2011_ lateral torsional buckling of rectangular beams using variational iteration method

7/27/2019 2011_ Lateral Torsional Buckling of Rectangular Beams Using Variational Iteration Method

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Scientific Research and Essays Vol. 6(6), pp. 1445-1457, 18 March, 2011Available online at http://www.academicjournals.org/SREISSN 1992-2248 2011 Academic Journals

Full Length Research Paper

Lateral torsional buckling of rectangular beams usingvariational iteration method

Seval Pinarbasi

Department of Civil Engineering, Kocaeli University, Umuttepe Campus, 41380, Kocaeli, Turkey.E-mail: [email protected]. Tel: +90 262 303 3274.

Accepted 15 March, 2011

Lateral torsional buckling is the main failure mode that controls the design of slender beams; that is,the beams which have greater major axis bending stiffness than minor axis bending stiffness or thebeams which have considerably large laterally unsupported lengths. Since the buckling equations for

beams are usually much more complex than those for columns, most of the analytical studies inliterature on beam buckling are concentrated on simple cases. This paper shows that complex beambuckling problems, such as lateral torsional buckling of narrow rectangular cantilever beams whoseminor axis flexural and torsional rigidities vary exponentially along their lengths, can successfully besolved using variational iteration method (VIM). The paper also investigates the effectiveness of threeVIM algorithms, two of which have been proposed very recently in solving lateral torsional bucklingequations. Analysis results show that all iteration algorithms yield exactly the same results in allstudied problems. As far as the computation times and spaces are concerned, however, one of thesealgorithms, called variational iteration algorithm II, is found to be superior than the others especially inlateral torsional buckling problems where the beam rigidities vary along the beam length.

Key words: Lateral torsional buckling, narrow rectangular beam, tapered beam, variable rigidity, variationaliteration algorithms, variational iteration method.

INTRODUCTION

Beams are main structural elements which primarilysupport transverse loads, that is, loads perpendicular totheir axes. They are also called flexural members sincethey are typically designed to carry flexural loads, that is,transverse shear forces and bending moments. As thenormal force in a beam is generally negligibly small, thedesign of a laterally-braced beam is usually straight-forward: select the most economical compact crosssection which has adequate major axis section modulus

to safely carry maximum bending moment in the beamwith adequate web area to carry maximum shear force.However, similar to a slender column which bucklesunder compressive loads, a laterally-unbraced slenderbeam can also buckle under flexural loads. This occursdue to the fact that when subjected to bending moments,compressive stresses develop at one part of the beam

Abbreviations: VIM, Variational iteration method; VIA,variational iteration algorithm.

cross section which tends to buckle in lateral (out-oplane) direction. In such a buckling mode, the beam noonly displaces laterally but also twists due to the fact thathe remaining part of the beam cross section, which issubjected to tensile stresses, resists against buckling. Fothis reason, this buckling mode is commonly calledlateral torsional buckling. Lateral torsional buckling, osimply lateral buckling, is the main limit state that has tobe considered in the design of slender beams, that is

the beams which have greater stiffness in in-plane (majoraxis) bending than in out-of-plane (minor axis) bending orthe beams which have considerably large laterally unsup-ported lengths. Unless properly braced against lateradeflection and/or torsion, a slender beam will buckle prioto the attainment of its major axis bending capacity.

Since determining the lateral torsional buckling load omoment of a slender beam is crucial in their designmany studies have been conducted on beam bucklingHowever, due to the fact that the beam bucklingequations are much more complex than the columnbuckling equations, most of the analytical studies in


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1446 Sci. Res. Essays

literature are concentrated on simple cases. The solutionsfor simple buckling problems can be obtained from well-known structural stability books, such as Timoshenko andGere (1961), Chajes (1974), Wang et al. (2005) andSimitses and Hodges (2006). In recent years, the advent ofcomputer-aided numerical techniques has enabled theresearchers to obtain solutions for more complex lateralbuckling problems.

The variational iteration method (VIM) is a kind ofnonlinear analytical technique which was proposed by He(1999) and developed fully in the following year. Thetechnique was successfully applied to various kinds ofnonlinear problems (He, 2000, 2007; He et al., 2007)since 1999. According to He et al. (2010), the number ofpublications on VIM has reached 130 in October, 2009,which clearly verifies the effectiveness of the technique insolving nonlinear problems. Very recently, VIM is alsoapplied to the buckling problems. Coskun and Atay(2009), Atay and Coskun (2009), Coskun (2010) andOkay et al. (2010) analyzed the elastic stability of Euler

columns with variable cross sections under differentloading and boundary conditions and verified that VIM isa very efficient and powerful method in analysis ofbuckling problems of columns with variable crosssections.

In this paper, this powerful analytical technique isapplied to two fundamental beam buckling problems:lateral buckling of (a) simply supported narrow rectangu-lar beams with variable rigidities under uniform momentand (b) narrow rectangular cantilever beams with variablerigidities carrying concentrated load at their free ends.Both linear and exponential variations are considered inminor axis flexural and torsional rigidities of the beams.

Exact solutions to these problems, some of which areconsiderably complex, are available in literature only forbeams of constant rigidities and some particular cases oflinearly tapered beams. To verify the effectiveness of VIMin solving lateral buckling problems, buckling loads/moments for uniform beams with constant rigidities arestudied first. Then, the nonuniform cases are studied foreach problem separately. In the paper, the effectivenessof the three VIM algorithms, two of which have recentlybeen proposed by He et al. (2010), in solving lateraltorsional buckling equations is also investigated.

LATERAL TORSIONAL BUCKLING OF BEAMS

In this section of the paper, first, the basic lateral buckling theory asgiven in Timoshenko and Gere (1961) will be summarized veryshortly and the notation that will be used in the study will beintroduced. Then, the differential equations for lateral torsionalbuckling of beams with two fundamental loading and restraintconditions will be derived.

General beam buckling equations

In order to derive general lateral torsional buckling equations fornarrow rectangular beams, we first consider a rectangular beam

subjected to arbitrary loading in y-zplane causing its strong-axisbending. The fixed x, y, z coordinate system defines theundeformed configuration of the beam, as shown in Figure 1a

Similarly, , , , coordinate system located at the centroid of thecross section at an arbitrary section of the beam along its lengthdefines the deformed configuration of the beam. As shown in Figure1b, the deformation of the beam can be defined by lateral (u) and

vertical (v) displacements of the centroid of the beam, which arepositive in the positive directions of xand y, respectively, and angle

of twist () of the cross section, which is positive about positive zaxis, obeying the right hand rule.

Hence, the displacements illustrated in Figure 1b, are bothnegative. For small deformations, the curvatures in xz and yzplanes can be taken, respectively, as d2u/dz2 and d2v/dz2 and thecosines of the angles between the axes can be taken as listed inTable 1. Since the warping rigidity of a narrow rectangular beamcan realistically be taken as zero, using positive directions ointernal moments defined in Figure 2, the equilibrium equations fothe buckled beam can be written as follows:

2

2

d vEI M

dz = ,

2

2

d uEI M

dz = and t

dGI M

dz

= (1)

representing the major axis bending, minor axis bending and

twisting of the beam, respectively. In Equation 1, EI and EIdenote the strong axis and weak axis flexural rigidities of the beamrespectively. Similarly, GItdenote the torsional rigidity of the beam.

Lateral buckling of simply supported rectangular beams inpure bending

Consider a simply supported rectangular beam with variable

rigidities ( )EI z , ( )EI z and ( )tGI z along its length L (Figure 3)If the beam is subjected to equal end moments Mo about x-axis, thebending and twisting moments at any cross section can be found by

determining the components of Mo about ,, axes. Consideringthe sign convention defined in Figure 2, and using Table 1, thesecomponents can be written as:

oM M = , oM M = and odu

M Mdz

=

(2)

Then, the equilibrium equations for the buckled beam (Figure 1bbecome

( )2

2 o

d vEI z M

dz

= , ( )2

2 o

d uEI z M

dz = and

( )t od du

GI z M dz dz

= (3)

It is apparent from Equation 3 that v is independent from uand which are coupled. Thus, in this problem, it is sufficient to consideronly the coupled equations. Differentiating the last equation inEquation 3 with respect to z and using the resulting equation toeliminate u in the second equation in Equation 3, the followingdifferential equation for the angle of twist of the beam is obtained:

( )

( ) ( ) ( )

22

2

10

t o

t t

d GI z Md d

dz dz dz GI z GI z EI z

+ + =

(4)


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Pinarbasi 1447

(a)

(b)

y

z

m

n

x

y

x

z

Side View

Top View

Section m-n

y

z

Side View

C

-v

x

Top View

C

-u

x

y

C

C

-v

-u

Section m-n

Figure 1. (a) Undeformed and (b) buckled shapes of a double symmetric beam loaded to bend about its majoraxis.

Since the ends of the simply supported beam are restrained againstrotation about zaxis, the boundary conditions for the problem are = 0 at both z= 0 and z= L.

Lateral buckling of rectangular cantilever beams with verticalend load

Consider a narrow rectangular cantilever beam of length L with

variable rigidities ( )EI z , ( )EI z and ( )tGI z . If the beamis subjected to a vertical load Ppassing through its centroid at itsfree end, as shown in Figure 4, the components of the moments ofthe load at an arbitrary section m-n about x, y, z axes are

( )xM P L z = , 0yM = and ( )1zM P u u = + (5)

where u1 is the lateral displacement of the loaded end of the beamas shown in Figure 4b.

Considering the sign convention defined in Figure 2, the bendingand twisting moments at this arbitrary section can be written as

( )M P L z = , ( )M P L z = and

( ) ( )1du

M P L z P u u dz

=

( ) ( )

2

2

d vEI z P L z

dz

= (6

Then, the equilibrium equations for the buckled beam become


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Figure 2. Positive directions for internal moments.

y

z

m

n

x

y

Section m-n

Mo Mo

L

Figure 3. Simply supported rectangular beam under pure bending.

Table 1. Cosine of angles between axes (Timoshenko and Gere,1961).

x y z

1 -du/dz

- 1 -dv/dz

du/dz dv/dz 1

( ) ( )

2

2

d u

EI z P L z dz

= and

( ) ( ) ( )1td du

GI z P L z P u u dz dz

=

(7)

Similar to the pure bending case, v is independent from uand .Differentiating the last equation in Equation 7 with respect to zandusing the resulting equation to eliminate uin the second equation in

Equation 7, the following differential equation is obtained for :

( )

( ) ( ) ( )( )

2 22

2

10

t

t t

d GI z d d PL z

dz dz dz GI z GI z EI z

+ + =

(8)

Since the fixed end of the beam is restrained against rotation andsince the twisting moment at the free end is zero, the boundaryconditions for this problem are

= 0 at z= 0 and d /dz= 0 at z= L.

VIM FORMULATIONS FOR THE BUCKLING PROBLEMS

In a recent paper, He et al. (2010) proposed three variationaiteration algorithms for solving various types of differentiaequations. The first algorithm is the classical VIM algorithm definedin He (1999). For a general homogeneous nonlinear differentiaequation,

( ) ( ) + = 0L z N z (9)

Where L is a linear operator and Nis a nonlinear operator, and thecorrection functional is

( ) ( ) ( ) ( ) ( ){ }

+ = + +

1

0

z

n n n n z z L N d (10)

In Equation 10, ( ) is a general Lagrange multiplier that can be

identified optimally via variational theory, n

is the n-th

approximate solution and n

denotes a restricted variation, that is

= 0n . The iteration algorithm in original VIM, called variationa

iteration algorithm I (VIA I), is as follows:

( ) ( ) ( ) ( ) ( ){ } + = + +10

z

n n n n z z L N d (11)

In the second algorithm proposed in He et al. (2010), calledvariational iteration algorithm II (VIA II), the iteration formula ismuch simpler:

( ) ( ) ( ) ( ){ } + = + 1 00

z

n nz z N d (12)

The main shortcoming of this algorithm is stated to be the

requirement that 0

be selected to satisfy the initial/boundary

conditions. The third algorithm in He et al. (2010), called variationaiteration algorithm III (VIA III), has the following i teration formula:

( ) ( ) ( ) ( ) ( ){ } + + += + 2 1 10

z

n n n n z z N N d (13

As summarized in He et al. (2010), for a second order differentiaequation such as the equations of the problems considered in this

paper, that is, Equation 4 and Equation 8, ( ) simply equals to

( ) = z (14)

Thus, the three VIM iteration algorithms for Equation 4 are:


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(a)

(b)

y

z

P

L

m

n

x

y

Section m-n

Side View

y

z

-v

x

y

C

C

-v

-u

Section m-n

x

z

-u1

-u

Top View

Figure 4. (a) Undeformed and (b) buckled shapes of a narrow rectangular cantilever beam carrying concentratedload at its free end.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ } + = + + +10

z

n n n n n z z z F FM d

( ) ( ) ( ) ( ) ( ) ( ) ( ){ } + = + +1 00

z

n n nz z z F FM d

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ } + + + + = + + 2 1 1 10

z

n n n n n n z z z F FM d

(15)

and those for Equation 8 are

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ } + = + + + 2

1

0

z

n n n n n z z z F FP L d

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ } + = + + 2

1 0

0

z

n n nz z z F FP L d

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ } + + + + = + + 2

2 1 1 1

0

z

n n n n n n z z z F FP L d

(16)

In Equation 15 and Equation 16,


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Table 2. Normalized buckling moments () for the first threemodes constant rigidities.

Mode # VIA I VIA II VIA III Exact

1 9.8696 9.8696 9.8696 9.8696

2 39.4787 39.4784 39.4784 39.4784

3 88.8264 88.8264 88.8264 88.8264

( )( )

( )

=

t

t

GI zF z

GI z

,( )

( ) ( )=

2

o

t

MFM z

GI z EI z

and( )

( ) ( )=

2

t

PFP z

GI z EI z

(17)

and prime denotes the differentiation of functions with respect totheir variables.

ANALYSIS OF RESULTS

Critical moment for pure bending case

Beams with constant rigidities

If the minor axis flexural and torsional rigidities of the

beam are constant; that is, ( ) =EI z EI and ( ) =t tGI z GI ,

then Equation 4 reduces to the following simplerequation:

+ =

22

120

d

dz(18)

Where

=2

2

1o

t

M

GI EI

The solution of Equation 18 is in the form

( ) ( ) = +1 1 2 1sin cosC z C z (19)

where C1 and C2 are constants to be determined fromboundary conditions. When the related boundary

conditions are used, Equation 19 leads to the followingcharacteristic equation

( ) =1sin 0L (20)

whose smallest root yields the first mode critical momentMcras

=

t

cr

GI EIM

L(21)

Exact values for the second and third modes can beobtained by determining larger roots of Equation 20which leads to four and nine times the first mode criticamoment, respectively.

In order to compare the efficiency of the three varia-tional iteration algorithms mentioned earlier, this case o

the problem is solved using all three VIM algorithms. Foreasier computations, the nondimensional form of thedifferential equation and the related boundary conditionsare written:

+ =

2

20

d

dz(22)

where,

=2 2

o

t

M L

GI EIwith ( ) =0 0 and

( ) =1 0

where = /z z L , = and is the nondimensionacritical moment. For all three algorithms, the initiaapproximation is chosen as a linear function withunknown coefficients as follows:

= +Az B (23)

The coefficients A and Bare determined by imposing theboundary conditions to the approximate solution obtainedat the end of the iterations, which leads to a characteristicequation whose roots give the buckling moments in

different modes. In order to see the convergence of theapproximate solutions to the exact solutions, seventeeniterations are conducted for each algorithm and criticamoments for the first three modes are computed. It isalso worth noting that the function given in Equation 23cannot satisfy both of the boundary conditionssimultaneously.

It is surprising that for this special case of the beambuckling problem, all iteration algorithms yield exactly thesame results for the normalized buckling moments, aslisted in Table 2, where exact results are also tabulatedVariation of percent errors with iterations plotted in Figure5 shows how VIM solutions converge to the exact

solution. Since all three VIM algorithms give identicaresults in all iterations, only one plot is presented inFigure 5. As shown in Table 2, VIM solutions are inexcellent agreement with the exact results. It is sufficiento perform only five iterations to obtain the exact value ofthe smallest critical moment. For higher mode values, thenumber of iterations has to be increased (Figure 5). Theexcellent match of VIM solutions with the exact resultsverifies that VIM is a powerful technique in predictingbuckling moments of simply supported rectangulabeams with constant rigidities under uniform moment andalso encourages its use in much more complex beam


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-100

-80

-60

-40

-20

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Iteration Number

Error

(%)

Mode 1

Mode 2

Mode 3

Figure 5. Variation of percent errors in VIM iterations for buckling moments of the first three

modes - constant rigidities.

buckling problems, like lateral buckling of beams withvariable rigidities.

Beams with linearly varying rigidities

If both the minor axis flexural and torsional rigidities of thebeam changes in linear form, that is, if

( )

= + 1t t

z

GI z GI b L and ( )

= + 1

z

EI z EI b L

(24)

where b is a constant determining the sharpness of thechanges in rigidities along the length of the beam, then,the buckling equation given in Equation 4 takes thefollowing form:

( )

+ + =

+ +

22 2

220

1 /

o

t

Md b d L

dz L bz dz GI EI bz L(25)

whose exact solution (Wang et al., 2005) is in thefollowing form:

( ) ( ) = +1 2sin ln cos lnC k C k (26)

where, = +1 /bz L and

=2 2

2

2

o

t

M Lk

GI EI b

When the related boundary conditions are used, thecharacteristic equation is obtained as follows:

( )( )+ =sin ln 1 0k b (27)

The smallest root of which yields the first mode criticamoment Mcras

( )

=+ln 1

t

cr

GI EIbM

b L(28)

Exact values for the second and third mode criticamoments can be obtained by determining the larger rootsof the characteristic equation, which leads to four andnine times the first mode value, respectively. Thenondimensional form of Equation 25 can be written as:

( )

+ + =

+ +

2

22

10

1 1

d b d

dz bz dz bz(29)

where = /z z L , = and is the nondimensionacritical moment, as defined in Equation 22.

Since in the case of constant rigidities, all VIM algorithmslead to the same results, the effectiveness of eachalgorithm is investigated further in this problem. For thispurpose, using all VIM algorithms, critical moments fothe first three modes are computed for b=0.3. In order tosee the convergence of the approximate solutions to theexact solutions, thirteen iterations are conducted for eachalgorithm. Similar to the case of constant rigidities, theiterations are started with the linear approximation givenin Equation 23. To simplify the integration processesvariable coefficients in the iteration integrals are


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Table 3. Normalized buckling moments () for the first three modes linearly varying rigidities (b=0.3).


1 12.9043 12.9043 12.9043 12.9043

2 51.6172 51.6172 51.6172 51.6172

3 115.9820 115.9820 115.9820 116.1387

-100

-80

-60-40

-20

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Iteration Number

Error(%)

Mode 1

Mode 2

Mode 3

Figure 6. Variation of percent errors with iterations for the first three buckling loads linearlyvarying rigidities (b=0.3).

expanded in series using nine terms.Similar to the case of constant rigidities, the different

VIM algorithms lead to the same results for bucklingmoments when b=0.3, as listed in Table 3. Variation ofpercent errors with iterations are also plotted in Figure 6to show how VIM solutions converge to the exactsolution. Since all three VIM algorithms give identicalresults, only one plot is presented. As shown in Table 3,the agreement between the VIM results and the exactresults is considerably good.

Even though all VIM algorithms yield exactly the sameresults, some important differences are observedbetween the iteration algorithms in this case as far as the

computation time and space are concerned. VIA II isobserved to complete the same number of iterations inmuch smaller amount of time than VIA I and VIA III, forwhich the computation times are almost identical. VIA II issuperior to VIA I and VIA III also in that, its output filetakes up less space. The sizes of the output files createdby VIA I and VIA III are almost four times larger than thatby VIA II. Thus, it can be concluded that VIA II is moreeffective in solving this kind of differential equations thanthe other two iteration algorithms. Such differences incomputation time and space may occur due to the fact

that VIA I necessitates, in each iteration, second ordedifferentiation of the solution obtained in the previousiteration and VIA III uses, in each iteration, the precedingtwo iteration results. Furthermore, in VIA I and VIA III, theintegral term has to be added to the solution obtained inthe previous iteration whereas it is added to the initiaapproximation in VIA II.

Using VIA II, the normalized buckling moments foother values of bare also computed and plotted in Figure7, where exact results are also shown. As it is seen fromthe figure, VIM solutions and exact results are in verygood agreement. It is worth noting that the smaldifferences between the results as b increases occurs

due to the fact that it is necessary to expand the variablecoefficients in the iteration integrals in series using moreterms when b is close to one. As the number of terms inseries is increased, VIM results are observed to convergeto exact results.

Beams with exponentially varying rigidities

If the minor axis bending and torsional rigidities of thebeam changes in the following exponential form:


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0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1

b

NondimensionalCriticalMoment

()

VIA II

Exact

b VIA II Exact

0 9.8696 9.8696

0.1 10.8648 10.8648

0.3 12.9043 12.9043

0.5 15.0083 15.0083

0.7 17.1744 17.1757

1 20.4462 20.5423

Figure 7. Variation of nondimensional critical moment with b values.

0

2

4

6

8

10

12

0 0.5 1 1.5 2a-values

Nondimensionalcriticalmoment()

a

0 9.8696

0.1 8.9230

0.3 7.2570

0.5 5.8631

0.7 4.7058

1 3.3428

1.2 2.6402

1.5 1.8319

1.7 1.4250

2 0.9671

Figure 8. Variation of nondimensional critical moment with a values.

( ) ( )

=/a z L

t tGI z GI e and ( ) ( )

=

/a z LEI z EI e (30)

where ais a positive constant determining the sharpnessof the changes in rigidities along the length of the beam,then, the nondimensionalized form of the bucklingequation in Equation 4 become

+ =

22

20az

d da e

dz dz (31)

where /z z L= , = and is the nondimensionacritical moment as defined in Equation 22.

Since in the linear case, VIA II is found to be moreeffective than VIA I and VIA III, Equation 31 is solvedusing VIA II for various a values, and the smallesnondimensional critical moments in the first bucklingmodes are obtained. The results are plotted in Figure 8which shows how critical moment of a simply supportedrectangular beam decreases as aincreases.


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Table 4. Normalized buckling loads () for the first three modes constant rigidities.


1 16.1010 16.1010 16.1010 16.1010

2 104.9830 104.9830 104.9830 104.9830

3 272.7750 272.7750 272.7750 272.7750

Even though it can be rather difficult to obtain exactanalytical solutions for this particular case of the beambuckling problem, VIM can effectively be used todetermine buckling moments in any kind of variations inrigidities, such as the exponential variation studied in thissection.

Critical load for cantilever case

Beams with constant rigidities

If the minor axis bending and torsional rigidities of thebeam are constant, then Equation 8 reduces to thefollowing simpler equation:

( )

+ =2

22

220

dL z

dz(32)

Where,

=2

2

2

t

P

GI EI

Introducing a new variable = s L z, Timoshenko andGere (1961) obtained the solution of Equation 32 as

( ) ( ) = + 2 2

1 1/ 4 2 2 1/ 4 2/ 2 / 2s A J s A J s (33)

Where 1/ 4J and 1/ 4J represent Bessel functions of the

first kind of order 1/4 and 1/4, respectively. When therelated boundary conditions are used to determine theconstants A1 and A2, the following characteristic equationis obtained:

( ) =21/ 4 2 / 2 0J L (34)

The smallest root of this equation is =22 / 2 2.0063L

leading to the first mode critical load Pcr

=

24.0126

t

cr

GI EIP

L(35)

observed that the numerical value in Equation 35

changes to 10.2461 and 16.5159 in the second and thirdmode critical load expressions, respectively. Thisbuckling problem is also solved using all three VIMalgorithms. For easier computations, the nondimensionaform of the differential equation and the related boundaryconditions are written:

( )

+ =2

2

21 0

dz

dz(36)

where

=2 4

t

P L

GI EIwith ( ) =0 0 and ( )

=1 0

d

dz

where = /z z L , = and is the nondimensionacritical load. For all three algorithms, the initiaapproximation is chosen as given in Equation 23. Thecoefficients A and B are obtained as defined in purebending case. Again, in order to see the convergence o

the approximate solutions to the exact solutionsseventeen iterations are conducted for each algorithmand critical moments for the first three modes arecomputed.

Similar to the pure bending case, exactly the samesolutions are obtained for critical loads from threedifferent VIM algorithms, as listed in Table 4. Variation ofpercent errors with iterations are also plotted in Figure 9to show how VIM solutions converge to the exactsolution.

As shown in Table 4 and Figure 9, VIM solutions are inexcellent agreement with the exact results. It is to benoted that while it is rather difficult to obtain the critica

load from the characteristic equation given in Equation34, which needs finding the roots of a Bessel function, itis much easier to solve the characteristic equationderived using VIM iterations which are in the form ofpolynomials. This is one of the advantages of using VIMin this problem, even in the case of constant rigidities.

Beams with linearly varying rigidities

If both the minor axis flexural and torsional rigidities of thebeam changes in linear form, that is, if

( )

= 1t tz

GI z GI b Land

( )

= 1

z

EI z EI b L(37)

where b is a positive constant that can take valuesbetween zero and one, then, the buckling equation givenin Equation 8 takes the following form:

( )

( )

+ =

22 2 2

22

1 /0

1 /t

z Ld b d P L

dz L bz dz GI EI bz L(38)

When larger roots of Equation 34 are obtained, it is


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-80

-60

-40

-20

020

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Iteration Number

Error(%)

Mode 1

Mode 2

Mode 3

Figure 9. Variation of percent errors with iterations for the first three buckling loads constantrigidities.

The nondimensional form of Equation 38 can be writtenas:

( )

( )

+ =

22

22

10

1 1

zd b d

dz bz dz bz(39)

where = /z z L , = and is the nondimensional

critical moment, as defined in Equation 36.

Since in the case of constant rigidities, all VIM algorithmslead to the same results, the effectiveness of eachalgorithm on solving Equation 39 is also investigated. Forb=0.5, the first mode critical moment is computed usingeach iteration algorithms separately. The iterations arestarted with the linear approximation given in Equation23. To simplify the integration processes, variablecoefficients in the iteration integrals are expanded inseries using twenty one terms. Similar to the earlierstudied cases, all algorithms lead to the same result.However, as in the pure moment case, VIA II is more

effective than VIA I and VIA III with regard to thecomputation time and space.As both the minor axis flexural rigidity and torsional

rigidity of a rectangular beam is directly proportional tothe height of the beam cross section, a special case oflinear variation in beam rigidities occurs when the heightof the beam is linearly tapered. Linearly tapered steelbeams have wide applications in many engineeringapplications. They lead to economical designs in laterally-braced cantilever beams with end loading, since undersuch a loading, the major axis bending momentdecreases linearly from the fixed end of the beam to the

free end. However, as shown in Figure 10, which isplotted using VIA II, the buckling load of a laterally-unbraced cantilever beam decreases considerably as bin other words, the slope of the tapering, increases.

Beams with exponentially varying rigidities

If the minor axis flexural and torsional rigidities of the

beam changes in exponential form as given in Equation30, the nondimensionalized form of the buckling equationin Equation 8 becomes

( )

+ =2

2 2

21 0az

d da e z

dz dz (40)

where = /z z L , = and is the nondimensionacritical moment, as defined in Equation 36. Since it hasalready been verified that VIA II is the most effectiveiteration algorithm for the buckling equations studied inthis paper, Equation 40 is solved using VIA II for variousvalues of a. The variation of normalized first modebuckling loads with a is shown in Figure 11. As shown inthe figure, the buckling load of a nonuniform cantilevebeam with end loading can drop as low as the quarter oits uniform value when ais equal to 2.

Conclusion

In this paper, two fundamental beam buckling problemslateral torsional buckling of (a) simply supported narrowrectangular beams under uniform moment and (b) narrow


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0

5

10

15

20

0 0.2 0.4 0.6 0.8 1

b-values

Nondimensionalcriticalload

()

b

0 16.1010

0.1 15.1633

0.3 13.2743

0.5 11.3497

0.7 9.3484

1 5.7879

Figure 10. Variation of nondimensional critical load with b values.

0

5

10

15

20

0 0.5 1 1.5 2

a-values

Non

dimensionalcriticalload()

a

0 16.1010

0.1 15.1835

0.3 13.4595

0.5 11.8788

0.7 10.4359

1 8.5168

1.2 7.3917

1.5 5.9186

1.7 5.0695

2 3.9772

Figure 11. Variation of nondimensional critical load with a values.

rectangular cantilever beams carrying concentrated loadat their free ends, are studied using variational iterationmethod (VIM). Exact solutions to these problems areavailable in literature only for beams of constant rigidities

and some particular cases of linearly tapered beams. Inorder to verify the effectiveness of VIM on solving beambuckling equations and to show the application of themethod; firstly, the problems with constant rigidities arestudied. The excellent match of the VIM results with theexact results verifies the efficiency of the technique in theanalysis of lateral torsional buckling problems. Then, thebuckling problems in which the minor axis flexural andtorsional rigidities of the beams vary along their lengthsare studied. Both linear and exponential variations areconsidered in nonuniform beams. For the case of variable

rigidities, the differential equations of the studied bucklingproblems have variable coefficients, which hinder thederivation of exact solutions for these types of problemsHowever, as shown in the paper, it is relatively easy to

write the variational iteration algorithms for thesedifferential equations, which lead to the bucklingload/moment of the beam after a few iterations evenwhen the rigidities of the beam change along its length, inexponential form. In the paper, the effectiveness of threeVIM algorithms, two of which have been proposed veryrecently by He et al. (2010), in solving lateral bucklingequations was also investigated. Analysis results showthat all iteration algorithms yielded exactly the sameresults in all studied problems. As far as the computationtimes and spaces are concerned, however, Variationa


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Iteration Algorithm II (VIA II) is found to be superior thanthe others especially in problems where the beamrigidities vary along the beam length.

REFERENCES

Atay MT, Coskun SB (2009). Elastic stability of Euler columns with acontinuous elastic restraint using variational iteration method.Comput. Math. Appl., 58: 2528-2534.

Chajes A (1974). Principles of structural stability theory. Prentice Hall,Englewood Cliffs. pp. 211-236.

Coskun SB, Atay MT (2009). Determination of critical buckling load forelastic columns of constant and variable cross-sections usingvariational iteration method, Comput. Math. Appl., 58: 2260-2266.

Coskun SB (2010). Analysis of tilt-buckling of Euler columns withvarying flexural stiffness using homotopy perturbation method. Math.Model. Anal., 15(3): 275-286.

He JH (1999). Variational iteration method - a kind of nonlinearanalytical technique: some examples, Int. J. Non Linear Mech.,34(4): 699-708.

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He JH (2000). A review on some new recently developed nonlinearanalytical techniques, Int. J. Nonlinear Sci. Numer. Simul., 1(1): 5170.

He JH (2007). Variational iteration method some recent results andnew interpretations., J. Comput. Appl. Math., 207 (1): 3-17.

He JH, Wazwaz AM, Xu L (2007). The variational iteration methodreliable, efficient and promising, Comput. Math. Appl., 54(7-8): 879880.

He JH, Wu GC, Austin F (2010). The variational iteration method whichshould be followed, Nonlinear Sci. Lett. A, 1(1): 1-30.

Okay F, Atay MT, Coskun SB (2010). Determination of buckling loadsand mode shapes of a heavy vertical column under its own weighusing the variational iteration method, Int. J. Nonlinear Sci. NumerSimul., 11(10): 851-857.

Simitses GJ, Hodges DH (2006). Fundamentals of structural stabilityElsevier. pp. 251-277.

Timoshenko SP, Gere JM (1961). Theory of elastic stability. SecondEdition, McGraw-Hill Book Company, New York.

Wang CM, Wang CY, Reddy JN (2005). Exact solutions for buckling ostructural members. CRC Press, Florida. pp. 77-85.

2011_ lateral torsional buckling of rectangular beams using variational iteration method

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