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Research ArticleComputation Analysis of Buckling Loads of Thin-WalledMembers with Open Sections

Lihua Huang1 Bin Li1 and Yuefang Wang2

1Faculty of Infrastructure Engineering Dalian University of Technology Dalian 116024 China2Department of Engineering Mechanics Dalian University of Technology Dalian 116024 China

Correspondence should be addressed to Lihua Huang lhhangdluteducn

Received 28 July 2016 Revised 5 October 2016 Accepted 24 October 2016

Academic Editor Fazal M Mahomed

Copyright copy 2016 Lihua Huang et al This 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

The computational methods for solving buckling loads of thin-walled members with open sections are not unique when differentconcerns are emphasized In this paper the buckling loads of thin-walledmembers in linear-elastic geometrically nonlinear-elasticandnonlinear-inelastic behaviors are investigated from the views ofmathematical formulation experiment andnumerical solutionThe differential equations and their solutions of linear-elastic and geometrically nonlinear-elastic buckling of thin-walled memberswith various constraints are derived Taking structural angle as an example numerical analysis of elastic and inelastic buckling iscarried out via ANSYS Elastic analyses for linearized buckling and nonlinear buckling are realized using finite elements of beamand shell and are compared with the theoretical results The effect of modeling of constraints on numerical results is studied whenshell element is appliedThe factors that influence the inelastic buckling load in numerical solution such as modeling of constraintloading pattern adding rib scale factor of initial defect and yield strength of material are studied The noteworthy problems andtheir solutions in numerically buckling analysis of thin-walled member with open section are pointed out

1 Introduction

Thin-walled members with open section are widely used instructures for their high ratio of strength to weight Theloading capacity of these structures is closely related to theoverall and local stability of thin-walled members leadingto the investigation of the elastic and inelastic bucklinganalysis of thesemembers In engineering practice the failuremodes of compressive thin-walled members are generallyanalyzed according to the type of stresses developed withinthe member at the time of failure For most long-slendermembers the failure of buckling is dominantly referred to aselastic instability since the compressive stress remains elasticFor intermediate compressive members inelastic bucklinggenerally occurs as a result of the compressive stresses insidethe members being greater than the materialsrsquo yield strengthBy contrast short-length compressive members also knownas stocky members do not become unstable rather thematerial simply yields or fractures

The computational methods of buckling for elastic andinelastic instability of thin-walled compressive member are

discussed in this paper Problems for linearized geomet-rically and physically nonlinear buckling are investigatedbased onmathematical formulation and numerical solutionsIn the classical theory of elastic buckling the couplingterms of translational displacements and axial rotation areignored in the regime of small deformation [1ndash3] Mohriet al derived governing equations of post flexural-torsionalbuckling of thin-walled compressive member with opensection supported by pins at two ends considering largedeformation and coupling terms of displacement [4] As tothe inelastic buckling problem Finite Element Method is themost effective way to obtain the buckling loads as well asthe primary and secondary equilibrium paths of memberswith material and geometrical nonlinear behavior Modelsof beam and shell elements are generally used to solve thehigher order nonlinear problems in instability analysis Thebuckling load of overall instability can be determined byadopting beam element and both local and overall instabilitycan be simulated via shell element By taking structuralangles with equal legs as an example in this paper elasticanalyses including linear buckling and nonlinear buckling are

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 8320469 9 pageshttpdxdoiorg10115520168320469

2 Mathematical Problems in Engineering

M

z

O y

x

w

C (yc zc)

Figure 1 Model of a thin-walled member with open section

realized using the elements of beam and shell The effectsof modeling of constraints on numerical results are studiedwhen shell elements are applied Analysis of linearized buck-ling analysis and nonlinearized buckling analysis of materialand geometrical nonlinearity are conducted and comparedwith experimental and theoretical results The factors thataffect the inelastic buckling loads in numerical analysis suchas modeling of constraints scale factors of initial defectsadding rib and loadingmodes are discussedThe noteworthyproblems and their solutions in numerical buckling analysisof thin-walled member with open section are pointed out

2 Mathematical Formulation

A thin-walled compressive member with an arbitrary opensection is shown in Figure 1 where point 119874 is the centroid ofthe section and point119862(119910119888 119911119888) is the shear center119874119909119910119911 is thecentroidal principal axis

21 Equations of Classical Linear-Elastic Buckling The equa-tions of classical stability for a thin-walled member are aset of fourth-order linear differential equations with constantcoefficients [1]

119864119868119911V(4) + 119875V10158401015840 + 11987511991111988812057910158401015840119909 = 0119864119868119910119908(4) + 11987511990810158401015840 minus 11987511991011988812057910158401015840119909 = 0119875119911119862V10158401015840 minus 11987511991011986211990810158401015840 + 119864119868120596120579(4)119909 + (1198751198680 minus 119866119869) 1205792119909 = 01198680 = (119868119910 + 119868119911)

119860 + 1199102119888 + 1199112119888

(1)

where 119864 and 119866 are the elastic constants 119875 is an axiallycompressive force applied at the centroid The displacementsof shear center in 119910 and 119911 directions are denoted by Vand 119908 respectively The rotation of the cross section isrepresented by 120579119909 119869 is free torsionalmoment of inertia and 119868120596denotes the principal warping moment of inertiaThe criticalload is determined through solving the eigenvalue problemsyielding

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119875119911 minus 119875 0 minus1198751199111198880 119875119910 minus 119875 119875119910119888

minus119875119911119888 119875119910119888 1198751205961198680 minus 1198751198680

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (2)

in which 119875119909 = 1198641198681199091205822 119875119910 = 1198641198681199101205822 and 119875120596 = (1198641198681205961205822 + 119866119869)1198680120582 = 119899120587119871 for member pinned at two ends where 119899 =1 2 3 and 119871 denotes the length of the member 120582 =2119899120587119871 for two fixed ends For a member fixed at one endand free at the other end 120582 = 119899120587(2119871) For a member fixed atone end and pinned at the other end 120582 = 449311987122 Formulation of Geometrically Nonlinear-Elastic-BucklingOn the basis of literature [4] the transversal displacementsV119872 and 119908119872 of any point 119872 on the cross section when themember is twisted at an angle 120579119909 are expressed by

V119872 = V minus (119911 minus 119911119862) sin 120579119909 minus (119910 minus 119910119862) (1 minus cos 120579119909) 119908119872 = 119908 + (119910 minus 119910119862) sin 120579119909 minus (119911 minus 119911119862) (1 minus cos 120579119909) (3)

The axial displacement 119906119872 is expressed by

119906119872 = 119906 minus 119910 (V1015840 cos 120579119909 + 1199081015840 sin 120579119909)minus 119911 (1199081015840 cos 120579119909 minus V1015840 sin 120579119909) + 1205961205791015840119909

(4)

where 119906 V and 119908 are the displacements of shear center in 119909119910 and 119911 directionsThe Green strain tensor that incorporates large displace-

ments is defined by

120576119894119895 = 12 (

120597119906119894120597119909119895 +120597119906119895120597119909119894 +

120597119906119896120597119909119894120597119906119896120597119909119895) (5)

Hence the strains of point119872 can be expressed through

120576119909119909 = 1199061015840 minus 119910 (V10158401015840 cos 120579119909 + 11990810158401015840 sin 120579119909)minus 119911 (11990810158401015840 cos 120579119909 minus V10158401015840 sin 120579119909) + 12059612057910158401015840119909+ 12 (V10158402 + 11990810158402 + 119877212057910158402119909 )

minus 1199101198621205791015840119909 (1199081015840 cos 120579119909 minus V1015840 sin 120579119909)+ 1199111198621205791015840119909 (V1015840 cos 120579119909 + 1199081015840 sin 120579119909)

120576119909119910 = minus12 (119911 minus 119911119862 minus120597120596120597119910 ) 1205791015840119909

120576119909119911 = 12 (119910 minus 119910119862 +

120597120596120597119911 ) 1205791015840119909

(6)

where

1198772 = (119910 minus 119910119862)2 + (119911 minus 119911119862)2 (7)

Based on the principle of virtual work one obtains

120575 (119880 minus119882) = 0 (8)

where

120575119880 = int119871int119860(120590119909119909120575120576119909119909 + 2120590119909119910120575120576119909119910 + 2120590119909119911120575120576119909119911) d119860 d119909

120575119882 = int (119902119909120575119906 + 119902119910120575V + 119902119911120575119908 + 119898119909120575120579119909) d119909(9)

Mathematical Problems in Engineering 3

Consequently the governing equations considering largedeformation and coupling terms are derived as follows

119873 = minus119875 (10)

119864119868119910 (119908(4) + 3119908101584011990810158401015840119908101584010158401015840 + 119908101584010158403 + 119908(4)119908101584022 ) minus 119873(11990810158401015840

minus 11991011988812057910158401015840119909 + 119911119888 (12057911990912057910158401015840119909 + 12057910158402119909 )) + (119864119868119911 minus 119864119868119910) (V(4)120579119909+ 2V1015840101584010158401205791015840119909 + V1015840101584012057910158401015840119909 + 119908(4)1205792119909 + 41199081015840101584010158401205791199091205791015840119909 + 21199081015840101584012057911990912057910158401015840119909+ 21199081015840101584012057910158402119909 ) = 0

(11)

119864119868119911 (V(4) + 3V1015840V10158401015840V101584010158401015840 + V101584010158403 + V(4)V10158402

2 ) minus 119873(V10158401015840

+ 11991111988812057910158401015840119909 + 119910119888 (12057911990912057910158401015840119909 + 12057910158402119909 )) + (119864119868119911 minus 119864119868119910) (119908(4)120579119909+ 21199081015840101584010158401205791015840119909 + 1199081015840101584012057910158401015840119909 minus V(4)1205792119909 minus 4V1015840101584010158401205791199091205791015840119909 minus 2V1015840101584012057911990912057910158401015840119909minus 2V1015840101584012057910158402119909 ) = 0

(12)

119864119868120596120579(4)119909 minus 11986611986912057910158401015840119909 minus 3211986411986811990512057910158402119909 12057910158401015840119909 minus 119873(119868012057910158401015840119909

minus 119910119888 (11990810158401015840 minus V10158401015840120579119909) + 119911119888 (V10158401015840 + 11990810158401015840120579119909)) minus (119864119868119911minus 119864119868119910) (V1015840101584011990810158401015840 minus V101584010158402120579119909 + 119908101584010158402120579119909) = 0

(13)

where

119868119905 = int119860((119910 minus 119910119888)2 + (119911 minus 119911119888)2)2 d119860 minus 11986011986820 (14)

Equations (11) through (13) can be precisely solved using thebackward difference formulation [5] For simply supportedmember under free warping the displacement functions inbending and torsion are defined by sinusoidal function

VV0= 1199081199080 =

1205791199091205790 = sin 120587119909119871 (15)

where V0 1199080 and 1205790 are the associated displacement ampli-tudes Substituting (15) into (10) through (13) and employingGalerkinrsquos approximation method yield

119875119911 (V0 + 11990528 V30) minus 119875(V0 + 1199111198881205790 +

4312058711991011988812057920)

+ (119875119911 minus 119875119910) ( 83120587119908012057920 minus

34V012057920) = 0

119875119910 (1199080 + 11990528 11990830) minus 119875(1199080 minus 1199101198881205790 +

4312058711991111988812057920)

+ (119875119911 minus 119875119910) ( 83120587V012057920 +

34119908012057920) = 0

1198751205791205790 + 311986411986811990581198680 120579301199052

minus 119875(1205790 minus 1199101198881198680 1199080 +1199111198881198680 V0 +

8120579031205871198680 (1199080119911119888 + V0119910119888))

minus 119875119911 minus 1198751199101198680 (8V011990803120587 minus 3

4V201205790 +34119908201205790) = 0

(16)

where 119905 = 120587119871 119875119910 = 12058721198641198681199101198712 119875119911 = 12058721198641198681199111198712 and 119875120579 =(1198641198681205961205872 + 119866119869)11986801198712Equations (16) are high order nonlinear algebraic equa-

tions of four unknowns 119875 V0 1199080 and 1205790 Different fromthe Newton-Raphson method used by Mohri et al [4]the homotopy continuation method is adopted to solve theequations in this paper Through setting a series of 1205790 in thesoftware HOM4PS in literature [6] the corresponding load-displacement curves can be obtained and the critical loadscan be figured out

Members with other boundary conditions have beenanalyzed similarly and the derivedmathematical formulationis as below

For the member with two fixed ends the approximatedisplacement function is defined by

VV0= 1199081199080 =

1205791199091205790 = cos 2120587119909119871 minus 1 (17)

For the member fixed at one end and free at the other endthe approximate displacement function is written as

VV0= 1199081199080 =

1205791199091205790 = 1 minus cos 1205871199092119871 (18)

However for the member fixed at one end and pinned at theother end the approximate displacement function is definedas

VV0

= 1199081199080 =

1205791199091205790 =sin (449119909119871)

449 minus cos 449119909119871 + 119871 minus 119909119871 (19)

23 Nonlinearly Inelastic Buckling The analytic solution ofnonlinearly inelastic buckling load is difficult to be derivedbecause elastic and plastic zones of sections are variablealong longitudinal axis of compressive member Location ofshear center of each cross section is not constant in thiscase Generally the approximate solution is given by thetheory of tangential modulus of elasticity Finite ElementMethod is believed to be themost effectivemethod for findingeigenvalue buckling load and nonlinearly inelastic bucklingload [2 3]


z

O

y

C

180

mm

07mm

158mm

158

mm

Figure 2 Profile of the angle

3 Experimental Analysis

Experiment for investigating nonlinear-inelastic bucklingloads and buckling modes of axially compressive thin-walledequilateral angles with pinned and fixed ends was carriedout The angles are made of cold rolled sheet of S235 steelwith material properties of 119864 = 206GPa V = 03 and120590119910 = 360MPa respectively The compressive member is180mm long with angle section as shown in Figure 2 Forthe occurrence of inelastic buckling the slenderness of themember is 120582 = 18012058331717 where 120583 is the effective lengthfactor Under the coordinate axes 119910 and 119911 and shear point 119862(0 minus5478mm) shown in Figure 2 geometric parameters aredetermined namely 119868119910 = 2183mm4 119868119911 = 8693mm4 119860 =217mm2 1198680 = 8012mm2 119868120596 = 0mm6 119869 = 357mm4 and119868119905 = 111345mm6 respectively Strain gauges were mountedon the surface of each edge at the position of 1198714 1198712 and31198714 to the bottom where 119871 is the length of the memberTheaxial load is measured by sensor of TGZ-100 at support anddeflections in the middle span are recorded by dial indicator

In the experiment the angle only bent in one directionwithout torsion when it is pinned at two ends The primaryfailure mode is flexural-torsional inelastic buckling as shownin Figure 3 The local buckling caused by torsion occurredfirst in the lower part of the member and then overallbuckling happened From the slenderness ratio of 5675 italso can be predicted that inelastic buckling would happenfor the specimen The critical buckling loads and the load-displacement curves of the specimen are obtained as shownin Figure 4

4 Finite Element Analysis

Buckling load and buckling mode can be identified by finiteelement software through either eigenvalue buckling analysisor nonlinear buckling analysis The eigenvalue bucklinganalysis is applicable to determine the buckling strength ofan ideally linear-elastic structure where nonlinear terms ofmaterial and geometry are linearized to keep the stiffnessmatrix unchanged during load progressing Buckling of thin-walled compressive members is usually analyzed by usingbeam and shell elements Mohri [7 8] has developed the 3D-beam element with two nodes and seven degrees of freedomper node to analyze the linear and nonlinear-elastic bucklingbehavior of thin-walled beams with arbitrary cross sectionsand verified the accuracy by themodel of shell element Study

Figure 3 Axial compressive test of an angle

2 4 6 80Displacement (mm)

0

1

2

3

4

Load

(kN

)

Two pinned endsTwo fixed ends

Figure 4 Load-displacement curves of the specimen

by Zhang and Tong [9 10] showed that the Wagner effectof bending could not precisely calculated by beam elementsused in ANSYS and ABAQUS and a proposed thin shellelement model is superior to the beam element for flexural-torsional buckling analysis of thin-walled beams and beam-columns of monosymmetric section and tapered section

41 Nonlinearly Elastic Buckling Analysis The specimen usedin the experiment is of 180mm length with small slendernesstherefore the inelastic buckling occurs In this section forinvestigating the numerical solution of elastic buckling thelength of the computational model of the angle is deliberatelyassigned to be 1000mm Elements of Beam189 and Shell93 inANSYS are used to solve elastic buckling load as below

411 Beam Model According to the analytical solution theforegoing linear-elastic equation (2) and nonlinear-elasticequations (16) the 119875-1205790 curves of the angles with fourdifferent constraints are obtained as shown in Figures 5ndash8It can be seen that the postbuckling behavior can be revealedby the nonlinear analytical solution and the buckling loads

Mathematical Problems in Engineering 5P

(N)

NonlinearLinear

40

50

60

70

80

90

100

110

120

002 004 006 008 0100001205790 (rad)

Figure 5 119875-1205790 curve of member with fixed and free ends

P(N

)

NonlinearLinear

100

200

300

400

500

01 02 03 04001205790 (rad)

Figure 6 119875-1205790 curve of member with two pinned ends

in the extreme point instability are different from the first-order eigenvalue in linear solution and numerical solutionwith beam element

In the buckling analysis with Beam189 in ANSYS it isdefault that the centroid of the section is the coordinate ori-ginAxially compressive load is applied along the longitudinalcentroidal axis Constraint is applied at the centroid of eachend

The numerical buckling loads of the angles with differentend-constraints are compared with the analytic values andlisted in Table 1 It can be seen that the first-order eigenvaluesfrom numerical analysis with beam element are close tothe linear analytical results and the buckling loads con-sidering geometrically nonlinear instability are larger thanthose obtained from the classical eigenvalue buckling analysissince the translationrotation coupling higher order terms ofdisplacements in governing equations and the shorter effect

NonlinearLinear

P(N

)

01 02 03001205790 (rad)

900

910

920

930

940

950

960

Figure 7 119875-1205790 curve of member with fixed and pinned ends

NonlinearLinear

P(N

)

01 02 03 04001205790 (rad)

1700

1800

1900

2000

2100

2200

Figure 8 119875-1205790 curve of member with two fixed ends

are all counted It also can be found that the elastic bucklingloads of the member with 1000mm length listed in Table 1are much smaller than the inelastic buckling loads obtainedin the experiment where the length of themember is 180mm

412 ShellModel When the compressive angle is discretizedwith Shell93 element the axial load cannot be placed directlyat the centroid for it is out of the material of the cross sectionIn this case an evenly distributed load equivalent to axial loadis applied on the end section to avoid stress concentration

For the member with angle section the boundary con-dition is achieved by fixing transverse displacements of thecentroid [11 12] the corner points or the entire clusterof nodes on the end sections [13] When the transversedisplacement of the centroid is constrained an auxiliary platehas to be added at the end cross section for transfer of theload A hard point with fixed transverse displacements should


Table 1 Comparison of buckling loads obtained from analytical and numerical solution (kN)

Constraints Analytical results Numerical results with beam elements (NB) ErrorLinear (TL) Nonlinear (TN) (NBTL) (NBTN)

Fixed and free ends 01109 01111 01109 0 minus018Two pinned ends 04438 04457 04436 minus005 minus047Fixed and pinned ends 09080 09594 09068 minus013 minus548Two fixed ends 17753 21649 17714 minus022 minus1818

Table 2 Buckling loads from shell model with different boundary treatments

Constraints Beam elements Centroid constrained Error Corner constrained Error All nodes constrained ErrorFixed and free ends 01109 01183 667 01183 667 01183 667Two pinned ends 04436 0473 663 0472 640 0965 11754Fixed and pinned ends 09068 0966 653 0966 653 0965 642Two fixed ends 17714 1771 minus002 1092 minus3885 1092 minus3885

Fixed and free endsTwo pinned ends

Fixed and pinned endsTwo fixed ends

25

30

35

40

Fixe

d an

d fre

e end

s (kN

)

300 400 500200Yield limit (MPa)

Figure 9 Effect of yield stress on buckling load

be created at the position of the centroid on the plate If themember is fixed at ends the rotational angles of the hardpoint should be further restricted

Thenonlinearly elastic buckling loads obtainedwith thesethree treatments of boundary conditions are presented andcompared with those from the beam model as shown inTable 2 It is found that the critical loads determined fromthe model of centroid constrained are bigger than those fromthe beammodel since the rigidity of the member is improvedby adding plates at the ends If all nodes on the two endsof the angle are pinned constraints the stronger boundarytreatment results in higher buckling load The critical loadsobtained from fixed centroid and fixed corner points areclose though the former treatment of boundary is moreideal than the latter Therefore it is suggested that restrictionof the centroid with auxiliary end plates should be used innonlinearly elastic buckling analysis

42 Inelastic Buckling Analysis The angles tested in theexperiment are modeled in the inelastic buckling analysis

since the actual inelastic buckling occurred Shell element isusually applied when geometrically and physically nonlinear-inelastic buckling is analyzed Effects of scale factor of initialdefect yield strength loading mode treatment of constraintand rib on numerically buckling analysis are studied as wellIn the environment of ANSYS solution control the optionof large deformation is specified Initial defects are imposedthrough the command UPGEOM The first-order bucklingmode of displacement obtained from eigenvalue bucklinganalysis is multiplied by a scale factor and is imposed onthe nonlinear model afterwards The arc-length method isapplied in the solution [14]

421 Effects of Yield Stress Limit and Scale Factor Inelasticbuckling load is affected by yield strength of material Thebuckling load improves with the increase of yield strengthas shown in Figure 9 For comparison of the numerical andexperimental results the constitutive model of material isselected to be bilinear isotropic hardening with yield strengthof 360MPa in the following computation

The influence of the scale factor on the buckling loadis analyzed and illustrated in Figure 10 It is noted that thebuckling load decreases with the increase of scale factor butis not remarkable Based on the current codes for designof steel structures an equivalent length of the member asthe initial geometrical defect is considered when the overallstability of compressive steel member is designed [15] In thecorresponding Chinese Code it is suggested that the initialgeometrical defect is given in the shape of the first-ordermodal of elastic buckling where we let the initial deflectionnamely the production of the scale factor with the maximumdeflection under the first-order modal be equal to 1198711000[16]Themaximum deflections are different for the memberswith different end-constraints but the buckling loads are littleaffected by the small value of scale factor which is about 0008in the discussed problem

422 Effects of Other Factors As previously mentioned theboundary condition of the compressive angle can be treatedby constraining the centroid corner nodes or the entire




1

2

3

4Bu

cklin

g lo

ad (k

N)

05 1000Scale factors

Figure 10 Effect of scale factor on buckling load

Figure 11 The pinned constraint in computation

nodes on the end section when shell element is applied Inthe inelastic buckling analysis it is found that the bucklingload of 3479 kN is greater than the theoretical 3172 kNand serious stress concentration occurs as the centroid ofthe added plate is constrained When the corner nodes areconstrained the buckling load of 1482 kN is much smallerthan the theoretical value which indicates that the treatmentof boundary condition is too weak to represent the realpinned supportThrough trial calculations the buckling loadis the closest to the theoretical value and the one with thebeam model when nine uniform nodes are constrained asshown in Figure 11

A convenient measure preventing local buckling is to adda transverse rib in the longitudinal direction of the angleZhang proposed modeling the rib separately from the angleand the transverse degrees of freedom at joint were coupled toavoid increasing stiffness of the structure [10] To investigatethe overall instability one rib is added in the same way atthe middle of the member as shown in Figure 12 based onthe slenderness ratio of the angle The computational results

Jan 25 2015144431

Figure 12 Angle with a rib in the middle

reveal that there is neither obvious change in buckling loadnor occurrence of locally large deformation

The computational buckling load is also affected by theways of loadings The nodal forces on the end section arenot uniform when certain displacement is assigned to aspecific node the descending segment of load-displacementcurve can be generated By contrast there is no obviousdescending segment of load-displacement curve when auniformly distributed force is applied on each node Theinflection point on the curve of load-displacement can befound whose ordinate gives the ultimate load of overallbuckling

423 Discussion of Numerical Results Based on the fore-going investigation the inelastic critical loads of the testedangle are obtained via nonlinear buckling analysis of shellelement in ANSYS as shown in Table 3 The theoreticalnonlinear-elastic results the loads of eigenvalue bucklingdetermined using beam and shell models are presented aswell for comparison The following can be found

(1) The tested buckling loads are bigger than the eigen-value buckling loads determined by shell model closeto beam model due to the pinned support withtorsional and flexural restriction in the experimentThe theoretical elastic buckling loads are smallerthan the first-order eigenvalue buckling loads sincethe nonlinear coupling terms are ignored in thetheoretical analysis even though the twomethods aresimilar

(2) Shell element is appropriate for analyzing overallstability of thin-walled member from the point ofgeometric similarity The eigenvalue buckling loadsthrough shell model are smaller than those of beammodel since the Wangerrsquos effect in moment of thin-walledmember cannot be admitted into beammodelleading the numerical results bigger than the realwhich is not suitable for buckling analysis for amember with monosymmetric section

(3) The buckling load determined through nonlinearbuckling analysis is different from eigenvalue anal-ysis via shell model Load-displacement curves of


Table 3 Nonlinear buckling load obtained by different methodologies (kN)

Constraints Test Analytic results Beam element (eigenvalue) Shell elements (eigenvalue) Shell elements (nonlinear)Fixed and free ends 2157 31844 2998 2852Two pinned ends 313 3172 3447 3283 3093Fixed and pinned ends 3359 3581 3323 3231Two fixed ends 340 3444 363 3528 3733

Two fixed endsTwo pinned ends

Fixed and free endsFixed and pinned ends

005 010000Displacement (mm)

0

1

2

3

Load

(kN

)

Figure 13 Load-displacement curve of nonlinearized inelasticbuckling

the angles with various constraints in nonlinear-inelastic buckling analysis are obtained as shown inFigure 13 This shows that the stability of the anglesimproves with the strengthening of constraints Thenonlinearized inelastic buckling loads may be biggeror smaller than those of eigenvalue buckling loadsfor the members with different constraints and scalefactors see the last column in Table 3 which gives thesimilar tendency as [17]

5 Conclusion

The computational methods of buckling loads of thin-walledmembers with open sections have been carried out in thispaper From the analysis of nonlinearly elastic and inelasticbuckling load computation the following is found

Nonlinearly elastic buckling loads determined frommathematical formula and beam model are close and pos-sess high precision Mathematical solution of higher orderdifferential equations is complicated So for the member withsimple deflection the method in this paper is acceptableotherwise differencemethod is generally appliedWhen shellmodel is used in the computation the restriction of thecentroid with auxiliary end plates is more ideal than othertreatments of boundary conditions

Numerical solution is the most effective method for thenonlinearly inelastic overall buckling analysis The effects ofloading mode treatment of boundary condition adding rib

initial defect and yield strength on computational resultsshould be noted when shell element is applied It has beenfound that the eigenvalue buckling analysis through shellmodel possesses high precision from the study of this paperThe nonlinear buckling load calculated by shell element maybe bigger or smaller than eigenvalue buckling load whichis influenced by the yield strength of the material and scalefactor of initial defect

Nomenclature

119860 Cross section area119864 Youngrsquos modulus119866 Shear modulus1198680 Polar moment of inertia about shear center119868119877 Fourth moment of inertia about shearcenter119868119905 Higher order shortening constant119868119910 119868119911 Principle moment of inertia about 119910 and 119911axes119868120596 Warping constant119869 St-Vant torsion constant119871 Member length119875 Compression axial load119880 Stress energy119882 Work of conservative loads119906 V 119908 Shear displacement component of shearcenter 119862 in 119909 119910 and 119911 axes119906119872 V119872 119908119872 Displacement components of119872 in 119909 119910and 119911 axes

V0 1199080 Displacement amplitudes of V and 119908 innumerical analysis119909 119910 119911 Principal coordinate of119872 in globalreference119910119888 119911119888 Shear coordinate of 119874119910119911 reference120576119909119909 120576119909119910 120576119909119911 Axial and shear strain components120583 Effective length factor120596 Warping coordinate1205790 Torsion angle120579119909 Torsion amplitude in numerical analysis

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


Acknowledgments

The authors would like to acknowledge the financial supportof the Natural Science Fund of Liaoning Province China(2014020008)

References

[1] S H Bao and J Zhou Structural Mechanics of Thin-Walled BarChina Architecture amp Building Press Beijing China 1991

[2] J Chen and H Chen Stability of Steel Structures Theory andDesign China Electric Power Press Beijing China 2009

[3] GM Liu R X Zhang andT S ZhangApplication of StructuralStability Calculation Science Press Beijing China 2004

[4] F Mohri L Azrar and M Potier-Ferry ldquoFlexural-torsionalpost-buckling analysis of thin-walled elements with open sec-tionsrdquoThin-Walled Structures vol 39 no 11 pp 907ndash938 2001

[5] E J Sapountzakis and V J Tsipiras ldquoWarping shear stressesin nonlinear nonuniform torsional vibrations of bars by BEMrdquoEngineering Structures vol 32 no 3 pp 741ndash752 2010

[6] T L Lee T Y Li and C H Tsai ldquoHOM4PS-20 a softwarepackage for solving polynomial systems by the polyhedralhomotopy continuation methodrdquo Computing vol 83 no 2-3pp 109ndash133 2008

[7] F Mohri A Eddinari N Damil and M Potier Ferry ldquoA beamfinite element for non-linear analyses of thin-walled elementsrdquoThin-Walled Structures vol 46 no 7ndash9 pp 981ndash990 2008

[8] F Mohri A Brouki and J C Roth ldquoTheoretical and numericalstability analyses of unrestrainedmono-symmetric thin-walledbeamsrdquo Journal of Constructional Steel Research vol 59 no 1pp 63ndash90 2003

[9] L Zhang and G S Tong ldquoFinite element modelling of thin-walledmembers in overall stability analysisrdquo Journal of ZhejiangUniversity (Engineering Science) vol 45 no 3 pp 531ndash538 2011

[10] L Zhang A Theory of Stability for Thin-Walled MembersConsidering the Effects of Transverse Stresses and Its ApplicationsZhejiang University Hangzhou China 2005

[11] J Chen Buckling Behavior of Single Cold-Formed Angle underAxial Load Xirsquoan University of Architecture and TechnologyXirsquoan China 2004

[12] H-Y Ban G Shi and Y-J Shi ldquoInvestigation on designmethodof overall buckling behaviour for Q420 high strength steelequal-leg anglemembers under axial compressionrdquoEngineeringMechanics vol 31 no 3 pp 63ndash71 2014

[13] X X Hong The Stability of the Cold-Formed 60∘ EquilateralSingle Angle under Compression Changrsquoan University XirsquoanChina 2008

[14] J F Zhu and H P Yang ldquoAccelerated arc-length method forsolving postbuckling path of structuresrdquo Journal of DalianUniversity of Technology vol 34 no 1 pp 17ndash22 1994

[15] G Shi Y Q Shi and Y Q Wang ldquoAnalysis on overall bucklingbehavior of ultra-high strength steel columns by ANSYSrdquoJournal of Jilin University (Engineering and Technology Edition)vol 39 no 1 pp 113ndash118 2009

[16] National Standard Specification Code for Design of Steel Struc-tures (GB 50017-2003 PRC) China Plan Press Beijing China2003 (Chinese)

[17] Q H Han H Jin J Ai and X Y Liu ldquoAnalysis of the overallbuckling load for engineering structurerdquo Journal of TianjinUniversity vol 38 no 12 pp 1051ndash1057 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


M

z

O y

x

w

C (yc zc)

Figure 1 Model of a thin-walled member with open section

realized using the elements of beam and shell The effectsof modeling of constraints on numerical results are studiedwhen shell elements are applied Analysis of linearized buck-ling analysis and nonlinearized buckling analysis of materialand geometrical nonlinearity are conducted and comparedwith experimental and theoretical results The factors thataffect the inelastic buckling loads in numerical analysis suchas modeling of constraints scale factors of initial defectsadding rib and loadingmodes are discussedThe noteworthyproblems and their solutions in numerical buckling analysisof thin-walled member with open section are pointed out

2 Mathematical Formulation

A thin-walled compressive member with an arbitrary opensection is shown in Figure 1 where point 119874 is the centroid ofthe section and point119862(119910119888 119911119888) is the shear center119874119909119910119911 is thecentroidal principal axis

21 Equations of Classical Linear-Elastic Buckling The equa-tions of classical stability for a thin-walled member are aset of fourth-order linear differential equations with constantcoefficients [1]

119864119868119911V(4) + 119875V10158401015840 + 11987511991111988812057910158401015840119909 = 0119864119868119910119908(4) + 11987511990810158401015840 minus 11987511991011988812057910158401015840119909 = 0119875119911119862V10158401015840 minus 11987511991011986211990810158401015840 + 119864119868120596120579(4)119909 + (1198751198680 minus 119866119869) 1205792119909 = 01198680 = (119868119910 + 119868119911)

119860 + 1199102119888 + 1199112119888

(1)

where 119864 and 119866 are the elastic constants 119875 is an axiallycompressive force applied at the centroid The displacementsof shear center in 119910 and 119911 directions are denoted by Vand 119908 respectively The rotation of the cross section isrepresented by 120579119909 119869 is free torsionalmoment of inertia and 119868120596denotes the principal warping moment of inertiaThe criticalload is determined through solving the eigenvalue problemsyielding

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119875119911 minus 119875 0 minus1198751199111198880 119875119910 minus 119875 119875119910119888

minus119875119911119888 119875119910119888 1198751205961198680 minus 1198751198680

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (2)

in which 119875119909 = 1198641198681199091205822 119875119910 = 1198641198681199101205822 and 119875120596 = (1198641198681205961205822 + 119866119869)1198680120582 = 119899120587119871 for member pinned at two ends where 119899 =1 2 3 and 119871 denotes the length of the member 120582 =2119899120587119871 for two fixed ends For a member fixed at one endand free at the other end 120582 = 119899120587(2119871) For a member fixed atone end and pinned at the other end 120582 = 449311987122 Formulation of Geometrically Nonlinear-Elastic-BucklingOn the basis of literature [4] the transversal displacementsV119872 and 119908119872 of any point 119872 on the cross section when themember is twisted at an angle 120579119909 are expressed by

V119872 = V minus (119911 minus 119911119862) sin 120579119909 minus (119910 minus 119910119862) (1 minus cos 120579119909) 119908119872 = 119908 + (119910 minus 119910119862) sin 120579119909 minus (119911 minus 119911119862) (1 minus cos 120579119909) (3)

The axial displacement 119906119872 is expressed by

119906119872 = 119906 minus 119910 (V1015840 cos 120579119909 + 1199081015840 sin 120579119909)minus 119911 (1199081015840 cos 120579119909 minus V1015840 sin 120579119909) + 1205961205791015840119909

(4)

where 119906 V and 119908 are the displacements of shear center in 119909119910 and 119911 directionsThe Green strain tensor that incorporates large displace-

ments is defined by

120576119894119895 = 12 (

120597119906119894120597119909119895 +120597119906119895120597119909119894 +

120597119906119896120597119909119894120597119906119896120597119909119895) (5)

Hence the strains of point119872 can be expressed through

120576119909119909 = 1199061015840 minus 119910 (V10158401015840 cos 120579119909 + 11990810158401015840 sin 120579119909)minus 119911 (11990810158401015840 cos 120579119909 minus V10158401015840 sin 120579119909) + 12059612057910158401015840119909+ 12 (V10158402 + 11990810158402 + 119877212057910158402119909 )

minus 1199101198621205791015840119909 (1199081015840 cos 120579119909 minus V1015840 sin 120579119909)+ 1199111198621205791015840119909 (V1015840 cos 120579119909 + 1199081015840 sin 120579119909)

120576119909119910 = minus12 (119911 minus 119911119862 minus120597120596120597119910 ) 1205791015840119909

120576119909119911 = 12 (119910 minus 119910119862 +

120597120596120597119911 ) 1205791015840119909

(6)

where

1198772 = (119910 minus 119910119862)2 + (119911 minus 119911119862)2 (7)

Based on the principle of virtual work one obtains

120575 (119880 minus119882) = 0 (8)

where

120575119880 = int119871int119860(120590119909119909120575120576119909119909 + 2120590119909119910120575120576119909119910 + 2120590119909119911120575120576119909119911) d119860 d119909

120575119882 = int (119902119909120575119906 + 119902119910120575V + 119902119911120575119908 + 119898119909120575120579119909) d119909(9)



119873 = minus119875 (10)

119864119868119910 (119908(4) + 3119908101584011990810158401015840119908101584010158401015840 + 119908101584010158403 + 119908(4)119908101584022 ) minus 119873(11990810158401015840

minus 11991011988812057910158401015840119909 + 119911119888 (12057911990912057910158401015840119909 + 12057910158402119909 )) + (119864119868119911 minus 119864119868119910) (V(4)120579119909+ 2V1015840101584010158401205791015840119909 + V1015840101584012057910158401015840119909 + 119908(4)1205792119909 + 41199081015840101584010158401205791199091205791015840119909 + 21199081015840101584012057911990912057910158401015840119909+ 21199081015840101584012057910158402119909 ) = 0

(11)

119864119868119911 (V(4) + 3V1015840V10158401015840V101584010158401015840 + V101584010158403 + V(4)V10158402

2 ) minus 119873(V10158401015840


(12)



(13)

where



VV0= 1199081199080 =

1205791199091205790 = sin 120587119909119871 (15)


119875119911 (V0 + 11990528 V30) minus 119875(V0 + 1199111198881205790 +

4312058711991011988812057920)

+ (119875119911 minus 119875119910) ( 83120587119908012057920 minus

34V012057920) = 0

119875119910 (1199080 + 11990528 11990830) minus 119875(1199080 minus 1199101198881205790 +

4312058711991111988812057920)

+ (119875119911 minus 119875119910) ( 83120587V012057920 +

34119908012057920) = 0

1198751205791205790 + 311986411986811990581198680 120579301199052

minus 119875(1205790 minus 1199101198881198680 1199080 +1199111198881198680 V0 +

8120579031205871198680 (1199080119911119888 + V0119910119888))


4V201205790 +34119908201205790) = 0

(16)





VV0= 1199081199080 =

1205791199091205790 = cos 2120587119909119871 minus 1 (17)


VV0= 1199081199080 =

1205791199091205790 = 1 minus cos 1205871199092119871 (18)


VV0

= 1199081199080 =

1205791199091205790 =sin (449119909119871)

449 minus cos 449119909119871 + 119871 minus 119909119871 (19)



z

O

y

C

180

mm

07mm

158mm

158

mm









0

1

2

3

4

Load

(kN

)







(N)

NonlinearLinear

40

50

60

70

80

90

100

110

120

002 004 006 008 0100001205790 (rad)


P(N

)

NonlinearLinear

100

200

300

400

500

01 02 03 04001205790 (rad)





NonlinearLinear

P(N

)

01 02 03001205790 (rad)

900

910

920

930

940

950

960


NonlinearLinear

P(N

)

01 02 03 04001205790 (rad)

1700

1800

1900

2000

2100

2200













25

30

35

40

Fixe

d an

d fre

e end

s (kN

)













1

2

3

4Bu

cklin

g lo

ad (k

N)






Jan 25 2015144431














0

1

2

3

Load

(kN

)



5 Conclusion





Nomenclature



Competing Interests



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119873 = minus119875 (10)

119864119868119910 (119908(4) + 3119908101584011990810158401015840119908101584010158401015840 + 119908101584010158403 + 119908(4)119908101584022 ) minus 119873(11990810158401015840

minus 11991011988812057910158401015840119909 + 119911119888 (12057911990912057910158401015840119909 + 12057910158402119909 )) + (119864119868119911 minus 119864119868119910) (V(4)120579119909+ 2V1015840101584010158401205791015840119909 + V1015840101584012057910158401015840119909 + 119908(4)1205792119909 + 41199081015840101584010158401205791199091205791015840119909 + 21199081015840101584012057911990912057910158401015840119909+ 21199081015840101584012057910158402119909 ) = 0

(11)

119864119868119911 (V(4) + 3V1015840V10158401015840V101584010158401015840 + V101584010158403 + V(4)V10158402

2 ) minus 119873(V10158401015840


(12)



(13)

where



VV0= 1199081199080 =

1205791199091205790 = sin 120587119909119871 (15)


119875119911 (V0 + 11990528 V30) minus 119875(V0 + 1199111198881205790 +

4312058711991011988812057920)

+ (119875119911 minus 119875119910) ( 83120587119908012057920 minus

34V012057920) = 0

119875119910 (1199080 + 11990528 11990830) minus 119875(1199080 minus 1199101198881205790 +

4312058711991111988812057920)

+ (119875119911 minus 119875119910) ( 83120587V012057920 +

34119908012057920) = 0

1198751205791205790 + 311986411986811990581198680 120579301199052

minus 119875(1205790 minus 1199101198881198680 1199080 +1199111198881198680 V0 +

8120579031205871198680 (1199080119911119888 + V0119910119888))


4V201205790 +34119908201205790) = 0

(16)





VV0= 1199081199080 =

1205791199091205790 = cos 2120587119909119871 minus 1 (17)


VV0= 1199081199080 =

1205791199091205790 = 1 minus cos 1205871199092119871 (18)


VV0

= 1199081199080 =

1205791199091205790 =sin (449119909119871)

449 minus cos 449119909119871 + 119871 minus 119909119871 (19)



z

O

y

C

180

mm

07mm

158mm

158

mm









0

1

2

3

4

Load

(kN

)







(N)

NonlinearLinear

40

50

60

70

80

90

100

110

120

002 004 006 008 0100001205790 (rad)


P(N

)

NonlinearLinear

100

200

300

400

500

01 02 03 04001205790 (rad)





NonlinearLinear

P(N

)

01 02 03001205790 (rad)

900

910

920

930

940

950

960


NonlinearLinear

P(N

)

01 02 03 04001205790 (rad)

1700

1800

1900

2000

2100

2200













25

30

35

40

Fixe

d an

d fre

e end

s (kN

)













1

2

3

4Bu

cklin

g lo

ad (k

N)






Jan 25 2015144431














0

1

2

3

Load

(kN

)



5 Conclusion





Nomenclature



Competing Interests



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










z

O

y

C

180

mm

07mm

158mm

158

mm









0

1

2

3

4

Load

(kN

)







(N)

NonlinearLinear

40

50

60

70

80

90

100

110

120

002 004 006 008 0100001205790 (rad)


P(N

)

NonlinearLinear

100

200

300

400

500

01 02 03 04001205790 (rad)





NonlinearLinear

P(N

)

01 02 03001205790 (rad)

900

910

920

930

940

950

960


NonlinearLinear

P(N

)

01 02 03 04001205790 (rad)

1700

1800

1900

2000

2100

2200













25

30

35

40

Fixe

d an

d fre

e end

s (kN

)













1

2

3

4Bu

cklin

g lo

ad (k

N)






Jan 25 2015144431














0

1

2

3

Load

(kN

)



5 Conclusion





Nomenclature



Competing Interests



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(N)

NonlinearLinear

40

50

60

70

80

90

100

110

120

002 004 006 008 0100001205790 (rad)


P(N

)

NonlinearLinear

100

200

300

400

500

01 02 03 04001205790 (rad)





NonlinearLinear

P(N

)

01 02 03001205790 (rad)

900

910

920

930

940

950

960


NonlinearLinear

P(N

)

01 02 03 04001205790 (rad)

1700

1800

1900

2000

2100

2200













25

30

35

40

Fixe

d an

d fre

e end

s (kN

)













1

2

3

4Bu

cklin

g lo

ad (k

N)






Jan 25 2015144431














0

1

2

3

Load

(kN

)



5 Conclusion





Nomenclature



Competing Interests



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















25

30

35

40

Fixe

d an

d fre

e end

s (kN

)













1

2

3

4Bu

cklin

g lo

ad (k

N)






Jan 25 2015144431














0

1

2

3

Load

(kN

)



5 Conclusion





Nomenclature



Competing Interests



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1

2

3

4Bu

cklin

g lo

ad (k

N)






Jan 25 2015144431














0

1

2

3

Load

(kN

)



5 Conclusion





Nomenclature



Competing Interests



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















0

1

2

3

Load

(kN

)



5 Conclusion





Nomenclature



Competing Interests



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Computation Analysis of Buckling Loads of ...downloads.hindawi.com/journals/mpe/2016/8320469.pdf · Research Article Computation Analysis of Buckling Loads of Thin-Walled