times 109

C R3

y = 374456xndash1

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025C R

4y = 1176ndash6x +

1087ndash11

(d)









RTM2

minus 6 bL3ky +

6b2L2ky + 64Ebt

31113872 1113873 6L

4ky + 974Ebt

31113872 1113873

1113969

79bt3E

(23)

RTM3

minus 2 bL3ky minus

b2L2ky + 16Ebt

31113872 1113873 4L

4ky + 974Ebt

31113872 1113873

1113969

395bt3E

(24)



M(E)crR R

TMlimM


GJtL

b (25)





RTMlim

RTM2lim asymp

L

17957b

RTM3lim asymp

L

2035b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)







Lb L(mm)

t(mm) RT

Mlimkymin (RT

M 105)(Nmm2)

kymax (RTM 095RT

Mlim)(Nmm2)




kymin asympJtL

L

1C2min

(27)

with

C2min max(1 967L minus 105473) (28)


kymax asympJtL

L

1C2max

(29)

with


D2max max 1 045L

bminus 3151113874 1113875

(30)


kymin (Nmm2)

0

2000

4000

6000

8000

J tL


L = 3000mm20001000

(a)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 1000mm

Monolithict = var

Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 2000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

2000

4000

6000

8000J t

LL

L = 3000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(d)




kymin asympJtL

L

1C3min

(31)

where


kymax asympJtL

L

1C3max

(32)

with


D3max max 1 145L

bminus 13521113874 1113875

(33)






Mqmax

cqqL2

8 72 kNm (34)




kymin (Nmm2)

1

10

RMT


L = 3000mm b = 300mm t = 10mm

105

kymin

kymax

095RMlim

RMT RMlim

T

T



0

3

6

9

12

15

18

RTM

10 20 30 400Lb

ky le kymin


y = x2035

y = x178

ng = 2 (Gint = 0)ng = 3 (Gint = 0)

RMlimT





RMcM


RMvcMv

2449MPa

(35)



(36)



qint 044MPa ampe


teq 1856mm



(37)



λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)


would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where


(41)

and thus

Mq












Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400































10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 100mm

RMT RMlim

T

(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

t = 5mm101520

304050

L = 1000mmb = 300mm

RMT RMlim

T

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 100mm RM

T RMlimT

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

1

3

5

7

9

RMT

t = 5mm101520

304050

L = 2000mmb = 300mm

RMT RMlim

T

(d)











RTM minus A + B

CD

radic (14a)

with


ky (Nmm2)

0

20

40

60

80

100

R

Mli

m

L = 3000mmb = 300m

102040

t (mm)

T

(a)


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mmb = 300m

52550

75100

RMlimT

(b)


ky (Nmm2)

10

20

30

40

t (m

m)

RMlim25

L = 3000mm20001000

Lb = 10

T RMlim50

T RMlim75

T


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10

(c)



A minus 6bL

3ky

395bt3E

(14b)

B 1

395bt3E

(14c)

C 3b2L2ky + 16Ebt

3 (14d)

D 12L4ky + 974Ebt

3 (14e)



RTM

minus CR1L3

t3 +

CR2

bt3

b bL2

+ CR3t3

1113872 1113873 CR4L4

+ bt3

1113872 1113873

1113969

(15a)


CR1 CR4 f ky1113872 1113873 (15b)

CR2 fky

11139691113874 1113875 (15c)

CR3 f1ky

1113888 1113889 (15d)







kymax ≔ D1

D2k2y + D3ky + D4

1113969


(16)




Spring local axis

b

My

lMesh

L0

Ky

t

xz y

xprime zprimeyprime






kymin ≔ D6

D2k2y + D3ky + D4

1113969

minus D7ky 105 (17)


10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

L = 3000mm b = 300mmt = 10mm



(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(c)






kymin asympJt

L

cmin




1113872 1113873 (19)





kymax asympJt

L

1Cmax

1113888 11138891



Cmax DmaxL

b

minus 351113888 1113889 (21)

where

Dmax 0026L

2minus 597L + 45500 (Lle 3000mm)

432L minus 40250 (Lgt 3000mm)

⎧⎨

⎩

(22)



10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025

C R1


(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

000

005

010

015

020

025

C R2

y = 0001634x05

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0

109

2 times 109

3 times 109

4 times 109

C R3

y = 374456xndash1

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025C R

4y = 1176ndash6x +

1087ndash11

(d)









RTM2

minus 6 bL3ky +

6b2L2ky + 64Ebt

31113872 1113873 6L

4ky + 974Ebt

31113872 1113873

1113969

79bt3E

(23)

RTM3

minus 2 bL3ky minus

b2L2ky + 16Ebt

31113872 1113873 4L

4ky + 974Ebt

31113872 1113873

1113969

395bt3E

(24)



M(E)crR R

TMlimM


GJtL

b (25)





RTMlim

RTM2lim asymp

L

17957b

RTM3lim asymp

L

2035b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)







Lb L(mm)

t(mm) RT

Mlimkymin (RT

M 105)(Nmm2)

kymax (RTM 095RT

Mlim)(Nmm2)




kymin asympJtL

L

1C2min

(27)

with

C2min max(1 967L minus 105473) (28)


kymax asympJtL

L

1C2max

(29)

with


D2max max 1 045L

bminus 3151113874 1113875

(30)


kymin (Nmm2)

0

2000

4000

6000

8000

J tL


L = 3000mm20001000

(a)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 1000mm

Monolithict = var

Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 2000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

2000

4000

6000

8000J t

LL

L = 3000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(d)




kymin asympJtL

L

1C3min

(31)

where


kymax asympJtL

L

1C3max

(32)

with


D3max max 1 145L

bminus 13521113874 1113875

(33)






Mqmax

cqqL2

8 72 kNm (34)




kymin (Nmm2)

1

10

RMT


L = 3000mm b = 300mm t = 10mm

105

kymin

kymax

095RMlim

RMT RMlim

T

T



0

3

6

9

12

15

18

RTM

10 20 30 400Lb

ky le kymin


y = x2035

y = x178

ng = 2 (Gint = 0)ng = 3 (Gint = 0)

RMlimT





RMcM


RMvcMv

2449MPa

(35)



(36)



qint 044MPa ampe


teq 1856mm



(37)



λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)


would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where


(41)

and thus

Mq












Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400




































RTM minus A + B

CD

radic (14a)

with


ky (Nmm2)

0

20

40

60

80

100

R

Mli

m

L = 3000mmb = 300m

102040

t (mm)

T

(a)


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mmb = 300m

52550

75100

RMlimT

(b)


ky (Nmm2)

10

20

30

40

t (m

m)

RMlim25

L = 3000mm20001000

Lb = 10

T RMlim50

T RMlim75

T


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10


ky (Nmm2)

10

20

30

40

t (m

m)

L = 3000mm20001000

Lb = 10

(c)



A minus 6bL

3ky

395bt3E

(14b)

B 1

395bt3E

(14c)

C 3b2L2ky + 16Ebt

3 (14d)

D 12L4ky + 974Ebt

3 (14e)



RTM

minus CR1L3

t3 +

CR2

bt3

b bL2

+ CR3t3

1113872 1113873 CR4L4

+ bt3

1113872 1113873

1113969

(15a)


CR1 CR4 f ky1113872 1113873 (15b)

CR2 fky

11139691113874 1113875 (15c)

CR3 f1ky

1113888 1113889 (15d)







kymax ≔ D1

D2k2y + D3ky + D4

1113969


(16)




Spring local axis

b

My

lMesh

L0

Ky

t

xz y

xprime zprimeyprime






kymin ≔ D6

D2k2y + D3ky + D4

1113969

minus D7ky 105 (17)


10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

L = 3000mm b = 300mmt = 10mm



(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(c)






kymin asympJt

L

cmin




1113872 1113873 (19)





kymax asympJt

L

1Cmax

1113888 11138891



Cmax DmaxL

b

minus 351113888 1113889 (21)

where

Dmax 0026L

2minus 597L + 45500 (Lle 3000mm)

432L minus 40250 (Lgt 3000mm)

⎧⎨

⎩

(22)



10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025

C R1


(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

000

005

010

015

020

025

C R2

y = 0001634x05

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0

109

2 times 109

3 times 109

4 times 109

C R3

y = 374456xndash1

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025C R

4y = 1176ndash6x +

1087ndash11

(d)









RTM2

minus 6 bL3ky +

6b2L2ky + 64Ebt

31113872 1113873 6L

4ky + 974Ebt

31113872 1113873

1113969

79bt3E

(23)

RTM3

minus 2 bL3ky minus

b2L2ky + 16Ebt

31113872 1113873 4L

4ky + 974Ebt

31113872 1113873

1113969

395bt3E

(24)



M(E)crR R

TMlimM


GJtL

b (25)





RTMlim

RTM2lim asymp

L

17957b

RTM3lim asymp

L

2035b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)







Lb L(mm)

t(mm) RT

Mlimkymin (RT

M 105)(Nmm2)

kymax (RTM 095RT

Mlim)(Nmm2)




kymin asympJtL

L

1C2min

(27)

with

C2min max(1 967L minus 105473) (28)


kymax asympJtL

L

1C2max

(29)

with


D2max max 1 045L

bminus 3151113874 1113875

(30)


kymin (Nmm2)

0

2000

4000

6000

8000

J tL


L = 3000mm20001000

(a)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 1000mm

Monolithict = var

Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 2000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

2000

4000

6000

8000J t

LL

L = 3000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(d)




kymin asympJtL

L

1C3min

(31)

where


kymax asympJtL

L

1C3max

(32)

with


D3max max 1 145L

bminus 13521113874 1113875

(33)






Mqmax

cqqL2

8 72 kNm (34)




kymin (Nmm2)

1

10

RMT


L = 3000mm b = 300mm t = 10mm

105

kymin

kymax

095RMlim

RMT RMlim

T

T



0

3

6

9

12

15

18

RTM

10 20 30 400Lb

ky le kymin


y = x2035

y = x178

ng = 2 (Gint = 0)ng = 3 (Gint = 0)

RMlimT





RMcM


RMvcMv

2449MPa

(35)



(36)



qint 044MPa ampe


teq 1856mm



(37)



λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)


would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where


(41)

and thus

Mq












Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400





























A minus 6bL

3ky

395bt3E

(14b)

B 1

395bt3E

(14c)

C 3b2L2ky + 16Ebt

3 (14d)

D 12L4ky + 974Ebt

3 (14e)



RTM

minus CR1L3

t3 +

CR2

bt3

b bL2

+ CR3t3

1113872 1113873 CR4L4

+ bt3

1113872 1113873

1113969

(15a)


CR1 CR4 f ky1113872 1113873 (15b)

CR2 fky

11139691113874 1113875 (15c)

CR3 f1ky

1113888 1113889 (15d)







kymax ≔ D1

D2k2y + D3ky + D4

1113969


(16)




Spring local axis

b

My

lMesh

L0

Ky

t

xz y

xprime zprimeyprime






kymin ≔ D6

D2k2y + D3ky + D4

1113969

minus D7ky 105 (17)


10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

L = 3000mm b = 300mmt = 10mm



(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(c)






kymin asympJt

L

cmin




1113872 1113873 (19)





kymax asympJt

L

1Cmax

1113888 11138891



Cmax DmaxL

b

minus 351113888 1113889 (21)

where

Dmax 0026L

2minus 597L + 45500 (Lle 3000mm)

432L minus 40250 (Lgt 3000mm)

⎧⎨

⎩

(22)



10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025

C R1


(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

000

005

010

015

020

025

C R2

y = 0001634x05

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0

109

2 times 109

3 times 109

4 times 109

C R3

y = 374456xndash1

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025C R

4y = 1176ndash6x +

1087ndash11

(d)









RTM2

minus 6 bL3ky +

6b2L2ky + 64Ebt

31113872 1113873 6L

4ky + 974Ebt

31113872 1113873

1113969

79bt3E

(23)

RTM3

minus 2 bL3ky minus

b2L2ky + 16Ebt

31113872 1113873 4L

4ky + 974Ebt

31113872 1113873

1113969

395bt3E

(24)



M(E)crR R

TMlimM


GJtL

b (25)





RTMlim

RTM2lim asymp

L

17957b

RTM3lim asymp

L

2035b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)







Lb L(mm)

t(mm) RT

Mlimkymin (RT

M 105)(Nmm2)

kymax (RTM 095RT

Mlim)(Nmm2)




kymin asympJtL

L

1C2min

(27)

with

C2min max(1 967L minus 105473) (28)


kymax asympJtL

L

1C2max

(29)

with


D2max max 1 045L

bminus 3151113874 1113875

(30)


kymin (Nmm2)

0

2000

4000

6000

8000

J tL


L = 3000mm20001000

(a)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 1000mm

Monolithict = var

Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 2000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

2000

4000

6000

8000J t

LL

L = 3000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(d)




kymin asympJtL

L

1C3min

(31)

where


kymax asympJtL

L

1C3max

(32)

with


D3max max 1 145L

bminus 13521113874 1113875

(33)






Mqmax

cqqL2

8 72 kNm (34)




kymin (Nmm2)

1

10

RMT


L = 3000mm b = 300mm t = 10mm

105

kymin

kymax

095RMlim

RMT RMlim

T

T



0

3

6

9

12

15

18

RTM

10 20 30 400Lb

ky le kymin


y = x2035

y = x178

ng = 2 (Gint = 0)ng = 3 (Gint = 0)

RMlimT





RMcM


RMvcMv

2449MPa

(35)



(36)



qint 044MPa ampe


teq 1856mm



(37)



λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)


would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where


(41)

and thus

Mq












Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400
































kymin ≔ D6

D2k2y + D3ky + D4

1113969

minus D7ky 105 (17)


10ndash2 100 102 10410ndash4

ky (Nmm2)

1

2

3

4

5

RMT

L = 3000mm b = 300mmt = 10mm



(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

03

06

09

12

15

18

21

24

k1


(c)






kymin asympJt

L

cmin




1113872 1113873 (19)





kymax asympJt

L

1Cmax

1113888 11138891



Cmax DmaxL

b

minus 351113888 1113889 (21)

where

Dmax 0026L

2minus 597L + 45500 (Lle 3000mm)

432L minus 40250 (Lgt 3000mm)

⎧⎨

⎩

(22)



10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025

C R1


(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

000

005

010

015

020

025

C R2

y = 0001634x05

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0

109

2 times 109

3 times 109

4 times 109

C R3

y = 374456xndash1

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025C R

4y = 1176ndash6x +

1087ndash11

(d)









RTM2

minus 6 bL3ky +

6b2L2ky + 64Ebt

31113872 1113873 6L

4ky + 974Ebt

31113872 1113873

1113969

79bt3E

(23)

RTM3

minus 2 bL3ky minus

b2L2ky + 16Ebt

31113872 1113873 4L

4ky + 974Ebt

31113872 1113873

1113969

395bt3E

(24)



M(E)crR R

TMlimM


GJtL

b (25)





RTMlim

RTM2lim asymp

L

17957b

RTM3lim asymp

L

2035b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)







Lb L(mm)

t(mm) RT

Mlimkymin (RT

M 105)(Nmm2)

kymax (RTM 095RT

Mlim)(Nmm2)




kymin asympJtL

L

1C2min

(27)

with

C2min max(1 967L minus 105473) (28)


kymax asympJtL

L

1C2max

(29)

with


D2max max 1 045L

bminus 3151113874 1113875

(30)


kymin (Nmm2)

0

2000

4000

6000

8000

J tL


L = 3000mm20001000

(a)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 1000mm

Monolithict = var

Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 2000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

2000

4000

6000

8000J t

LL

L = 3000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(d)




kymin asympJtL

L

1C3min

(31)

where


kymax asympJtL

L

1C3max

(32)

with


D3max max 1 145L

bminus 13521113874 1113875

(33)






Mqmax

cqqL2

8 72 kNm (34)




kymin (Nmm2)

1

10

RMT


L = 3000mm b = 300mm t = 10mm

105

kymin

kymax

095RMlim

RMT RMlim

T

T



0

3

6

9

12

15

18

RTM

10 20 30 400Lb

ky le kymin


y = x2035

y = x178

ng = 2 (Gint = 0)ng = 3 (Gint = 0)

RMlimT





RMcM


RMvcMv

2449MPa

(35)



(36)



qint 044MPa ampe


teq 1856mm



(37)



λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)


would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where


(41)

and thus

Mq












Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400































kymin asympJt

L

cmin




1113872 1113873 (19)





kymax asympJt

L

1Cmax

1113888 11138891



Cmax DmaxL

b

minus 351113888 1113889 (21)

where

Dmax 0026L

2minus 597L + 45500 (Lle 3000mm)

432L minus 40250 (Lgt 3000mm)

⎧⎨

⎩

(22)



10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025

C R1


(a)

10ndash2 100 102 10410ndash4

ky (Nmm2)

000

005

010

015

020

025

C R2

y = 0001634x05

(b)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0

109

2 times 109

3 times 109

4 times 109

C R3

y = 374456xndash1

(c)

10ndash2 100 102 10410ndash4

ky (Nmm2)

0000

0005

0010

0015

0020

0025C R

4y = 1176ndash6x +

1087ndash11

(d)









RTM2

minus 6 bL3ky +

6b2L2ky + 64Ebt

31113872 1113873 6L

4ky + 974Ebt

31113872 1113873

1113969

79bt3E

(23)

RTM3

minus 2 bL3ky minus

b2L2ky + 16Ebt

31113872 1113873 4L

4ky + 974Ebt

31113872 1113873

1113969

395bt3E

(24)



M(E)crR R

TMlimM


GJtL

b (25)





RTMlim

RTM2lim asymp

L

17957b

RTM3lim asymp

L

2035b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)







Lb L(mm)

t(mm) RT

Mlimkymin (RT

M 105)(Nmm2)

kymax (RTM 095RT

Mlim)(Nmm2)




kymin asympJtL

L

1C2min

(27)

with

C2min max(1 967L minus 105473) (28)


kymax asympJtL

L

1C2max

(29)

with


D2max max 1 045L

bminus 3151113874 1113875

(30)


kymin (Nmm2)

0

2000

4000

6000

8000

J tL


L = 3000mm20001000

(a)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 1000mm

Monolithict = var

Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 2000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

2000

4000

6000

8000J t

LL

L = 3000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(d)




kymin asympJtL

L

1C3min

(31)

where


kymax asympJtL

L

1C3max

(32)

with


D3max max 1 145L

bminus 13521113874 1113875

(33)






Mqmax

cqqL2

8 72 kNm (34)




kymin (Nmm2)

1

10

RMT


L = 3000mm b = 300mm t = 10mm

105

kymin

kymax

095RMlim

RMT RMlim

T

T



0

3

6

9

12

15

18

RTM

10 20 30 400Lb

ky le kymin


y = x2035

y = x178

ng = 2 (Gint = 0)ng = 3 (Gint = 0)

RMlimT





RMcM


RMvcMv

2449MPa

(35)



(36)



qint 044MPa ampe


teq 1856mm



(37)



λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)


would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where


(41)

and thus

Mq












Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400



































RTM2

minus 6 bL3ky +

6b2L2ky + 64Ebt

31113872 1113873 6L

4ky + 974Ebt

31113872 1113873

1113969

79bt3E

(23)

RTM3

minus 2 bL3ky minus

b2L2ky + 16Ebt

31113872 1113873 4L

4ky + 974Ebt

31113872 1113873

1113969

395bt3E

(24)



M(E)crR R

TMlimM


GJtL

b (25)





RTMlim

RTM2lim asymp

L

17957b

RTM3lim asymp

L

2035b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)







Lb L(mm)

t(mm) RT

Mlimkymin (RT

M 105)(Nmm2)

kymax (RTM 095RT

Mlim)(Nmm2)




kymin asympJtL

L

1C2min

(27)

with

C2min max(1 967L minus 105473) (28)


kymax asympJtL

L

1C2max

(29)

with


D2max max 1 045L

bminus 3151113874 1113875

(30)


kymin (Nmm2)

0

2000

4000

6000

8000

J tL


L = 3000mm20001000

(a)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 1000mm

Monolithict = var

Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 2000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

2000

4000

6000

8000J t

LL

L = 3000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(d)




kymin asympJtL

L

1C3min

(31)

where


kymax asympJtL

L

1C3max

(32)

with


D3max max 1 145L

bminus 13521113874 1113875

(33)






Mqmax

cqqL2

8 72 kNm (34)




kymin (Nmm2)

1

10

RMT


L = 3000mm b = 300mm t = 10mm

105

kymin

kymax

095RMlim

RMT RMlim

T

T



0

3

6

9

12

15

18

RTM

10 20 30 400Lb

ky le kymin


y = x2035

y = x178

ng = 2 (Gint = 0)ng = 3 (Gint = 0)

RMlimT





RMcM


RMvcMv

2449MPa

(35)



(36)



qint 044MPa ampe


teq 1856mm



(37)



λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)


would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where


(41)

and thus

Mq












Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400






























kymin asympJtL

L

1C2min

(27)

with

C2min max(1 967L minus 105473) (28)


kymax asympJtL

L

1C2max

(29)

with


D2max max 1 045L

bminus 3151113874 1113875

(30)


kymin (Nmm2)

0

2000

4000

6000

8000

J tL


L = 3000mm20001000

(a)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 1000mm

Monolithict = var

Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

2000

4000

6000

8000

J tL

L

L = 2000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

2000

4000

6000

8000J t

LL

L = 3000mm

Monolithict = var

Lb = 5

Lb = 10

Lb = 20

(d)




kymin asympJtL

L

1C3min

(31)

where


kymax asympJtL

L

1C3max

(32)

with


D3max max 1 145L

bminus 13521113874 1113875

(33)






Mqmax

cqqL2

8 72 kNm (34)




kymin (Nmm2)

1

10

RMT


L = 3000mm b = 300mm t = 10mm

105

kymin

kymax

095RMlim

RMT RMlim

T

T



0

3

6

9

12

15

18

RTM

10 20 30 400Lb

ky le kymin


y = x2035

y = x178

ng = 2 (Gint = 0)ng = 3 (Gint = 0)

RMlimT





RMcM


RMvcMv

2449MPa

(35)



(36)



qint 044MPa ampe


teq 1856mm



(37)



λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)


would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where


(41)

and thus

Mq












Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400






























kymin asympJtL

L

1C3min

(31)

where


kymax asympJtL

L

1C3max

(32)

with


D3max max 1 145L

bminus 13521113874 1113875

(33)






Mqmax

cqqL2

8 72 kNm (34)




kymin (Nmm2)

1

10

RMT


L = 3000mm b = 300mm t = 10mm

105

kymin

kymax

095RMlim

RMT RMlim

T

T



0

3

6

9

12

15

18

RTM

10 20 30 400Lb

ky le kymin


y = x2035

y = x178

ng = 2 (Gint = 0)ng = 3 (Gint = 0)

RMlimT





RMcM


RMvcMv

2449MPa

(35)



(36)



qint 044MPa ampe


teq 1856mm



(37)



λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)


would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where


(41)

and thus

Mq












Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400






























RMcM


RMvcMv

2449MPa

(35)



(36)



qint 044MPa ampe


teq 1856mm



(37)



λqLT

Wxfgkst

M(E)cr

1113971

1114 (38)

with

M(E)cr k1

πL

EJzeq GJttot

1113969 2702 middot 107 Nmm (39)


would give

χLT 1

Φ +

Φ2 minus λ2LT1113969 0561 (40)

where


(41)

and thus

Mq












Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400





























Mq

E d

Mq

bR d

7201042

069le 1 (43)


RTM2lim asymp

L

17957b 495 (44)








kymin (Nmm2)

0

200

400

600

800J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

200

400

600

800

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400





























Mq

E d

Mq

bR d

043le 1 (45)




kymin (Nmm2)

0

400

800

1200J t

LL

L = 3000mm20001000


(a)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 1000mm


Lb = 333

Lb = 5

Lb = 10

(b)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 2000mm


Lb = 5

Lb = 10

Lb = 20

(c)


kymax (Nmm2)

0

400

800

1200

J tL

L

L = 3000mm


Lb = 5

Lb = 10

Lb = 20

(d)



6 Conclusions




Data Availability




References


















200m

400





























6 Conclusions




Data Availability




References


















200m

400























































Documents

SimplifiedLateralTorsionalBuckling(LTB)AnalysisofGlass