Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 341063 8 pageshttpdxdoiorg1011552013341063

Research ArticleBuckling of Euler Columns with a Continuous Elastic Restraintvia Homotopy Analysis Method

Aytekin EryJlmaz1 M TarJk Atay2 Safa B CoGkun3 and Musa BaGbuumlk1

1 Department of Mathematics Nevsehir University 50300 Nevsehir Turkey2Department of Mathematics Nigde University 51100 Nigde Turkey3 Department of Civil Engineering Kocaeli University 41380 Kocaeli Turkey

Correspondence should be addressed to Aytekin Eryılmaz eryilmazaytekingmailcom

Received 12 October 2012 Revised 10 December 2012 Accepted 10 December 2012

Academic Editor Fazlollah Soleymani

Copyright copy 2013 Aytekin Eryılmaz et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Homotopy Analysis Method (HAM) is applied to find the critical buckling load of the Euler columns with continuous elasticrestraints HAM has been successfully applied to many linear and nonlinear ordinary and partial differential equations integralequations and difference equations In this study we presented the application of HAM to the critical buckling loads for Eulercolumns with five different support cases continuous elastic restraints The results are compared with the analytic solutions

1 Introduction

The research area of buckling of nonuniform columns hasbeen one of the important topics of extensive studies basedon the reality that is closely related to the fields of structuralmechanical and aeronautical engineering Determinationof practical load carrying capacity of a structural memberrequires a detailed stability analysis in theoretical and com-putational manner Columns are one of the most used basicstructural elements and there are extensive studies relatedto the elastic stability of columns with different propertiesin shape and of material and to their static and dynamicbehaviors Many types of structures and structural memberscan be defined as a uniform andor non-uniform column ina simplified state with different end conditions for bucklinganalysis On the other hand it is difficult to determine theexact analytical solutions for these buckling problems ofvarious column types with arbitrary distributions of flexuralstiffness and various end conditions Conducting research onbuckling of columns has become the center point of studyfor many researchers and studying this subject becomesmore and more systematic during the last decades As astarting point of this line of research topic Eulerrsquos early study

of buckling of columns under their own weight [1] can becounted Afterwards Greenhill [2] made remarkable contri-butions to this field In this field of study generally the closedform solutions are extremely hard to establish Howeversolutions for simple cases are found by Dinnik [3] Karmanand Biot [4] Timoshenko and Gere [5] and others Wang etal [6] established exact solutions for buckling of structuralmembers including various cases of columns beams archesrings plates and shells In addition to this line of research thecolumns with variable cross-section some exact solutions aregiven in terms of logarithmic and trigonometric functions byBleich [7] in terms of Bessel functions by Dinnik [8] and interms of Lommel functions by Elishakoff and Pellegrini [9ndash11] Exact solution by series representation for buckling loadfor variable cross section columns with variable axial forceswas established by Eisenberger [12] Exact buckling solutionsfor several special types of tapered columns with simpleboundary conditions were given by Gere and Carter [13] withBessel functions Moreover solutions for a problem of thebuckling of elastic columns with step varying thicknesses aregiven by Arbabi and Li [14] Siginer [15] conducted researchon the stability of a column whose flexural rigidity has a con-tinuous linear variation along the column Furthermore the

2 Journal of Applied Mathematics

x

P

L

w

Elasticrestraints

Figure 1 Column with continuous elastic restraints

exact analytical solutions of a one-step bar and multistep barwith varying cross section under the action of concentratedand variably distributed axial loads were obtained by Li etal [16ndash18] Sampaio and Hundhausen [19] gave the solutionfor the problem of buckling behavior of inclined beamcolumn using energy method They formulated the exactsolution using generalized hypergeometric functions More-over the researchers who studied the mechanical behaviorof beamscolumns can be given as Keller [20] Tadjbakhshand Keller [21] and Taylor [22] A number of researches onthis topic have been made by Atay and Coskun to investigatethe elastic stability of a homogenous and nonhomogenousEuler beam by using variational iteration method and homo-topy perturbation method [23ndash29] The problem of stabilityanalysis of non-uniform rectangular beams such as lateraltorsional buckling of rectangular beams was solved by usinghomotopy perturbation method by Pinarbasi [30] By trans-forming the governing equation with varying coefficientsto linear algebraic equations and also by using various endboundary conditions critical buckling loads of beams witharbitrarily axial inhomogeneity are solved by Huang and Luo[31] Recently Yuan and Wang [32] used a new differentialquadrature based iterative numerical integration method tosolve postbuckling differential equations of extensible beamcolumns with six different cases

Liao [33] introducedHomotopyAnalysisMethod (HAM)to obtain series solutions of various linear and nonlinearproblems HAM is an efficient method that presents usacceptable analytical results with convenient convergence[33] In opposition to the perturbation techniques thisapproach is independent of any small parameters and HAMprovides us with a simple procedure to obtain the conver-gence of series of solutions so that one can obtain accurateenough approximations by auxiliary convergence controllerparameter ℎ Liao solvedmany linear and nonlinear problemsby HAM In his book especially he points out the basic ideasof theHAM [33ndash36] Recently this technique has successfully

been applied to several nonlinear problems such as the vis-cous flows of non-Newtonian fluids [37 38] nonlinear heattransfer [39] nonlinear Fredholm integral equations [40]the KdV-type equations [41] differential difference equations[42] time-dependent Emden-Fowler type equations [43]Laplace equation with Dirichlet and Neumann conditions[44] and multipantograph equations [45]

In this study we apply Homotopy Analysis Method(HAM) to find the critical buckling load of elastic columnswith continuous restraints This problem has been solved bydifferent approaches such as Variational Iteration Method(VIM) but Ham has some advantages such as being basedon a generalized concept of the homotopy in topology theHAM has the following advantagesThe HAM is always validno matter whether there exist small physical parameters ornot the method provides a convenient way to guarantee theconvergence of approximation series and also the methodprovides great freedom to choose the equation type of linearsubproblems and the base functions of solutions As a resultthe HAM overcomes the restrictions of all other analyticapproximation methods mentioned above and is valid forhighly nonlinear problems [33]

2 Buckling of Elastic Columns withContinuous Restraints

A uniform homogeneous column which is continuouslyrestrained along its length with flexural rigidity119864119868 and length119871 is investigated The restraint consists of lateral springs ofstiffness 119896 per unit length

Governing equation for the buckling of an Euler columnwith continuous elastic restraints in Figure 1 is given by

1198892

1198891199092(119864119868

1198892119908

1198891199092) + 119875

1198892119908

1198891199092+ 119896119908 = 0 (1)

If the governing equation (1) is divided by 119864119868 then it isnormalized by defining nondimensional displacement andlength 119908 = 119908119871 119909 = 119909119871 respectively Then normalizedgoverning equation becomes

1198894119908

1198891199094+ 120572

1198892119908

1198891199092+ 120573119908 = 0 where 120572 =

1198751198712

119864119868and 120573 =

1198961198714

119864119868

(2)

Investigation of buckling loads for continuously restrainedelastic columns with five different support cases will beconducted via HAM throughout the study

The general solution of governing equation and stabilitycriteria for the columns with different end conditions aregiven in Wang et al [6] These end conditions can be seenin Figure 2

The stability criteria for the columns considered in thisstudy [6] are given in Table 1

Journal of Applied Mathematics 3

Table 1 The stability criteria for continuously restrained elastic columns

Column Stability criteriaC-F column [120572 (119878

2+ 1198792) minus 2119878

21198792] cos119879 cos 119878 minus 120572(119878

2+ 1198792) + (119878

4+ 1198794) + 119878119879[2120572 minus (119878

2+ 1198792)] sin119879 sin 119878 = 0

P-P column sin119879 = 0

C-P column 119879 cos119879 sin 119878 minus 119878 sin119879 cos 119878 = 0

C-C column 2119878119879[cos119879 cos 119878 minus 1] + (1198782+ 1198792) sin119879 sin 119878 = 0

C-S column 119879 sin119879 cos 119878 minus 119878 cos119879 sin 119878 = 0

C-Fcolumn

P-Pcolumn

C-Pcolumn

C-Ccolumn

C-Scolumn

Figure 2 Various end conditions for restrained columns

Where

119878 = radic 120572

2minus radic(

120572

2)

2

minus 120573 (3)

119879 = radic120572

2+ radic(

120572

2)

2

minus 120573 (4)

The classical boundary conditions [6] are as follows

Clamped (Fixed)End 119908 = 0119889119908

119889119909= 0

Pinned End 119908 = 01198892119908

1198891199092= 0

Free End 1198892119908

1198891199092= 0

1198893119908

1198891199093+ 120572

119889119908

119889119909= 0

Sliding Restraint 119889119908

119889119909= 0

1198893119908

1198891199093+ 120572

119889119908

119889119909= 0

(5)

3 HAM Formulation of the Problem

To solve (3) by means of homotopy analysis method let usdefine119873[120601(119909 119902)] as follows

(1 minus 119902) 119871 [120601 (119909 119902) minus 1199080(119909)] = 119902ℎ119867 (119909)119873 [120601 (119909 119902)] (6)

where 120601(119909 119902) is an unknown function to be determined and119873[120601(119909 119902)] is given by

119873[120601 (119909 119902)] = 120601(120484119907)

(119909 119902) + 12057212060110158401015840(119909 119902) + 120573120601 (119909 119902) (7)

The high-order deformation equation is as follows

119908119898(119909) = 120594

119898119908119898minus1

(119909)

+ ℎint

119909

0

int

120591

0

int

120577

0

int

120595

0

[119908(120484119907)

119898minus1(120585) + 120572119908

10158401015840

119898minus1(120585)

+120573119908119898minus1

(120585) ] 119889120585119889120595119889120577119889120591

(8)

where

120594119898=

0 119898 le 1

1 119898 gt 1(9)

Starting with 1199080(119909) successively 119908

119894(119909) 119894 = 1 2 3 are

determined by the so-called high-order deformation equa-tion (8) then the solution is

119908 (119909) = 1199080(119909) +

infin

sum

119898=1

119908119898(119909) (10)

4 Critical Buckling Loads for ContinuouslyRestrained Elastic Columns

A cubic polynomial 1199080(119909) = 119886119909

3+ 1198871199092+ 119888119909 + 119889 is chosen

as an initial approximation due to four boundary conditionsfor each case This polynomial has been successfully used inprevious studies employing different analytical approximatetechniques This polynomial has also been used as the inter-polation function in the finite element analysis of Euler beamsor columns Hence the initial approximation is expectedto produce good results The approximation includes fourunknown coefficients which will be found by substitutingfour boundary conditions into the solution We successively

4 Journal of Applied Mathematics

obtain 119908119894(119909) 119894 = 1 2 3 by the 119898th-order deformation

equation (3)

1199081(119909) =

1

12119887ℎ1205721199094+

1

20119886ℎ1205721199095+

1

24119889ℎ1205731199094+

1

120119888ℎ1205731199095

+1

360119887ℎ1205731199096+

1

840119886ℎ1205731199097

1199082(119909) =

1

12119887ℎ1205721199094+

1

12119887ℎ21205721199094+

1

20119886ℎ1205721199095+

1

20119886ℎ21205721199095

+1

360119887ℎ212057221199096+

1

840119886ℎ212057221199097+

1

24119889ℎ1205731199094

+1

24119889ℎ21205731199094+

1

120119888ℎ1205731199095+

1

120119888ℎ21205731199095

+1

360119887ℎ1205731199096+

1

360119887ℎ21205731199096+

1

840119886ℎ1205731199097

+1

840119886ℎ21205731199097+

1

720119889ℎ21205721205731199096+

1

5040119888ℎ21205721205731199097

+1

10080119887ℎ21205721205731199098+

1

30240119886ℎ21205721205731199099

+1

40320119889ℎ212057321199098+

1

362880119888ℎ212057321199099

+1

1814400119887ℎ2120573211990910

+1

6652800119886ℎ2120573211990911

(11)

Ten iterations are conducted In this way we get the finalapproximation as follows

11988210

(119909 ℎ) =

10

sum

119899=0

119908119899(119909)

= 1199080(119909) + 119908

1(119909) + 119908

2(119909) + sdot sdot sdot + 119908

10(119909)

(12)

By substituting (12) into the boundary conditions weobtained four homogeneous equations By representing theseequations in the matrix form by coefficient matrix [119862 (120572 120573)]we obtained the following equation in matrix form

[119862 (120572 120573)][[[

[

119886

119887

119888

119889

]]]

]

=[[[

[

0

0

0

0

]]]

]

(13)

where a b c and 119889 are the unknown constants which hasbeen introduced in the initial approximation For a nontrivialsolution the determinant of the coefficient matrix [119862(120572 120573)]

must vanish Then the problem takes the following form

Det [119862 (120572 120573)] = 0 (14)

The smallest positive real root of (14) is the normalized criti-cal buckling loadThenext positive real root is the normalizedbuckling load for second mode and so on Equation (14)

0

0

10

20

30

ħ

minus2 minus15 minus1 minus05

minus10

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 3 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-S

column

depends on the stability parameter 120572 the restraint stiffnessparameter 120573 and the convergence control parameter ℎ Thenwe define the function 119880(120572 120573 ℎ) as follows

119880 (120572 120573 ℎ) = Det [119862 (120572 120573)] (15)

And then we plot the ℎ curves of the 1198801015840

10(120572 120573 ℎ) and

11988010158401015840

10(120572 120573 ℎ) in order to find convergence region of the ℎ

where prime denotes derivatives of 119880(120572 120573 ℎ) with respect to120572

41 Clamped-Free (C-F) Column Substituting the 10th-orderapproximation 119882

10(120572 120573 ℎ) into the boundary conditions of

C-F column we get coefficientmatrix [119862119862-119865(120572 120573)]We define

the function 119880(120572 120573 ℎ) as follows

119880(120572 120573 ℎ) = Det [119862119862-119865 (120572 120573)] (16)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 3 and the valid region of ℎ is approximatedas minus1 5 lt ℎ lt minus0 3

Finally the critical buckling load obtained from (14) forℎ = minus0 99 is 24674

42 Pinned-Pinned (P-P) Column Substituting the 10th-order approximation 119882

10(120572 120573 ℎ) into the boundary condi-

tions of P-P column we get coefficient matrix [119862119875-119875(120572 120573)]

We define the function 119880(120572 120573 ℎ) as follows

119880 (120572 120573 ℎ) = Det [119862119875-119875 (120572 120573)] (17)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 4 and the valid region of ℎ is about minus1 6 lt ℎ lt

minus0 4Finally the critical buckling load obtained from (14) for

ℎ = minus0 995 is 98696

Journal of Applied Mathematics 5

0 05

ħ

minus2

minus2

minus4

minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

0

2

4

6

8

10

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 4 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for P-P

column

0

ħ

minus2minus5

minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

0

5

10

15

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 5 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-P

column

43 Clamped-Pinned (C-P) Column Substituting the 10thorder approximation 119882

10(120572 120573 ℎ) into the boundary condi-

tions of C-P column we get coefficient matrix [119862119862-119875(120572 120573)]

We define the function 119880(120572 120573 ℎ) as follows

119880(120572 120573 ℎ) = Det [119862119862-119875 (120572 120573)] (18)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 5 and the valid region of ℎ is as follows minus1 75 lt

ℎ lt minus0 3Finally the critical buckling load obtained from (14) for

ℎ = minus0 97 is 201907

44 Clamped-Clamped (C-C) Column Substituting the 10th-order approximation 119882

10(120572 120573 ℎ) into the boundary condi-

tions of C-C column we get coefficient matrix [119862119862-119862(120572 120573)]

We define the function 119880(120572 120573 ℎ) as follows

119880 (120572 120573 ℎ) = Det [119862119862-119862 (120572 120573)] (19)

0

5

10

15

0 05

ħ

minus2minus5

minus25 minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 6 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-C

column

0

10

20

30

0

ħ

minus2 minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

minus10

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 7 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-S

column

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 6 and the valid region of ℎ can be minus1 8 lt ℎ lt minus0 1Finally the critical buckling load obtained from (14) for

ℎ = minus0 96868 is 394784

45 Clamped-Sliding Restraint (C-S) Column Substitutingthe 10th-order approximation 119882

10(120572 120573 ℎ) into the bound-

ary conditions of C-S column we get coefficient matrix[119862119862-119878(120572 120573)] We define the function 119880(120572 120573 ℎ) as follows

119880(120572 120573 ℎ) = Det [119862119862-119878 (120572 120573)] (20)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 7 and the valid region of ℎ can be minus1 55 lt ℎ lt

minus0 37Finally the critical buckling load obtained from (14) for

ℎ = minus0 9909 is 98696The exact solutions for the presented cases are ob-

tained via stability criteria provided by Wang et al [6] In

6 Journal of Applied Mathematics

Table 2 Comparison of exact and HAM solutions of the buckling loads for 120573 = 0 with ten iterations

Mode C-F column P-P column C-P column C-C column C-S columnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 24674 24674 98696 98696 201907 201907 394784 394784 98696 986962 222066 222066 394784 394779 596795 596813 807629 809336 394784 394784

Table 3 Comparison of exact and HAM solutions of the buckling loads for 120573 = 50 with ten iterations

Mode C-F Column P-P Column C-P Column C-C Column C-S ColumnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 88614 88614 149357 149357 242852 242852 432606 432606 235717 2357172 330879 330873 407449 407442 610966 611003 817943 820059 444104 444053

Table 4 Comparison of exact and HAM solutions of the buckling loads for 120573 = 100 with ten iterations

Mode C-F column P-P column C-P column C-C column C-S columnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 119964 119964 200017 200017 283066 283066 470066 470066 326690 3266902 452659 452574 420114 420105 625613 625678 828246 830859 528965 528105

Table 5 Absolute errors of HAM approximations for the first buckling mode

C-F column P-P column C-P column C-C column C-S column120573 = 0 130562119864 minus 13 103727119864 minus 12 000000 000000 539109119864 minus 11

120573 = 50 981515119864 minus 10 126009119864 minus 10 109165119864 minus 7 464191119864 minus 5 150249119864 minus 7

120573 = 100 304503119864 minus 8 165444119864 minus 8 737428119864 minus 7 59300119864 minus 7 181936119864 minus 5

Table 6 Percent relative errors of HAM approximations for the second buckling mode

C-F column P-P column C-P column C-C column C-S column120573 = 0 000000621 000124745 000303663 021128177 000008617120573 = 50 000163762 000173402 000599296 025870613 001135007120573 = 100 001874692 000227055 001038389 031551889 016247717

0

5

10

15

20

25

0 2 4 6 8 10

Nor

mal

ized

buc

klin

g lo

ad

Iteration number

Convergence for mode 1

β = 100

β = 50

β = 0

Figure 8 Convergence of solutions through the iterations for thefirst mode of P-P column

the following tables the buckling loads for the first twomodesare given and compared with exact results

From Tables 2ndash4 it is observed that normalized bucklingloads for the first mode which is the critical buckling load arein excellent agreement with analytical results Absolute errorsfor these cases are given in Table 5

The differences in the results of the second mode existdue to lack of iterations provided by the method Additionaliterations would improve the results for the second modeHowever the results for the second mode are still in goodagreement with the exact results Relative errors for this caseare provided in the following Table 6

These percent relative errors show that the presented solu-tion is in good agreement with analytical ones Convergenceof solutions for the PP column is simulated in Figures 8 and 9Solid lines show the exact solutions for different normalizedspring stiffnesses

Same convergence behaviors are observed for all casesconsidered in this study Furthermore Figure 8 shows that atleast 6 iterations are required for the first mode and Figure 9shows that at least 10 iterations are required for the secondmode to obtain satisfactory results from the analysis

5 Conclusions

In this work a reliable algorithm based on the HAM toobtain the normalized buckling loads of the Euler columns

Journal of Applied Mathematics 7

β = 100

β = 50

β = 0

36

38

40

42

44

0 2 4 6 8 10

Nor

mal

ized

buc

klin

g lo

ad

Iteration number

Convergence for mode 2

Figure 9 Convergence of solutions through the iterations for thesecond mode of P-P column

with constant flexural stiffness is presented Several cases aregiven to illustrate the validity and accuracy of this procedureThe series solutions of (3) by HAM contain the auxiliaryparameter ℎ In general by means of the so-called ℎ-curveit is straightforward to choose a proper value of ℎ whichensures that the series solution is convergent Figures 2 34 5 and 6 show the ℎ-curves obtained from the 119898th-orderHAM approximation solutions From these figures the validregions of ℎ correspond to the line segments nearly parallelto the horizontal axis By the use of the proper value withthis parameter two buckling loads for the first and secondmodes are obtained The method will provide the results forthe following modes if additional iterations are introducedin the analysis The buckling loads are positive real roots ofthe characteristic equationswhich are obtained consecutivelyThis is a huge advantage because it is still very difficult toobtain those roots consecutively even with a mathematicssoftware As a result HAM is an efficient powerful andaccurate tool for determining the buckling loads of Eulercolumns

References

[1] L Euler ldquoDie altitudine colomnarum sub proprio pondere cor-ruentiumrdquo Acta Academiae Scientiarum Imperialis Petropoli-tanae 1778 in Latin

[2] A G Greenhill ldquoDetermination of the greatest height consis-tent with stability that a vertical pole on mast can be made andof the greatest height to which a tree of given proportions cangrowrdquo in Proceedings of the Cambridge Philosophical Society IV1883

[3] A N Dinnik ldquoDesign of columns of varying cross-sectionrdquoTransactions of the ASME vol 51 1929

[4] T R Karman and M A Biot Mathematical Methods in Engi-neering McGraw Hill New York NY USA 1940

[5] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw Hill New York NY USA 1961

[6] C M Wang C Y Wang and J N Reddy Exact Solutions forBuckling of Structural Members CRC Press LLC Florida FlaUSA 2005

[7] H Bleich Buckling Strength of Metal Structures McGraw-HillNew York NY USA 1952

[8] A N Dinnik ldquoDesign of columns of varying cross-sectionrdquoTransactions of the ASME Applied Mechanics vol 51 1932

[9] I Elishakoff and F Pellegrini ldquoExact and effective approximatesolution solutions of some divergent type non-conservativeproblemsrdquo Journal of Sound and Vibration vol 114 pp 144ndash1481987

[10] I Elishakoff and F Pellegrini ldquoApplication of bessel and lommelfunctions and the undetermined multiplier galerkin methodversion for instability of non-uniform columnrdquo Journal ofSound and Vibration vol 115 no 1 pp 182ndash186 1987

[11] I Elishakoff and F Pellegrini ldquoExact solutions for buckling ofsome divergence-type nonconservative systems in terms of bes-sel and lommel functionsrdquo Computer Methods in AppliedMechanics and Engineering vol 66 no 1 pp 107ndash119 1988

[12] M Eisenberger ldquoBuckling loads for variable cross-sectionmembers with variable axial forcesrdquo International Journal ofSolids and Structures vol 27 no 2 pp 135ndash143 1991

[13] J M Gere andW O Carter ldquoCritical buckling loads for taperedcolumnsrdquo Journal of Structural Engineering ASCE vol 88 no 1pp 1ndash11 1962

[14] F Arbabi and F Li ldquoBuckling of variable cross-section columnsintegral equation approachrdquo Journal of Structural EngineeringASCE vol 117 no 8 1991

[15] A Siginer ldquoBuckling of columns of variable flexural rigidityrdquoJournal of Engineering Mechanics ASCE vol 118 no 3 pp 543ndash640 1992

[16] Q Li H Cao and G Li ldquoStability analysis of a bar with multi-segments of varying cross-sectionrdquo Computers and Structuresvol 53 no 5 pp 1085ndash1089 1994

[17] L Qiusheng CHong and L Guiqing ldquoStability analysis of barswith varying cross-sectionrdquo International Journal of Solids andStructures vol 32 no 21 pp 3217ndash3228 1995

[18] Q Li H Cao and G Li ldquoStatic and dynamic analysis of straightbars with variable cross-sectionrdquoComputers and Structures vol59 no 6 pp 1185ndash1191 1996

[19] J H B Sampaio and J R Hundhausen ldquoA mathematical modeland analytical solution for buckling of inclined beam-columnsrdquoApplied Mathematical Modelling vol 22 no 6 pp 405ndash4211998

[20] J B Keller ldquoThe shape of the strongest columnrdquo Archive forRationalMechanics and Analysis vol 5 no 1 pp 275ndash285 1960

[21] I Tadjbakhsh and J B Keller ldquoStrongest columns and isoperi-metric inequalities for eigenvaluesrdquo Journal of Applied Mechan-ics ASME vol 29 pp 159ndash164 1962

[22] J E Taylor ldquoThe strongest columnmdashan energy approachrdquoJournal of Applied Mechanics ASME vol 34 pp 486ndash487 1967

[23] M T Atay and S B Coskun ldquoElastic stability of Euler columnswith a continuous elastic restraint using variational iterationmethodrdquoComputers andMathematics with Applications vol 58no 11-12 pp 2528ndash2534 2009

[24] S B Coskun andM T Atay ldquoDetermination of critical bucklingload for elastic columns of constant and variable cross-sectionsusing variational iteration methodrdquo Computers and Mathemat-ics with Applications vol 58 no 11-12 pp 2260ndash2266 2009

[25] M T Atay ldquoDetermination of buckling loads of tilted buckledcolumn with varying flexural rigidity using variational itera-tion methodrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 11 no 2 pp 97ndash103 2010

[26] F Okay M T Atay and S B Cockun ldquoDetermination of buck-ling loads and mode shapes of a heavy vertical column underits own weight using the variational iteration methodrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 11 no 10 pp 851ndash857 2010

8 Journal of Applied Mathematics

[27] S B Coskun ldquoDetermination of critical buckling loads for eulercolumns of variable flexural stiffness with a continuous elasticrestraint using homotopy perturbation methodrdquo InternationalJournal of Nonlinear Sciences and Numerical Simulation vol 10no 2 pp 191ndash197 2009

[28] MTAtay ldquoDetermination of critical buckling loads for variablestiffness euler columns using homotopy perturbation methodrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 10 no 2 pp 199ndash206 2009

[29] S B Coskun ldquoAnalysis of tilt-buckling of euler columns withvarying flexural stiffness using homotopy perturbation meth-odrdquoMathematicalModelling andAnalysis vol 15 no 3 pp 275ndash286 2010

[30] S Pinarbasi ldquoStability analysis of non-uniform rectangularbeams using homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2012 Article ID 197483 18 pages2012

[31] YHuang andQ Z Luo ldquoA simplemethod to determine the crit-ical buckling loads for axially inhomogeneous beams with elas-tic restraintrdquoComputers andMathematics withApplications vol61 no 9 pp 2510ndash2517 2011

[32] Z X Yuan and XWWang ldquoBuckling and post-buckling analy-sis of extensible beam-columns by using the differential quadra-ture methodrdquo Computers ampMathematics with Applications vol62 no 12 pp 4499ndash4513 2011

[33] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method CRC Press 2004

[34] S J Liao ldquoHomotopy analysis method a new analytic methodfor nonlinear problemsrdquo Applied Mathematics and Mechanicsvol 19 no 10 pp 957ndash962 1998

[35] S J Liao ldquoA uniformly valid analytic solution of two-dimen-sional viscous flow over a semi-infinite flat platerdquo Journal ofFluid Mechanics vol 385 pp 101ndash128 1999

[36] S J Liao ldquoAn explicit analytic solution to the Thomas-Fermiequationrdquo Applied Mathematics and Computation vol 144 no2-3 pp 495ndash506 2003

[37] T Hayat T Javed and M Sajid ldquoAnalytic solution for rotatingflow and heat transfer analysis of a third-grade fluidrdquo ActaMechanica vol 191 no 3-4 pp 219ndash229 2007

[38] T Hayat S B Khan M Sajid and S Asghar ldquoRotating flow of athird grade fluid in a porous space with Hall currentrdquoNonlinearDynamics vol 49 no 1-2 pp 83ndash91 2007

[39] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007

[40] S Abbasbandy and E Shivanian ldquoA new analytical technique tosolve Fredholmrsquos integral equationsrdquoNumerical Algorithms vol56 no 1 pp 27ndash43 2011

[41] S Abbasbandy and F S Zakaria ldquoSoliton solutions for the fifth-order KdV equation with the homotopy analysis methodrdquoNon-linear Dynamics vol 51 no 1-2 pp 83ndash87 2008

[42] Z Wang L Zou and H Zhang ldquoApplying homotopy analysismethod for solving differential-difference equationrdquo PhysicsLetters A vol 369 no 1-2 pp 77ndash84 2007

[43] A S Bataineh M S M Noorani and I Hashim ldquoSolutions oftime-dependent Emden-Fowler type equations by homotopyanalysis methodrdquo Physics Letters A vol 371 no 1-2 pp 72ndash822007

[44] M Inc ldquoOn exact solution of Laplace equation with Dirichletand Neumann boundary conditions by the homotopy analysismethodrdquo Physics Letters A vol 365 no 5-6 pp 412ndash415 2007

[45] F Awawdeh A Adawi and S Al-Shara ldquoAnalytic solution ofmulti pantograph equationrdquo Journal of Applied Mathematicsand Decision Sciences vol 2008 Article ID 605064 11 pages2008

Submit your manuscripts athttpwwwhindawicom

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

x

P

L

w

Elasticrestraints

Figure 1 Column with continuous elastic restraints

exact analytical solutions of a one-step bar and multistep barwith varying cross section under the action of concentratedand variably distributed axial loads were obtained by Li etal [16ndash18] Sampaio and Hundhausen [19] gave the solutionfor the problem of buckling behavior of inclined beamcolumn using energy method They formulated the exactsolution using generalized hypergeometric functions More-over the researchers who studied the mechanical behaviorof beamscolumns can be given as Keller [20] Tadjbakhshand Keller [21] and Taylor [22] A number of researches onthis topic have been made by Atay and Coskun to investigatethe elastic stability of a homogenous and nonhomogenousEuler beam by using variational iteration method and homo-topy perturbation method [23ndash29] The problem of stabilityanalysis of non-uniform rectangular beams such as lateraltorsional buckling of rectangular beams was solved by usinghomotopy perturbation method by Pinarbasi [30] By trans-forming the governing equation with varying coefficientsto linear algebraic equations and also by using various endboundary conditions critical buckling loads of beams witharbitrarily axial inhomogeneity are solved by Huang and Luo[31] Recently Yuan and Wang [32] used a new differentialquadrature based iterative numerical integration method tosolve postbuckling differential equations of extensible beamcolumns with six different cases

Liao [33] introducedHomotopyAnalysisMethod (HAM)to obtain series solutions of various linear and nonlinearproblems HAM is an efficient method that presents usacceptable analytical results with convenient convergence[33] In opposition to the perturbation techniques thisapproach is independent of any small parameters and HAMprovides us with a simple procedure to obtain the conver-gence of series of solutions so that one can obtain accurateenough approximations by auxiliary convergence controllerparameter ℎ Liao solvedmany linear and nonlinear problemsby HAM In his book especially he points out the basic ideasof theHAM [33ndash36] Recently this technique has successfully

been applied to several nonlinear problems such as the vis-cous flows of non-Newtonian fluids [37 38] nonlinear heattransfer [39] nonlinear Fredholm integral equations [40]the KdV-type equations [41] differential difference equations[42] time-dependent Emden-Fowler type equations [43]Laplace equation with Dirichlet and Neumann conditions[44] and multipantograph equations [45]

In this study we apply Homotopy Analysis Method(HAM) to find the critical buckling load of elastic columnswith continuous restraints This problem has been solved bydifferent approaches such as Variational Iteration Method(VIM) but Ham has some advantages such as being basedon a generalized concept of the homotopy in topology theHAM has the following advantagesThe HAM is always validno matter whether there exist small physical parameters ornot the method provides a convenient way to guarantee theconvergence of approximation series and also the methodprovides great freedom to choose the equation type of linearsubproblems and the base functions of solutions As a resultthe HAM overcomes the restrictions of all other analyticapproximation methods mentioned above and is valid forhighly nonlinear problems [33]

2 Buckling of Elastic Columns withContinuous Restraints

A uniform homogeneous column which is continuouslyrestrained along its length with flexural rigidity119864119868 and length119871 is investigated The restraint consists of lateral springs ofstiffness 119896 per unit length

Governing equation for the buckling of an Euler columnwith continuous elastic restraints in Figure 1 is given by

1198892

1198891199092(119864119868

1198892119908

1198891199092) + 119875

1198892119908

1198891199092+ 119896119908 = 0 (1)

If the governing equation (1) is divided by 119864119868 then it isnormalized by defining nondimensional displacement andlength 119908 = 119908119871 119909 = 119909119871 respectively Then normalizedgoverning equation becomes

1198894119908

1198891199094+ 120572

1198892119908

1198891199092+ 120573119908 = 0 where 120572 =

1198751198712

119864119868and 120573 =

1198961198714

119864119868

(2)

Investigation of buckling loads for continuously restrainedelastic columns with five different support cases will beconducted via HAM throughout the study

The general solution of governing equation and stabilitycriteria for the columns with different end conditions aregiven in Wang et al [6] These end conditions can be seenin Figure 2

The stability criteria for the columns considered in thisstudy [6] are given in Table 1

Journal of Applied Mathematics 3

Table 1 The stability criteria for continuously restrained elastic columns

Column Stability criteriaC-F column [120572 (119878

2+ 1198792) minus 2119878

21198792] cos119879 cos 119878 minus 120572(119878

2+ 1198792) + (119878

4+ 1198794) + 119878119879[2120572 minus (119878

2+ 1198792)] sin119879 sin 119878 = 0

P-P column sin119879 = 0

C-P column 119879 cos119879 sin 119878 minus 119878 sin119879 cos 119878 = 0

C-C column 2119878119879[cos119879 cos 119878 minus 1] + (1198782+ 1198792) sin119879 sin 119878 = 0

C-S column 119879 sin119879 cos 119878 minus 119878 cos119879 sin 119878 = 0

C-Fcolumn

P-Pcolumn

C-Pcolumn

C-Ccolumn

C-Scolumn

Figure 2 Various end conditions for restrained columns

Where

119878 = radic 120572

2minus radic(

120572

2)

2

minus 120573 (3)

119879 = radic120572

2+ radic(

120572

2)

2

minus 120573 (4)

The classical boundary conditions [6] are as follows

Clamped (Fixed)End 119908 = 0119889119908

119889119909= 0

Pinned End 119908 = 01198892119908

1198891199092= 0

Free End 1198892119908

1198891199092= 0

1198893119908

1198891199093+ 120572

119889119908

119889119909= 0

Sliding Restraint 119889119908

119889119909= 0

1198893119908

1198891199093+ 120572

119889119908

119889119909= 0

(5)

3 HAM Formulation of the Problem

To solve (3) by means of homotopy analysis method let usdefine119873[120601(119909 119902)] as follows

(1 minus 119902) 119871 [120601 (119909 119902) minus 1199080(119909)] = 119902ℎ119867 (119909)119873 [120601 (119909 119902)] (6)

where 120601(119909 119902) is an unknown function to be determined and119873[120601(119909 119902)] is given by

119873[120601 (119909 119902)] = 120601(120484119907)

(119909 119902) + 12057212060110158401015840(119909 119902) + 120573120601 (119909 119902) (7)

The high-order deformation equation is as follows

119908119898(119909) = 120594

119898119908119898minus1

(119909)

+ ℎint

119909

0

int

120591

0

int

120577

0

int

120595

0

[119908(120484119907)

119898minus1(120585) + 120572119908

10158401015840

119898minus1(120585)

+120573119908119898minus1

(120585) ] 119889120585119889120595119889120577119889120591

(8)

where

120594119898=

0 119898 le 1

1 119898 gt 1(9)

Starting with 1199080(119909) successively 119908

119894(119909) 119894 = 1 2 3 are

determined by the so-called high-order deformation equa-tion (8) then the solution is

119908 (119909) = 1199080(119909) +

infin

sum

119898=1

119908119898(119909) (10)

4 Critical Buckling Loads for ContinuouslyRestrained Elastic Columns

A cubic polynomial 1199080(119909) = 119886119909

3+ 1198871199092+ 119888119909 + 119889 is chosen

as an initial approximation due to four boundary conditionsfor each case This polynomial has been successfully used inprevious studies employing different analytical approximatetechniques This polynomial has also been used as the inter-polation function in the finite element analysis of Euler beamsor columns Hence the initial approximation is expectedto produce good results The approximation includes fourunknown coefficients which will be found by substitutingfour boundary conditions into the solution We successively

4 Journal of Applied Mathematics

obtain 119908119894(119909) 119894 = 1 2 3 by the 119898th-order deformation

equation (3)

1199081(119909) =

1

12119887ℎ1205721199094+

1

20119886ℎ1205721199095+

1

24119889ℎ1205731199094+

1

120119888ℎ1205731199095

+1

360119887ℎ1205731199096+

1

840119886ℎ1205731199097

1199082(119909) =

1

12119887ℎ1205721199094+

1

12119887ℎ21205721199094+

1

20119886ℎ1205721199095+

1

20119886ℎ21205721199095

+1

360119887ℎ212057221199096+

1

840119886ℎ212057221199097+

1

24119889ℎ1205731199094

+1

24119889ℎ21205731199094+

1

120119888ℎ1205731199095+

1

120119888ℎ21205731199095

+1

360119887ℎ1205731199096+

1

360119887ℎ21205731199096+

1

840119886ℎ1205731199097

+1

840119886ℎ21205731199097+

1

720119889ℎ21205721205731199096+

1

5040119888ℎ21205721205731199097

+1

10080119887ℎ21205721205731199098+

1

30240119886ℎ21205721205731199099

+1

40320119889ℎ212057321199098+

1

362880119888ℎ212057321199099

+1

1814400119887ℎ2120573211990910

+1

6652800119886ℎ2120573211990911

(11)

Ten iterations are conducted In this way we get the finalapproximation as follows

11988210

(119909 ℎ) =

10

sum

119899=0

119908119899(119909)

= 1199080(119909) + 119908

1(119909) + 119908

2(119909) + sdot sdot sdot + 119908

10(119909)

(12)

By substituting (12) into the boundary conditions weobtained four homogeneous equations By representing theseequations in the matrix form by coefficient matrix [119862 (120572 120573)]we obtained the following equation in matrix form

[119862 (120572 120573)][[[

[

119886

119887

119888

119889

]]]

]

=[[[

[

0

0

0

0

]]]

]

(13)

where a b c and 119889 are the unknown constants which hasbeen introduced in the initial approximation For a nontrivialsolution the determinant of the coefficient matrix [119862(120572 120573)]

must vanish Then the problem takes the following form

Det [119862 (120572 120573)] = 0 (14)

The smallest positive real root of (14) is the normalized criti-cal buckling loadThenext positive real root is the normalizedbuckling load for second mode and so on Equation (14)

0

0

10

20

30

ħ

minus2 minus15 minus1 minus05

minus10

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 3 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-S

column

depends on the stability parameter 120572 the restraint stiffnessparameter 120573 and the convergence control parameter ℎ Thenwe define the function 119880(120572 120573 ℎ) as follows

119880 (120572 120573 ℎ) = Det [119862 (120572 120573)] (15)

And then we plot the ℎ curves of the 1198801015840

10(120572 120573 ℎ) and

11988010158401015840

10(120572 120573 ℎ) in order to find convergence region of the ℎ

where prime denotes derivatives of 119880(120572 120573 ℎ) with respect to120572

41 Clamped-Free (C-F) Column Substituting the 10th-orderapproximation 119882

10(120572 120573 ℎ) into the boundary conditions of

C-F column we get coefficientmatrix [119862119862-119865(120572 120573)]We define

the function 119880(120572 120573 ℎ) as follows

119880(120572 120573 ℎ) = Det [119862119862-119865 (120572 120573)] (16)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 3 and the valid region of ℎ is approximatedas minus1 5 lt ℎ lt minus0 3

Finally the critical buckling load obtained from (14) forℎ = minus0 99 is 24674

42 Pinned-Pinned (P-P) Column Substituting the 10th-order approximation 119882

10(120572 120573 ℎ) into the boundary condi-

tions of P-P column we get coefficient matrix [119862119875-119875(120572 120573)]

We define the function 119880(120572 120573 ℎ) as follows

119880 (120572 120573 ℎ) = Det [119862119875-119875 (120572 120573)] (17)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 4 and the valid region of ℎ is about minus1 6 lt ℎ lt

minus0 4Finally the critical buckling load obtained from (14) for

ℎ = minus0 995 is 98696

Journal of Applied Mathematics 5

0 05

ħ

minus2

minus2

minus4

minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

0

2

4

6

8

10

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 4 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for P-P

column

0

ħ

minus2minus5

minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

0

5

10

15

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 5 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-P

column

43 Clamped-Pinned (C-P) Column Substituting the 10thorder approximation 119882

10(120572 120573 ℎ) into the boundary condi-

tions of C-P column we get coefficient matrix [119862119862-119875(120572 120573)]

We define the function 119880(120572 120573 ℎ) as follows

119880(120572 120573 ℎ) = Det [119862119862-119875 (120572 120573)] (18)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 5 and the valid region of ℎ is as follows minus1 75 lt

ℎ lt minus0 3Finally the critical buckling load obtained from (14) for

ℎ = minus0 97 is 201907

44 Clamped-Clamped (C-C) Column Substituting the 10th-order approximation 119882

10(120572 120573 ℎ) into the boundary condi-

tions of C-C column we get coefficient matrix [119862119862-119862(120572 120573)]

We define the function 119880(120572 120573 ℎ) as follows

119880 (120572 120573 ℎ) = Det [119862119862-119862 (120572 120573)] (19)

0

5

10

15

0 05

ħ

minus2minus5

minus25 minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 6 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-C

column

0

10

20

30

0

ħ

minus2 minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

minus10

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 7 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-S

column

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 6 and the valid region of ℎ can be minus1 8 lt ℎ lt minus0 1Finally the critical buckling load obtained from (14) for

ℎ = minus0 96868 is 394784

45 Clamped-Sliding Restraint (C-S) Column Substitutingthe 10th-order approximation 119882

10(120572 120573 ℎ) into the bound-

ary conditions of C-S column we get coefficient matrix[119862119862-119878(120572 120573)] We define the function 119880(120572 120573 ℎ) as follows

119880(120572 120573 ℎ) = Det [119862119862-119878 (120572 120573)] (20)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 7 and the valid region of ℎ can be minus1 55 lt ℎ lt

minus0 37Finally the critical buckling load obtained from (14) for

ℎ = minus0 9909 is 98696The exact solutions for the presented cases are ob-

tained via stability criteria provided by Wang et al [6] In

6 Journal of Applied Mathematics

Table 2 Comparison of exact and HAM solutions of the buckling loads for 120573 = 0 with ten iterations

Mode C-F column P-P column C-P column C-C column C-S columnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 24674 24674 98696 98696 201907 201907 394784 394784 98696 986962 222066 222066 394784 394779 596795 596813 807629 809336 394784 394784

Table 3 Comparison of exact and HAM solutions of the buckling loads for 120573 = 50 with ten iterations

Mode C-F Column P-P Column C-P Column C-C Column C-S ColumnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 88614 88614 149357 149357 242852 242852 432606 432606 235717 2357172 330879 330873 407449 407442 610966 611003 817943 820059 444104 444053

Table 4 Comparison of exact and HAM solutions of the buckling loads for 120573 = 100 with ten iterations

Mode C-F column P-P column C-P column C-C column C-S columnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 119964 119964 200017 200017 283066 283066 470066 470066 326690 3266902 452659 452574 420114 420105 625613 625678 828246 830859 528965 528105

Table 5 Absolute errors of HAM approximations for the first buckling mode

C-F column P-P column C-P column C-C column C-S column120573 = 0 130562119864 minus 13 103727119864 minus 12 000000 000000 539109119864 minus 11

120573 = 50 981515119864 minus 10 126009119864 minus 10 109165119864 minus 7 464191119864 minus 5 150249119864 minus 7

120573 = 100 304503119864 minus 8 165444119864 minus 8 737428119864 minus 7 59300119864 minus 7 181936119864 minus 5

Table 6 Percent relative errors of HAM approximations for the second buckling mode

C-F column P-P column C-P column C-C column C-S column120573 = 0 000000621 000124745 000303663 021128177 000008617120573 = 50 000163762 000173402 000599296 025870613 001135007120573 = 100 001874692 000227055 001038389 031551889 016247717

0

5

10

15

20

25

0 2 4 6 8 10

Nor

mal

ized

buc

klin

g lo

ad

Iteration number

Convergence for mode 1

β = 100

β = 50

β = 0

Figure 8 Convergence of solutions through the iterations for thefirst mode of P-P column

the following tables the buckling loads for the first twomodesare given and compared with exact results

From Tables 2ndash4 it is observed that normalized bucklingloads for the first mode which is the critical buckling load arein excellent agreement with analytical results Absolute errorsfor these cases are given in Table 5

The differences in the results of the second mode existdue to lack of iterations provided by the method Additionaliterations would improve the results for the second modeHowever the results for the second mode are still in goodagreement with the exact results Relative errors for this caseare provided in the following Table 6

These percent relative errors show that the presented solu-tion is in good agreement with analytical ones Convergenceof solutions for the PP column is simulated in Figures 8 and 9Solid lines show the exact solutions for different normalizedspring stiffnesses

Same convergence behaviors are observed for all casesconsidered in this study Furthermore Figure 8 shows that atleast 6 iterations are required for the first mode and Figure 9shows that at least 10 iterations are required for the secondmode to obtain satisfactory results from the analysis

5 Conclusions

In this work a reliable algorithm based on the HAM toobtain the normalized buckling loads of the Euler columns

Journal of Applied Mathematics 7

β = 100

β = 50

β = 0

36

38

40

42

44

0 2 4 6 8 10

Nor

mal

ized

buc

klin

g lo

ad

Iteration number

Convergence for mode 2

Figure 9 Convergence of solutions through the iterations for thesecond mode of P-P column

with constant flexural stiffness is presented Several cases aregiven to illustrate the validity and accuracy of this procedureThe series solutions of (3) by HAM contain the auxiliaryparameter ℎ In general by means of the so-called ℎ-curveit is straightforward to choose a proper value of ℎ whichensures that the series solution is convergent Figures 2 34 5 and 6 show the ℎ-curves obtained from the 119898th-orderHAM approximation solutions From these figures the validregions of ℎ correspond to the line segments nearly parallelto the horizontal axis By the use of the proper value withthis parameter two buckling loads for the first and secondmodes are obtained The method will provide the results forthe following modes if additional iterations are introducedin the analysis The buckling loads are positive real roots ofthe characteristic equationswhich are obtained consecutivelyThis is a huge advantage because it is still very difficult toobtain those roots consecutively even with a mathematicssoftware As a result HAM is an efficient powerful andaccurate tool for determining the buckling loads of Eulercolumns

References

[1] L Euler ldquoDie altitudine colomnarum sub proprio pondere cor-ruentiumrdquo Acta Academiae Scientiarum Imperialis Petropoli-tanae 1778 in Latin

[2] A G Greenhill ldquoDetermination of the greatest height consis-tent with stability that a vertical pole on mast can be made andof the greatest height to which a tree of given proportions cangrowrdquo in Proceedings of the Cambridge Philosophical Society IV1883

[3] A N Dinnik ldquoDesign of columns of varying cross-sectionrdquoTransactions of the ASME vol 51 1929

[4] T R Karman and M A Biot Mathematical Methods in Engi-neering McGraw Hill New York NY USA 1940

[5] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw Hill New York NY USA 1961

[6] C M Wang C Y Wang and J N Reddy Exact Solutions forBuckling of Structural Members CRC Press LLC Florida FlaUSA 2005

[7] H Bleich Buckling Strength of Metal Structures McGraw-HillNew York NY USA 1952

[8] A N Dinnik ldquoDesign of columns of varying cross-sectionrdquoTransactions of the ASME Applied Mechanics vol 51 1932

[9] I Elishakoff and F Pellegrini ldquoExact and effective approximatesolution solutions of some divergent type non-conservativeproblemsrdquo Journal of Sound and Vibration vol 114 pp 144ndash1481987

[10] I Elishakoff and F Pellegrini ldquoApplication of bessel and lommelfunctions and the undetermined multiplier galerkin methodversion for instability of non-uniform columnrdquo Journal ofSound and Vibration vol 115 no 1 pp 182ndash186 1987

[11] I Elishakoff and F Pellegrini ldquoExact solutions for buckling ofsome divergence-type nonconservative systems in terms of bes-sel and lommel functionsrdquo Computer Methods in AppliedMechanics and Engineering vol 66 no 1 pp 107ndash119 1988

[12] M Eisenberger ldquoBuckling loads for variable cross-sectionmembers with variable axial forcesrdquo International Journal ofSolids and Structures vol 27 no 2 pp 135ndash143 1991

[13] J M Gere andW O Carter ldquoCritical buckling loads for taperedcolumnsrdquo Journal of Structural Engineering ASCE vol 88 no 1pp 1ndash11 1962

[14] F Arbabi and F Li ldquoBuckling of variable cross-section columnsintegral equation approachrdquo Journal of Structural EngineeringASCE vol 117 no 8 1991

[15] A Siginer ldquoBuckling of columns of variable flexural rigidityrdquoJournal of Engineering Mechanics ASCE vol 118 no 3 pp 543ndash640 1992

[16] Q Li H Cao and G Li ldquoStability analysis of a bar with multi-segments of varying cross-sectionrdquo Computers and Structuresvol 53 no 5 pp 1085ndash1089 1994

[17] L Qiusheng CHong and L Guiqing ldquoStability analysis of barswith varying cross-sectionrdquo International Journal of Solids andStructures vol 32 no 21 pp 3217ndash3228 1995

[18] Q Li H Cao and G Li ldquoStatic and dynamic analysis of straightbars with variable cross-sectionrdquoComputers and Structures vol59 no 6 pp 1185ndash1191 1996

[19] J H B Sampaio and J R Hundhausen ldquoA mathematical modeland analytical solution for buckling of inclined beam-columnsrdquoApplied Mathematical Modelling vol 22 no 6 pp 405ndash4211998

[20] J B Keller ldquoThe shape of the strongest columnrdquo Archive forRationalMechanics and Analysis vol 5 no 1 pp 275ndash285 1960

[21] I Tadjbakhsh and J B Keller ldquoStrongest columns and isoperi-metric inequalities for eigenvaluesrdquo Journal of Applied Mechan-ics ASME vol 29 pp 159ndash164 1962

[22] J E Taylor ldquoThe strongest columnmdashan energy approachrdquoJournal of Applied Mechanics ASME vol 34 pp 486ndash487 1967

[23] M T Atay and S B Coskun ldquoElastic stability of Euler columnswith a continuous elastic restraint using variational iterationmethodrdquoComputers andMathematics with Applications vol 58no 11-12 pp 2528ndash2534 2009

[24] S B Coskun andM T Atay ldquoDetermination of critical bucklingload for elastic columns of constant and variable cross-sectionsusing variational iteration methodrdquo Computers and Mathemat-ics with Applications vol 58 no 11-12 pp 2260ndash2266 2009

[25] M T Atay ldquoDetermination of buckling loads of tilted buckledcolumn with varying flexural rigidity using variational itera-tion methodrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 11 no 2 pp 97ndash103 2010

[26] F Okay M T Atay and S B Cockun ldquoDetermination of buck-ling loads and mode shapes of a heavy vertical column underits own weight using the variational iteration methodrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 11 no 10 pp 851ndash857 2010

8 Journal of Applied Mathematics

[27] S B Coskun ldquoDetermination of critical buckling loads for eulercolumns of variable flexural stiffness with a continuous elasticrestraint using homotopy perturbation methodrdquo InternationalJournal of Nonlinear Sciences and Numerical Simulation vol 10no 2 pp 191ndash197 2009

[28] MTAtay ldquoDetermination of critical buckling loads for variablestiffness euler columns using homotopy perturbation methodrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 10 no 2 pp 199ndash206 2009

[29] S B Coskun ldquoAnalysis of tilt-buckling of euler columns withvarying flexural stiffness using homotopy perturbation meth-odrdquoMathematicalModelling andAnalysis vol 15 no 3 pp 275ndash286 2010

[30] S Pinarbasi ldquoStability analysis of non-uniform rectangularbeams using homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2012 Article ID 197483 18 pages2012

[31] YHuang andQ Z Luo ldquoA simplemethod to determine the crit-ical buckling loads for axially inhomogeneous beams with elas-tic restraintrdquoComputers andMathematics withApplications vol61 no 9 pp 2510ndash2517 2011

[32] Z X Yuan and XWWang ldquoBuckling and post-buckling analy-sis of extensible beam-columns by using the differential quadra-ture methodrdquo Computers ampMathematics with Applications vol62 no 12 pp 4499ndash4513 2011

[33] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method CRC Press 2004

[34] S J Liao ldquoHomotopy analysis method a new analytic methodfor nonlinear problemsrdquo Applied Mathematics and Mechanicsvol 19 no 10 pp 957ndash962 1998

[35] S J Liao ldquoA uniformly valid analytic solution of two-dimen-sional viscous flow over a semi-infinite flat platerdquo Journal ofFluid Mechanics vol 385 pp 101ndash128 1999

[36] S J Liao ldquoAn explicit analytic solution to the Thomas-Fermiequationrdquo Applied Mathematics and Computation vol 144 no2-3 pp 495ndash506 2003

[37] T Hayat T Javed and M Sajid ldquoAnalytic solution for rotatingflow and heat transfer analysis of a third-grade fluidrdquo ActaMechanica vol 191 no 3-4 pp 219ndash229 2007

[38] T Hayat S B Khan M Sajid and S Asghar ldquoRotating flow of athird grade fluid in a porous space with Hall currentrdquoNonlinearDynamics vol 49 no 1-2 pp 83ndash91 2007

[39] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007

[40] S Abbasbandy and E Shivanian ldquoA new analytical technique tosolve Fredholmrsquos integral equationsrdquoNumerical Algorithms vol56 no 1 pp 27ndash43 2011

[41] S Abbasbandy and F S Zakaria ldquoSoliton solutions for the fifth-order KdV equation with the homotopy analysis methodrdquoNon-linear Dynamics vol 51 no 1-2 pp 83ndash87 2008

[42] Z Wang L Zou and H Zhang ldquoApplying homotopy analysismethod for solving differential-difference equationrdquo PhysicsLetters A vol 369 no 1-2 pp 77ndash84 2007

[43] A S Bataineh M S M Noorani and I Hashim ldquoSolutions oftime-dependent Emden-Fowler type equations by homotopyanalysis methodrdquo Physics Letters A vol 371 no 1-2 pp 72ndash822007

[44] M Inc ldquoOn exact solution of Laplace equation with Dirichletand Neumann boundary conditions by the homotopy analysismethodrdquo Physics Letters A vol 365 no 5-6 pp 412ndash415 2007

[45] F Awawdeh A Adawi and S Al-Shara ldquoAnalytic solution ofmulti pantograph equationrdquo Journal of Applied Mathematicsand Decision Sciences vol 2008 Article ID 605064 11 pages2008

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

Table 1 The stability criteria for continuously restrained elastic columns

Column Stability criteriaC-F column [120572 (119878

2+ 1198792) minus 2119878

21198792] cos119879 cos 119878 minus 120572(119878

2+ 1198792) + (119878

4+ 1198794) + 119878119879[2120572 minus (119878

2+ 1198792)] sin119879 sin 119878 = 0

P-P column sin119879 = 0

C-P column 119879 cos119879 sin 119878 minus 119878 sin119879 cos 119878 = 0

C-C column 2119878119879[cos119879 cos 119878 minus 1] + (1198782+ 1198792) sin119879 sin 119878 = 0

C-S column 119879 sin119879 cos 119878 minus 119878 cos119879 sin 119878 = 0

C-Fcolumn

P-Pcolumn

C-Pcolumn

C-Ccolumn

C-Scolumn

Figure 2 Various end conditions for restrained columns

Where

119878 = radic 120572

2minus radic(

120572

2)

2

minus 120573 (3)

119879 = radic120572

2+ radic(

120572

2)

2

minus 120573 (4)

The classical boundary conditions [6] are as follows

Clamped (Fixed)End 119908 = 0119889119908

119889119909= 0

Pinned End 119908 = 01198892119908

1198891199092= 0

Free End 1198892119908

1198891199092= 0

1198893119908

1198891199093+ 120572

119889119908

119889119909= 0

Sliding Restraint 119889119908

119889119909= 0

1198893119908

1198891199093+ 120572

119889119908

119889119909= 0

(5)

3 HAM Formulation of the Problem

To solve (3) by means of homotopy analysis method let usdefine119873[120601(119909 119902)] as follows

(1 minus 119902) 119871 [120601 (119909 119902) minus 1199080(119909)] = 119902ℎ119867 (119909)119873 [120601 (119909 119902)] (6)

where 120601(119909 119902) is an unknown function to be determined and119873[120601(119909 119902)] is given by

119873[120601 (119909 119902)] = 120601(120484119907)

(119909 119902) + 12057212060110158401015840(119909 119902) + 120573120601 (119909 119902) (7)

The high-order deformation equation is as follows

119908119898(119909) = 120594

119898119908119898minus1

(119909)

+ ℎint

119909

0

int

120591

0

int

120577

0

int

120595

0

[119908(120484119907)

119898minus1(120585) + 120572119908

10158401015840

119898minus1(120585)

+120573119908119898minus1

(120585) ] 119889120585119889120595119889120577119889120591

(8)

where

120594119898=

0 119898 le 1

1 119898 gt 1(9)

Starting with 1199080(119909) successively 119908

119894(119909) 119894 = 1 2 3 are

determined by the so-called high-order deformation equa-tion (8) then the solution is

119908 (119909) = 1199080(119909) +

infin

sum

119898=1

119908119898(119909) (10)

4 Critical Buckling Loads for ContinuouslyRestrained Elastic Columns

A cubic polynomial 1199080(119909) = 119886119909

3+ 1198871199092+ 119888119909 + 119889 is chosen

as an initial approximation due to four boundary conditionsfor each case This polynomial has been successfully used inprevious studies employing different analytical approximatetechniques This polynomial has also been used as the inter-polation function in the finite element analysis of Euler beamsor columns Hence the initial approximation is expectedto produce good results The approximation includes fourunknown coefficients which will be found by substitutingfour boundary conditions into the solution We successively

4 Journal of Applied Mathematics

obtain 119908119894(119909) 119894 = 1 2 3 by the 119898th-order deformation

equation (3)

1199081(119909) =

1

12119887ℎ1205721199094+

1

20119886ℎ1205721199095+

1

24119889ℎ1205731199094+

1

120119888ℎ1205731199095

+1

360119887ℎ1205731199096+

1

840119886ℎ1205731199097

1199082(119909) =

1

12119887ℎ1205721199094+

1

12119887ℎ21205721199094+

1

20119886ℎ1205721199095+

1

20119886ℎ21205721199095

+1

360119887ℎ212057221199096+

1

840119886ℎ212057221199097+

1

24119889ℎ1205731199094

+1

24119889ℎ21205731199094+

1

120119888ℎ1205731199095+

1

120119888ℎ21205731199095

+1

360119887ℎ1205731199096+

1

360119887ℎ21205731199096+

1

840119886ℎ1205731199097

+1

840119886ℎ21205731199097+

1

720119889ℎ21205721205731199096+

1

5040119888ℎ21205721205731199097

+1

10080119887ℎ21205721205731199098+

1

30240119886ℎ21205721205731199099

+1

40320119889ℎ212057321199098+

1

362880119888ℎ212057321199099

+1

1814400119887ℎ2120573211990910

+1

6652800119886ℎ2120573211990911

(11)

Ten iterations are conducted In this way we get the finalapproximation as follows

11988210

(119909 ℎ) =

10

sum

119899=0

119908119899(119909)

= 1199080(119909) + 119908

1(119909) + 119908

2(119909) + sdot sdot sdot + 119908

10(119909)

(12)

By substituting (12) into the boundary conditions weobtained four homogeneous equations By representing theseequations in the matrix form by coefficient matrix [119862 (120572 120573)]we obtained the following equation in matrix form

[119862 (120572 120573)][[[

[

119886

119887

119888

119889

]]]

]

=[[[

[

0

0

0

0

]]]

]

(13)

where a b c and 119889 are the unknown constants which hasbeen introduced in the initial approximation For a nontrivialsolution the determinant of the coefficient matrix [119862(120572 120573)]

must vanish Then the problem takes the following form

Det [119862 (120572 120573)] = 0 (14)

The smallest positive real root of (14) is the normalized criti-cal buckling loadThenext positive real root is the normalizedbuckling load for second mode and so on Equation (14)

0

0

10

20

30

ħ

minus2 minus15 minus1 minus05

minus10

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 3 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-S

column

depends on the stability parameter 120572 the restraint stiffnessparameter 120573 and the convergence control parameter ℎ Thenwe define the function 119880(120572 120573 ℎ) as follows

119880 (120572 120573 ℎ) = Det [119862 (120572 120573)] (15)

And then we plot the ℎ curves of the 1198801015840

10(120572 120573 ℎ) and

11988010158401015840

10(120572 120573 ℎ) in order to find convergence region of the ℎ

where prime denotes derivatives of 119880(120572 120573 ℎ) with respect to120572

41 Clamped-Free (C-F) Column Substituting the 10th-orderapproximation 119882

10(120572 120573 ℎ) into the boundary conditions of

C-F column we get coefficientmatrix [119862119862-119865(120572 120573)]We define

the function 119880(120572 120573 ℎ) as follows

119880(120572 120573 ℎ) = Det [119862119862-119865 (120572 120573)] (16)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 3 and the valid region of ℎ is approximatedas minus1 5 lt ℎ lt minus0 3

Finally the critical buckling load obtained from (14) forℎ = minus0 99 is 24674

42 Pinned-Pinned (P-P) Column Substituting the 10th-order approximation 119882

10(120572 120573 ℎ) into the boundary condi-

tions of P-P column we get coefficient matrix [119862119875-119875(120572 120573)]

We define the function 119880(120572 120573 ℎ) as follows

119880 (120572 120573 ℎ) = Det [119862119875-119875 (120572 120573)] (17)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 4 and the valid region of ℎ is about minus1 6 lt ℎ lt

minus0 4Finally the critical buckling load obtained from (14) for

ℎ = minus0 995 is 98696

Journal of Applied Mathematics 5

0 05

ħ

minus2

minus2

minus4

minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

0

2

4

6

8

10

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 4 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for P-P

column

0

ħ

minus2minus5

minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

0

5

10

15

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 5 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-P

column

43 Clamped-Pinned (C-P) Column Substituting the 10thorder approximation 119882

10(120572 120573 ℎ) into the boundary condi-

tions of C-P column we get coefficient matrix [119862119862-119875(120572 120573)]

We define the function 119880(120572 120573 ℎ) as follows

119880(120572 120573 ℎ) = Det [119862119862-119875 (120572 120573)] (18)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 5 and the valid region of ℎ is as follows minus1 75 lt

ℎ lt minus0 3Finally the critical buckling load obtained from (14) for

ℎ = minus0 97 is 201907

44 Clamped-Clamped (C-C) Column Substituting the 10th-order approximation 119882

10(120572 120573 ℎ) into the boundary condi-

tions of C-C column we get coefficient matrix [119862119862-119862(120572 120573)]

We define the function 119880(120572 120573 ℎ) as follows

119880 (120572 120573 ℎ) = Det [119862119862-119862 (120572 120573)] (19)

0

5

10

15

0 05

ħ

minus2minus5

minus25 minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 6 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-C

column

0

10

20

30

0

ħ

minus2 minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

minus10

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 7 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-S

column

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 6 and the valid region of ℎ can be minus1 8 lt ℎ lt minus0 1Finally the critical buckling load obtained from (14) for

ℎ = minus0 96868 is 394784

45 Clamped-Sliding Restraint (C-S) Column Substitutingthe 10th-order approximation 119882

10(120572 120573 ℎ) into the bound-

ary conditions of C-S column we get coefficient matrix[119862119862-119878(120572 120573)] We define the function 119880(120572 120573 ℎ) as follows

119880(120572 120573 ℎ) = Det [119862119862-119878 (120572 120573)] (20)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 7 and the valid region of ℎ can be minus1 55 lt ℎ lt

minus0 37Finally the critical buckling load obtained from (14) for

ℎ = minus0 9909 is 98696The exact solutions for the presented cases are ob-

tained via stability criteria provided by Wang et al [6] In

6 Journal of Applied Mathematics

Table 2 Comparison of exact and HAM solutions of the buckling loads for 120573 = 0 with ten iterations

Mode C-F column P-P column C-P column C-C column C-S columnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 24674 24674 98696 98696 201907 201907 394784 394784 98696 986962 222066 222066 394784 394779 596795 596813 807629 809336 394784 394784

Table 3 Comparison of exact and HAM solutions of the buckling loads for 120573 = 50 with ten iterations

Mode C-F Column P-P Column C-P Column C-C Column C-S ColumnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 88614 88614 149357 149357 242852 242852 432606 432606 235717 2357172 330879 330873 407449 407442 610966 611003 817943 820059 444104 444053

Table 4 Comparison of exact and HAM solutions of the buckling loads for 120573 = 100 with ten iterations

Mode C-F column P-P column C-P column C-C column C-S columnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 119964 119964 200017 200017 283066 283066 470066 470066 326690 3266902 452659 452574 420114 420105 625613 625678 828246 830859 528965 528105

Table 5 Absolute errors of HAM approximations for the first buckling mode

C-F column P-P column C-P column C-C column C-S column120573 = 0 130562119864 minus 13 103727119864 minus 12 000000 000000 539109119864 minus 11

120573 = 50 981515119864 minus 10 126009119864 minus 10 109165119864 minus 7 464191119864 minus 5 150249119864 minus 7

120573 = 100 304503119864 minus 8 165444119864 minus 8 737428119864 minus 7 59300119864 minus 7 181936119864 minus 5

Table 6 Percent relative errors of HAM approximations for the second buckling mode

C-F column P-P column C-P column C-C column C-S column120573 = 0 000000621 000124745 000303663 021128177 000008617120573 = 50 000163762 000173402 000599296 025870613 001135007120573 = 100 001874692 000227055 001038389 031551889 016247717

0

5

10

15

20

25

0 2 4 6 8 10

Nor

mal

ized

buc

klin

g lo

ad

Iteration number

Convergence for mode 1

β = 100

β = 50

β = 0

Figure 8 Convergence of solutions through the iterations for thefirst mode of P-P column

the following tables the buckling loads for the first twomodesare given and compared with exact results

From Tables 2ndash4 it is observed that normalized bucklingloads for the first mode which is the critical buckling load arein excellent agreement with analytical results Absolute errorsfor these cases are given in Table 5

The differences in the results of the second mode existdue to lack of iterations provided by the method Additionaliterations would improve the results for the second modeHowever the results for the second mode are still in goodagreement with the exact results Relative errors for this caseare provided in the following Table 6

These percent relative errors show that the presented solu-tion is in good agreement with analytical ones Convergenceof solutions for the PP column is simulated in Figures 8 and 9Solid lines show the exact solutions for different normalizedspring stiffnesses

Same convergence behaviors are observed for all casesconsidered in this study Furthermore Figure 8 shows that atleast 6 iterations are required for the first mode and Figure 9shows that at least 10 iterations are required for the secondmode to obtain satisfactory results from the analysis

5 Conclusions

In this work a reliable algorithm based on the HAM toobtain the normalized buckling loads of the Euler columns

Journal of Applied Mathematics 7

β = 100

β = 50

β = 0

36

38

40

42

44

0 2 4 6 8 10

Nor

mal

ized

buc

klin

g lo

ad

Iteration number

Convergence for mode 2

Figure 9 Convergence of solutions through the iterations for thesecond mode of P-P column

with constant flexural stiffness is presented Several cases aregiven to illustrate the validity and accuracy of this procedureThe series solutions of (3) by HAM contain the auxiliaryparameter ℎ In general by means of the so-called ℎ-curveit is straightforward to choose a proper value of ℎ whichensures that the series solution is convergent Figures 2 34 5 and 6 show the ℎ-curves obtained from the 119898th-orderHAM approximation solutions From these figures the validregions of ℎ correspond to the line segments nearly parallelto the horizontal axis By the use of the proper value withthis parameter two buckling loads for the first and secondmodes are obtained The method will provide the results forthe following modes if additional iterations are introducedin the analysis The buckling loads are positive real roots ofthe characteristic equationswhich are obtained consecutivelyThis is a huge advantage because it is still very difficult toobtain those roots consecutively even with a mathematicssoftware As a result HAM is an efficient powerful andaccurate tool for determining the buckling loads of Eulercolumns

References

[1] L Euler ldquoDie altitudine colomnarum sub proprio pondere cor-ruentiumrdquo Acta Academiae Scientiarum Imperialis Petropoli-tanae 1778 in Latin

[2] A G Greenhill ldquoDetermination of the greatest height consis-tent with stability that a vertical pole on mast can be made andof the greatest height to which a tree of given proportions cangrowrdquo in Proceedings of the Cambridge Philosophical Society IV1883

[3] A N Dinnik ldquoDesign of columns of varying cross-sectionrdquoTransactions of the ASME vol 51 1929

[4] T R Karman and M A Biot Mathematical Methods in Engi-neering McGraw Hill New York NY USA 1940

[5] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw Hill New York NY USA 1961

[6] C M Wang C Y Wang and J N Reddy Exact Solutions forBuckling of Structural Members CRC Press LLC Florida FlaUSA 2005

[7] H Bleich Buckling Strength of Metal Structures McGraw-HillNew York NY USA 1952

[8] A N Dinnik ldquoDesign of columns of varying cross-sectionrdquoTransactions of the ASME Applied Mechanics vol 51 1932

[9] I Elishakoff and F Pellegrini ldquoExact and effective approximatesolution solutions of some divergent type non-conservativeproblemsrdquo Journal of Sound and Vibration vol 114 pp 144ndash1481987

[10] I Elishakoff and F Pellegrini ldquoApplication of bessel and lommelfunctions and the undetermined multiplier galerkin methodversion for instability of non-uniform columnrdquo Journal ofSound and Vibration vol 115 no 1 pp 182ndash186 1987

[11] I Elishakoff and F Pellegrini ldquoExact solutions for buckling ofsome divergence-type nonconservative systems in terms of bes-sel and lommel functionsrdquo Computer Methods in AppliedMechanics and Engineering vol 66 no 1 pp 107ndash119 1988

[12] M Eisenberger ldquoBuckling loads for variable cross-sectionmembers with variable axial forcesrdquo International Journal ofSolids and Structures vol 27 no 2 pp 135ndash143 1991

[13] J M Gere andW O Carter ldquoCritical buckling loads for taperedcolumnsrdquo Journal of Structural Engineering ASCE vol 88 no 1pp 1ndash11 1962

[14] F Arbabi and F Li ldquoBuckling of variable cross-section columnsintegral equation approachrdquo Journal of Structural EngineeringASCE vol 117 no 8 1991

[15] A Siginer ldquoBuckling of columns of variable flexural rigidityrdquoJournal of Engineering Mechanics ASCE vol 118 no 3 pp 543ndash640 1992

[16] Q Li H Cao and G Li ldquoStability analysis of a bar with multi-segments of varying cross-sectionrdquo Computers and Structuresvol 53 no 5 pp 1085ndash1089 1994

[17] L Qiusheng CHong and L Guiqing ldquoStability analysis of barswith varying cross-sectionrdquo International Journal of Solids andStructures vol 32 no 21 pp 3217ndash3228 1995

[18] Q Li H Cao and G Li ldquoStatic and dynamic analysis of straightbars with variable cross-sectionrdquoComputers and Structures vol59 no 6 pp 1185ndash1191 1996

[19] J H B Sampaio and J R Hundhausen ldquoA mathematical modeland analytical solution for buckling of inclined beam-columnsrdquoApplied Mathematical Modelling vol 22 no 6 pp 405ndash4211998

[20] J B Keller ldquoThe shape of the strongest columnrdquo Archive forRationalMechanics and Analysis vol 5 no 1 pp 275ndash285 1960

[21] I Tadjbakhsh and J B Keller ldquoStrongest columns and isoperi-metric inequalities for eigenvaluesrdquo Journal of Applied Mechan-ics ASME vol 29 pp 159ndash164 1962

[22] J E Taylor ldquoThe strongest columnmdashan energy approachrdquoJournal of Applied Mechanics ASME vol 34 pp 486ndash487 1967

[23] M T Atay and S B Coskun ldquoElastic stability of Euler columnswith a continuous elastic restraint using variational iterationmethodrdquoComputers andMathematics with Applications vol 58no 11-12 pp 2528ndash2534 2009

[24] S B Coskun andM T Atay ldquoDetermination of critical bucklingload for elastic columns of constant and variable cross-sectionsusing variational iteration methodrdquo Computers and Mathemat-ics with Applications vol 58 no 11-12 pp 2260ndash2266 2009

[25] M T Atay ldquoDetermination of buckling loads of tilted buckledcolumn with varying flexural rigidity using variational itera-tion methodrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 11 no 2 pp 97ndash103 2010

[26] F Okay M T Atay and S B Cockun ldquoDetermination of buck-ling loads and mode shapes of a heavy vertical column underits own weight using the variational iteration methodrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 11 no 10 pp 851ndash857 2010

8 Journal of Applied Mathematics

[27] S B Coskun ldquoDetermination of critical buckling loads for eulercolumns of variable flexural stiffness with a continuous elasticrestraint using homotopy perturbation methodrdquo InternationalJournal of Nonlinear Sciences and Numerical Simulation vol 10no 2 pp 191ndash197 2009

[28] MTAtay ldquoDetermination of critical buckling loads for variablestiffness euler columns using homotopy perturbation methodrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 10 no 2 pp 199ndash206 2009

[29] S B Coskun ldquoAnalysis of tilt-buckling of euler columns withvarying flexural stiffness using homotopy perturbation meth-odrdquoMathematicalModelling andAnalysis vol 15 no 3 pp 275ndash286 2010

[30] S Pinarbasi ldquoStability analysis of non-uniform rectangularbeams using homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2012 Article ID 197483 18 pages2012

[31] YHuang andQ Z Luo ldquoA simplemethod to determine the crit-ical buckling loads for axially inhomogeneous beams with elas-tic restraintrdquoComputers andMathematics withApplications vol61 no 9 pp 2510ndash2517 2011

[32] Z X Yuan and XWWang ldquoBuckling and post-buckling analy-sis of extensible beam-columns by using the differential quadra-ture methodrdquo Computers ampMathematics with Applications vol62 no 12 pp 4499ndash4513 2011

[33] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method CRC Press 2004

[34] S J Liao ldquoHomotopy analysis method a new analytic methodfor nonlinear problemsrdquo Applied Mathematics and Mechanicsvol 19 no 10 pp 957ndash962 1998

[35] S J Liao ldquoA uniformly valid analytic solution of two-dimen-sional viscous flow over a semi-infinite flat platerdquo Journal ofFluid Mechanics vol 385 pp 101ndash128 1999

[36] S J Liao ldquoAn explicit analytic solution to the Thomas-Fermiequationrdquo Applied Mathematics and Computation vol 144 no2-3 pp 495ndash506 2003

[37] T Hayat T Javed and M Sajid ldquoAnalytic solution for rotatingflow and heat transfer analysis of a third-grade fluidrdquo ActaMechanica vol 191 no 3-4 pp 219ndash229 2007

[38] T Hayat S B Khan M Sajid and S Asghar ldquoRotating flow of athird grade fluid in a porous space with Hall currentrdquoNonlinearDynamics vol 49 no 1-2 pp 83ndash91 2007

[39] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007

[40] S Abbasbandy and E Shivanian ldquoA new analytical technique tosolve Fredholmrsquos integral equationsrdquoNumerical Algorithms vol56 no 1 pp 27ndash43 2011

[41] S Abbasbandy and F S Zakaria ldquoSoliton solutions for the fifth-order KdV equation with the homotopy analysis methodrdquoNon-linear Dynamics vol 51 no 1-2 pp 83ndash87 2008

[42] Z Wang L Zou and H Zhang ldquoApplying homotopy analysismethod for solving differential-difference equationrdquo PhysicsLetters A vol 369 no 1-2 pp 77ndash84 2007

[43] A S Bataineh M S M Noorani and I Hashim ldquoSolutions oftime-dependent Emden-Fowler type equations by homotopyanalysis methodrdquo Physics Letters A vol 371 no 1-2 pp 72ndash822007

[44] M Inc ldquoOn exact solution of Laplace equation with Dirichletand Neumann boundary conditions by the homotopy analysismethodrdquo Physics Letters A vol 365 no 5-6 pp 412ndash415 2007

[45] F Awawdeh A Adawi and S Al-Shara ldquoAnalytic solution ofmulti pantograph equationrdquo Journal of Applied Mathematicsand Decision Sciences vol 2008 Article ID 605064 11 pages2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

obtain 119908119894(119909) 119894 = 1 2 3 by the 119898th-order deformation

equation (3)

1199081(119909) =

1

12119887ℎ1205721199094+

1

20119886ℎ1205721199095+

1

24119889ℎ1205731199094+

1

120119888ℎ1205731199095

+1

360119887ℎ1205731199096+

1

840119886ℎ1205731199097

1199082(119909) =

1

12119887ℎ1205721199094+

1

12119887ℎ21205721199094+

1

20119886ℎ1205721199095+

1

20119886ℎ21205721199095

+1

360119887ℎ212057221199096+

1

840119886ℎ212057221199097+

1

24119889ℎ1205731199094

+1

24119889ℎ21205731199094+

1

120119888ℎ1205731199095+

1

120119888ℎ21205731199095

+1

360119887ℎ1205731199096+

1

360119887ℎ21205731199096+

1

840119886ℎ1205731199097

+1

840119886ℎ21205731199097+

1

720119889ℎ21205721205731199096+

1

5040119888ℎ21205721205731199097

+1

10080119887ℎ21205721205731199098+

1

30240119886ℎ21205721205731199099

+1

40320119889ℎ212057321199098+

1

362880119888ℎ212057321199099

+1

1814400119887ℎ2120573211990910

+1

6652800119886ℎ2120573211990911

(11)

Ten iterations are conducted In this way we get the finalapproximation as follows

11988210

(119909 ℎ) =

10

sum

119899=0

119908119899(119909)

= 1199080(119909) + 119908

1(119909) + 119908

2(119909) + sdot sdot sdot + 119908

10(119909)

(12)

By substituting (12) into the boundary conditions weobtained four homogeneous equations By representing theseequations in the matrix form by coefficient matrix [119862 (120572 120573)]we obtained the following equation in matrix form

[119862 (120572 120573)][[[

[

119886

119887

119888

119889

]]]

]

=[[[

[

0

0

0

0

]]]

]

(13)

where a b c and 119889 are the unknown constants which hasbeen introduced in the initial approximation For a nontrivialsolution the determinant of the coefficient matrix [119862(120572 120573)]

must vanish Then the problem takes the following form

Det [119862 (120572 120573)] = 0 (14)

The smallest positive real root of (14) is the normalized criti-cal buckling loadThenext positive real root is the normalizedbuckling load for second mode and so on Equation (14)

0

0

10

20

30

ħ

minus2 minus15 minus1 minus05

minus10

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 3 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-S

column

depends on the stability parameter 120572 the restraint stiffnessparameter 120573 and the convergence control parameter ℎ Thenwe define the function 119880(120572 120573 ℎ) as follows

119880 (120572 120573 ℎ) = Det [119862 (120572 120573)] (15)

And then we plot the ℎ curves of the 1198801015840

10(120572 120573 ℎ) and

11988010158401015840

10(120572 120573 ℎ) in order to find convergence region of the ℎ

where prime denotes derivatives of 119880(120572 120573 ℎ) with respect to120572

41 Clamped-Free (C-F) Column Substituting the 10th-orderapproximation 119882

10(120572 120573 ℎ) into the boundary conditions of

C-F column we get coefficientmatrix [119862119862-119865(120572 120573)]We define

the function 119880(120572 120573 ℎ) as follows

119880(120572 120573 ℎ) = Det [119862119862-119865 (120572 120573)] (16)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 3 and the valid region of ℎ is approximatedas minus1 5 lt ℎ lt minus0 3

Finally the critical buckling load obtained from (14) forℎ = minus0 99 is 24674

42 Pinned-Pinned (P-P) Column Substituting the 10th-order approximation 119882

10(120572 120573 ℎ) into the boundary condi-

tions of P-P column we get coefficient matrix [119862119875-119875(120572 120573)]

We define the function 119880(120572 120573 ℎ) as follows

119880 (120572 120573 ℎ) = Det [119862119875-119875 (120572 120573)] (17)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 4 and the valid region of ℎ is about minus1 6 lt ℎ lt

minus0 4Finally the critical buckling load obtained from (14) for

ℎ = minus0 995 is 98696

Journal of Applied Mathematics 5

0 05

ħ

minus2

minus2

minus4

minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

0

2

4

6

8

10

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 4 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for P-P

column

0

ħ

minus2minus5

minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

0

5

10

15

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 5 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-P

column

43 Clamped-Pinned (C-P) Column Substituting the 10thorder approximation 119882

10(120572 120573 ℎ) into the boundary condi-

tions of C-P column we get coefficient matrix [119862119862-119875(120572 120573)]

We define the function 119880(120572 120573 ℎ) as follows

119880(120572 120573 ℎ) = Det [119862119862-119875 (120572 120573)] (18)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 5 and the valid region of ℎ is as follows minus1 75 lt

ℎ lt minus0 3Finally the critical buckling load obtained from (14) for

ℎ = minus0 97 is 201907

44 Clamped-Clamped (C-C) Column Substituting the 10th-order approximation 119882

10(120572 120573 ℎ) into the boundary condi-

tions of C-C column we get coefficient matrix [119862119862-119862(120572 120573)]

We define the function 119880(120572 120573 ℎ) as follows

119880 (120572 120573 ℎ) = Det [119862119862-119862 (120572 120573)] (19)

0

5

10

15

0 05

ħ

minus2minus5

minus25 minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 6 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-C

column

0

10

20

30

0

ħ

minus2 minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

minus10

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 7 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-S

column

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 6 and the valid region of ℎ can be minus1 8 lt ℎ lt minus0 1Finally the critical buckling load obtained from (14) for

ℎ = minus0 96868 is 394784

45 Clamped-Sliding Restraint (C-S) Column Substitutingthe 10th-order approximation 119882

10(120572 120573 ℎ) into the bound-

ary conditions of C-S column we get coefficient matrix[119862119862-119878(120572 120573)] We define the function 119880(120572 120573 ℎ) as follows

119880(120572 120573 ℎ) = Det [119862119862-119878 (120572 120573)] (20)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 7 and the valid region of ℎ can be minus1 55 lt ℎ lt

minus0 37Finally the critical buckling load obtained from (14) for

ℎ = minus0 9909 is 98696The exact solutions for the presented cases are ob-

tained via stability criteria provided by Wang et al [6] In

6 Journal of Applied Mathematics

Table 2 Comparison of exact and HAM solutions of the buckling loads for 120573 = 0 with ten iterations

Mode C-F column P-P column C-P column C-C column C-S columnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 24674 24674 98696 98696 201907 201907 394784 394784 98696 986962 222066 222066 394784 394779 596795 596813 807629 809336 394784 394784

Table 3 Comparison of exact and HAM solutions of the buckling loads for 120573 = 50 with ten iterations

Mode C-F Column P-P Column C-P Column C-C Column C-S ColumnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 88614 88614 149357 149357 242852 242852 432606 432606 235717 2357172 330879 330873 407449 407442 610966 611003 817943 820059 444104 444053

Table 4 Comparison of exact and HAM solutions of the buckling loads for 120573 = 100 with ten iterations

Mode C-F column P-P column C-P column C-C column C-S columnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 119964 119964 200017 200017 283066 283066 470066 470066 326690 3266902 452659 452574 420114 420105 625613 625678 828246 830859 528965 528105

Table 5 Absolute errors of HAM approximations for the first buckling mode

C-F column P-P column C-P column C-C column C-S column120573 = 0 130562119864 minus 13 103727119864 minus 12 000000 000000 539109119864 minus 11

120573 = 50 981515119864 minus 10 126009119864 minus 10 109165119864 minus 7 464191119864 minus 5 150249119864 minus 7

120573 = 100 304503119864 minus 8 165444119864 minus 8 737428119864 minus 7 59300119864 minus 7 181936119864 minus 5

Table 6 Percent relative errors of HAM approximations for the second buckling mode

C-F column P-P column C-P column C-C column C-S column120573 = 0 000000621 000124745 000303663 021128177 000008617120573 = 50 000163762 000173402 000599296 025870613 001135007120573 = 100 001874692 000227055 001038389 031551889 016247717

0

5

10

15

20

25

0 2 4 6 8 10

Nor

mal

ized

buc

klin

g lo

ad

Iteration number

Convergence for mode 1

β = 100

β = 50

β = 0

Figure 8 Convergence of solutions through the iterations for thefirst mode of P-P column

the following tables the buckling loads for the first twomodesare given and compared with exact results

From Tables 2ndash4 it is observed that normalized bucklingloads for the first mode which is the critical buckling load arein excellent agreement with analytical results Absolute errorsfor these cases are given in Table 5

The differences in the results of the second mode existdue to lack of iterations provided by the method Additionaliterations would improve the results for the second modeHowever the results for the second mode are still in goodagreement with the exact results Relative errors for this caseare provided in the following Table 6

These percent relative errors show that the presented solu-tion is in good agreement with analytical ones Convergenceof solutions for the PP column is simulated in Figures 8 and 9Solid lines show the exact solutions for different normalizedspring stiffnesses

Same convergence behaviors are observed for all casesconsidered in this study Furthermore Figure 8 shows that atleast 6 iterations are required for the first mode and Figure 9shows that at least 10 iterations are required for the secondmode to obtain satisfactory results from the analysis

5 Conclusions

In this work a reliable algorithm based on the HAM toobtain the normalized buckling loads of the Euler columns

Journal of Applied Mathematics 7

β = 100

β = 50

β = 0

36

38

40

42

44

0 2 4 6 8 10

Nor

mal

ized

buc

klin

g lo

ad

Iteration number

Convergence for mode 2

Figure 9 Convergence of solutions through the iterations for thesecond mode of P-P column

with constant flexural stiffness is presented Several cases aregiven to illustrate the validity and accuracy of this procedureThe series solutions of (3) by HAM contain the auxiliaryparameter ℎ In general by means of the so-called ℎ-curveit is straightforward to choose a proper value of ℎ whichensures that the series solution is convergent Figures 2 34 5 and 6 show the ℎ-curves obtained from the 119898th-orderHAM approximation solutions From these figures the validregions of ℎ correspond to the line segments nearly parallelto the horizontal axis By the use of the proper value withthis parameter two buckling loads for the first and secondmodes are obtained The method will provide the results forthe following modes if additional iterations are introducedin the analysis The buckling loads are positive real roots ofthe characteristic equationswhich are obtained consecutivelyThis is a huge advantage because it is still very difficult toobtain those roots consecutively even with a mathematicssoftware As a result HAM is an efficient powerful andaccurate tool for determining the buckling loads of Eulercolumns

References

[1] L Euler ldquoDie altitudine colomnarum sub proprio pondere cor-ruentiumrdquo Acta Academiae Scientiarum Imperialis Petropoli-tanae 1778 in Latin

[2] A G Greenhill ldquoDetermination of the greatest height consis-tent with stability that a vertical pole on mast can be made andof the greatest height to which a tree of given proportions cangrowrdquo in Proceedings of the Cambridge Philosophical Society IV1883

[3] A N Dinnik ldquoDesign of columns of varying cross-sectionrdquoTransactions of the ASME vol 51 1929

[4] T R Karman and M A Biot Mathematical Methods in Engi-neering McGraw Hill New York NY USA 1940

[5] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw Hill New York NY USA 1961

[6] C M Wang C Y Wang and J N Reddy Exact Solutions forBuckling of Structural Members CRC Press LLC Florida FlaUSA 2005

[7] H Bleich Buckling Strength of Metal Structures McGraw-HillNew York NY USA 1952

[8] A N Dinnik ldquoDesign of columns of varying cross-sectionrdquoTransactions of the ASME Applied Mechanics vol 51 1932

[9] I Elishakoff and F Pellegrini ldquoExact and effective approximatesolution solutions of some divergent type non-conservativeproblemsrdquo Journal of Sound and Vibration vol 114 pp 144ndash1481987

[10] I Elishakoff and F Pellegrini ldquoApplication of bessel and lommelfunctions and the undetermined multiplier galerkin methodversion for instability of non-uniform columnrdquo Journal ofSound and Vibration vol 115 no 1 pp 182ndash186 1987

[11] I Elishakoff and F Pellegrini ldquoExact solutions for buckling ofsome divergence-type nonconservative systems in terms of bes-sel and lommel functionsrdquo Computer Methods in AppliedMechanics and Engineering vol 66 no 1 pp 107ndash119 1988

[12] M Eisenberger ldquoBuckling loads for variable cross-sectionmembers with variable axial forcesrdquo International Journal ofSolids and Structures vol 27 no 2 pp 135ndash143 1991

[13] J M Gere andW O Carter ldquoCritical buckling loads for taperedcolumnsrdquo Journal of Structural Engineering ASCE vol 88 no 1pp 1ndash11 1962

[14] F Arbabi and F Li ldquoBuckling of variable cross-section columnsintegral equation approachrdquo Journal of Structural EngineeringASCE vol 117 no 8 1991

[15] A Siginer ldquoBuckling of columns of variable flexural rigidityrdquoJournal of Engineering Mechanics ASCE vol 118 no 3 pp 543ndash640 1992

[16] Q Li H Cao and G Li ldquoStability analysis of a bar with multi-segments of varying cross-sectionrdquo Computers and Structuresvol 53 no 5 pp 1085ndash1089 1994

[17] L Qiusheng CHong and L Guiqing ldquoStability analysis of barswith varying cross-sectionrdquo International Journal of Solids andStructures vol 32 no 21 pp 3217ndash3228 1995

[18] Q Li H Cao and G Li ldquoStatic and dynamic analysis of straightbars with variable cross-sectionrdquoComputers and Structures vol59 no 6 pp 1185ndash1191 1996

[19] J H B Sampaio and J R Hundhausen ldquoA mathematical modeland analytical solution for buckling of inclined beam-columnsrdquoApplied Mathematical Modelling vol 22 no 6 pp 405ndash4211998

[20] J B Keller ldquoThe shape of the strongest columnrdquo Archive forRationalMechanics and Analysis vol 5 no 1 pp 275ndash285 1960

[21] I Tadjbakhsh and J B Keller ldquoStrongest columns and isoperi-metric inequalities for eigenvaluesrdquo Journal of Applied Mechan-ics ASME vol 29 pp 159ndash164 1962

[22] J E Taylor ldquoThe strongest columnmdashan energy approachrdquoJournal of Applied Mechanics ASME vol 34 pp 486ndash487 1967

[23] M T Atay and S B Coskun ldquoElastic stability of Euler columnswith a continuous elastic restraint using variational iterationmethodrdquoComputers andMathematics with Applications vol 58no 11-12 pp 2528ndash2534 2009

[24] S B Coskun andM T Atay ldquoDetermination of critical bucklingload for elastic columns of constant and variable cross-sectionsusing variational iteration methodrdquo Computers and Mathemat-ics with Applications vol 58 no 11-12 pp 2260ndash2266 2009

[25] M T Atay ldquoDetermination of buckling loads of tilted buckledcolumn with varying flexural rigidity using variational itera-tion methodrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 11 no 2 pp 97ndash103 2010

[26] F Okay M T Atay and S B Cockun ldquoDetermination of buck-ling loads and mode shapes of a heavy vertical column underits own weight using the variational iteration methodrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 11 no 10 pp 851ndash857 2010

8 Journal of Applied Mathematics

[27] S B Coskun ldquoDetermination of critical buckling loads for eulercolumns of variable flexural stiffness with a continuous elasticrestraint using homotopy perturbation methodrdquo InternationalJournal of Nonlinear Sciences and Numerical Simulation vol 10no 2 pp 191ndash197 2009

[28] MTAtay ldquoDetermination of critical buckling loads for variablestiffness euler columns using homotopy perturbation methodrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 10 no 2 pp 199ndash206 2009

[29] S B Coskun ldquoAnalysis of tilt-buckling of euler columns withvarying flexural stiffness using homotopy perturbation meth-odrdquoMathematicalModelling andAnalysis vol 15 no 3 pp 275ndash286 2010

[30] S Pinarbasi ldquoStability analysis of non-uniform rectangularbeams using homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2012 Article ID 197483 18 pages2012

[31] YHuang andQ Z Luo ldquoA simplemethod to determine the crit-ical buckling loads for axially inhomogeneous beams with elas-tic restraintrdquoComputers andMathematics withApplications vol61 no 9 pp 2510ndash2517 2011

[32] Z X Yuan and XWWang ldquoBuckling and post-buckling analy-sis of extensible beam-columns by using the differential quadra-ture methodrdquo Computers ampMathematics with Applications vol62 no 12 pp 4499ndash4513 2011

[33] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method CRC Press 2004

[34] S J Liao ldquoHomotopy analysis method a new analytic methodfor nonlinear problemsrdquo Applied Mathematics and Mechanicsvol 19 no 10 pp 957ndash962 1998

[35] S J Liao ldquoA uniformly valid analytic solution of two-dimen-sional viscous flow over a semi-infinite flat platerdquo Journal ofFluid Mechanics vol 385 pp 101ndash128 1999

[36] S J Liao ldquoAn explicit analytic solution to the Thomas-Fermiequationrdquo Applied Mathematics and Computation vol 144 no2-3 pp 495ndash506 2003

[37] T Hayat T Javed and M Sajid ldquoAnalytic solution for rotatingflow and heat transfer analysis of a third-grade fluidrdquo ActaMechanica vol 191 no 3-4 pp 219ndash229 2007

[38] T Hayat S B Khan M Sajid and S Asghar ldquoRotating flow of athird grade fluid in a porous space with Hall currentrdquoNonlinearDynamics vol 49 no 1-2 pp 83ndash91 2007

[39] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007

[40] S Abbasbandy and E Shivanian ldquoA new analytical technique tosolve Fredholmrsquos integral equationsrdquoNumerical Algorithms vol56 no 1 pp 27ndash43 2011

[41] S Abbasbandy and F S Zakaria ldquoSoliton solutions for the fifth-order KdV equation with the homotopy analysis methodrdquoNon-linear Dynamics vol 51 no 1-2 pp 83ndash87 2008

[42] Z Wang L Zou and H Zhang ldquoApplying homotopy analysismethod for solving differential-difference equationrdquo PhysicsLetters A vol 369 no 1-2 pp 77ndash84 2007

[43] A S Bataineh M S M Noorani and I Hashim ldquoSolutions oftime-dependent Emden-Fowler type equations by homotopyanalysis methodrdquo Physics Letters A vol 371 no 1-2 pp 72ndash822007

[44] M Inc ldquoOn exact solution of Laplace equation with Dirichletand Neumann boundary conditions by the homotopy analysismethodrdquo Physics Letters A vol 365 no 5-6 pp 412ndash415 2007

[45] F Awawdeh A Adawi and S Al-Shara ldquoAnalytic solution ofmulti pantograph equationrdquo Journal of Applied Mathematicsand Decision Sciences vol 2008 Article ID 605064 11 pages2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

0 05

ħ

minus2

minus2

minus4

minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

0

2

4

6

8

10

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 4 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for P-P

column

0

ħ

minus2minus5

minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

0

5

10

15

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 5 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-P

column

43 Clamped-Pinned (C-P) Column Substituting the 10thorder approximation 119882

10(120572 120573 ℎ) into the boundary condi-

tions of C-P column we get coefficient matrix [119862119862-119875(120572 120573)]

We define the function 119880(120572 120573 ℎ) as follows

119880(120572 120573 ℎ) = Det [119862119862-119875 (120572 120573)] (18)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 5 and the valid region of ℎ is as follows minus1 75 lt

ℎ lt minus0 3Finally the critical buckling load obtained from (14) for

ℎ = minus0 97 is 201907

44 Clamped-Clamped (C-C) Column Substituting the 10th-order approximation 119882

10(120572 120573 ℎ) into the boundary condi-

tions of C-C column we get coefficient matrix [119862119862-119862(120572 120573)]

We define the function 119880(120572 120573 ℎ) as follows

119880 (120572 120573 ℎ) = Det [119862119862-119862 (120572 120573)] (19)

0

5

10

15

0 05

ħ

minus2minus5

minus25 minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 6 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-C

column

0

10

20

30

0

ħ

minus2 minus15 minus1 minus05

Uprime10 (1 1 ħ)

Uprimeprime10 (1 1 ħ)

minus10

Uprimeprime 10

(1 1

ħ)

and

Uprime 10

(1 1

ħ)

Figure 7 The ℎ curves of 1198801015840

10(1 1 ℎ) and 119880

10158401015840

10(1 1 ℎ) for C-S

column

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 6 and the valid region of ℎ can be minus1 8 lt ℎ lt minus0 1Finally the critical buckling load obtained from (14) for

ℎ = minus0 96868 is 394784

45 Clamped-Sliding Restraint (C-S) Column Substitutingthe 10th-order approximation 119882

10(120572 120573 ℎ) into the bound-

ary conditions of C-S column we get coefficient matrix[119862119862-119878(120572 120573)] We define the function 119880(120572 120573 ℎ) as follows

119880(120572 120573 ℎ) = Det [119862119862-119878 (120572 120573)] (20)

Then the ℎ curves of119880101584010(1 1 ℎ) and119880

10158401015840

10(1 1 ℎ) are obtained

in Figure 7 and the valid region of ℎ can be minus1 55 lt ℎ lt

minus0 37Finally the critical buckling load obtained from (14) for

ℎ = minus0 9909 is 98696The exact solutions for the presented cases are ob-

tained via stability criteria provided by Wang et al [6] In

6 Journal of Applied Mathematics

Table 2 Comparison of exact and HAM solutions of the buckling loads for 120573 = 0 with ten iterations

Mode C-F column P-P column C-P column C-C column C-S columnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 24674 24674 98696 98696 201907 201907 394784 394784 98696 986962 222066 222066 394784 394779 596795 596813 807629 809336 394784 394784

Table 3 Comparison of exact and HAM solutions of the buckling loads for 120573 = 50 with ten iterations

Mode C-F Column P-P Column C-P Column C-C Column C-S ColumnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 88614 88614 149357 149357 242852 242852 432606 432606 235717 2357172 330879 330873 407449 407442 610966 611003 817943 820059 444104 444053

Table 4 Comparison of exact and HAM solutions of the buckling loads for 120573 = 100 with ten iterations

Mode C-F column P-P column C-P column C-C column C-S columnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 119964 119964 200017 200017 283066 283066 470066 470066 326690 3266902 452659 452574 420114 420105 625613 625678 828246 830859 528965 528105

Table 5 Absolute errors of HAM approximations for the first buckling mode

C-F column P-P column C-P column C-C column C-S column120573 = 0 130562119864 minus 13 103727119864 minus 12 000000 000000 539109119864 minus 11

120573 = 50 981515119864 minus 10 126009119864 minus 10 109165119864 minus 7 464191119864 minus 5 150249119864 minus 7

120573 = 100 304503119864 minus 8 165444119864 minus 8 737428119864 minus 7 59300119864 minus 7 181936119864 minus 5

Table 6 Percent relative errors of HAM approximations for the second buckling mode

C-F column P-P column C-P column C-C column C-S column120573 = 0 000000621 000124745 000303663 021128177 000008617120573 = 50 000163762 000173402 000599296 025870613 001135007120573 = 100 001874692 000227055 001038389 031551889 016247717

0

5

10

15

20

25

0 2 4 6 8 10

Nor

mal

ized

buc

klin

g lo

ad

Iteration number

Convergence for mode 1

β = 100

β = 50

β = 0

Figure 8 Convergence of solutions through the iterations for thefirst mode of P-P column

the following tables the buckling loads for the first twomodesare given and compared with exact results

From Tables 2ndash4 it is observed that normalized bucklingloads for the first mode which is the critical buckling load arein excellent agreement with analytical results Absolute errorsfor these cases are given in Table 5

The differences in the results of the second mode existdue to lack of iterations provided by the method Additionaliterations would improve the results for the second modeHowever the results for the second mode are still in goodagreement with the exact results Relative errors for this caseare provided in the following Table 6

These percent relative errors show that the presented solu-tion is in good agreement with analytical ones Convergenceof solutions for the PP column is simulated in Figures 8 and 9Solid lines show the exact solutions for different normalizedspring stiffnesses

Same convergence behaviors are observed for all casesconsidered in this study Furthermore Figure 8 shows that atleast 6 iterations are required for the first mode and Figure 9shows that at least 10 iterations are required for the secondmode to obtain satisfactory results from the analysis

5 Conclusions

In this work a reliable algorithm based on the HAM toobtain the normalized buckling loads of the Euler columns

Journal of Applied Mathematics 7

β = 100

β = 50

β = 0

36

38

40

42

44

0 2 4 6 8 10

Nor

mal

ized

buc

klin

g lo

ad

Iteration number

Convergence for mode 2

Figure 9 Convergence of solutions through the iterations for thesecond mode of P-P column

with constant flexural stiffness is presented Several cases aregiven to illustrate the validity and accuracy of this procedureThe series solutions of (3) by HAM contain the auxiliaryparameter ℎ In general by means of the so-called ℎ-curveit is straightforward to choose a proper value of ℎ whichensures that the series solution is convergent Figures 2 34 5 and 6 show the ℎ-curves obtained from the 119898th-orderHAM approximation solutions From these figures the validregions of ℎ correspond to the line segments nearly parallelto the horizontal axis By the use of the proper value withthis parameter two buckling loads for the first and secondmodes are obtained The method will provide the results forthe following modes if additional iterations are introducedin the analysis The buckling loads are positive real roots ofthe characteristic equationswhich are obtained consecutivelyThis is a huge advantage because it is still very difficult toobtain those roots consecutively even with a mathematicssoftware As a result HAM is an efficient powerful andaccurate tool for determining the buckling loads of Eulercolumns

References

[1] L Euler ldquoDie altitudine colomnarum sub proprio pondere cor-ruentiumrdquo Acta Academiae Scientiarum Imperialis Petropoli-tanae 1778 in Latin

[2] A G Greenhill ldquoDetermination of the greatest height consis-tent with stability that a vertical pole on mast can be made andof the greatest height to which a tree of given proportions cangrowrdquo in Proceedings of the Cambridge Philosophical Society IV1883

[3] A N Dinnik ldquoDesign of columns of varying cross-sectionrdquoTransactions of the ASME vol 51 1929

[4] T R Karman and M A Biot Mathematical Methods in Engi-neering McGraw Hill New York NY USA 1940

[5] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw Hill New York NY USA 1961

[6] C M Wang C Y Wang and J N Reddy Exact Solutions forBuckling of Structural Members CRC Press LLC Florida FlaUSA 2005

[7] H Bleich Buckling Strength of Metal Structures McGraw-HillNew York NY USA 1952

[8] A N Dinnik ldquoDesign of columns of varying cross-sectionrdquoTransactions of the ASME Applied Mechanics vol 51 1932

[9] I Elishakoff and F Pellegrini ldquoExact and effective approximatesolution solutions of some divergent type non-conservativeproblemsrdquo Journal of Sound and Vibration vol 114 pp 144ndash1481987

[10] I Elishakoff and F Pellegrini ldquoApplication of bessel and lommelfunctions and the undetermined multiplier galerkin methodversion for instability of non-uniform columnrdquo Journal ofSound and Vibration vol 115 no 1 pp 182ndash186 1987

[11] I Elishakoff and F Pellegrini ldquoExact solutions for buckling ofsome divergence-type nonconservative systems in terms of bes-sel and lommel functionsrdquo Computer Methods in AppliedMechanics and Engineering vol 66 no 1 pp 107ndash119 1988

[12] M Eisenberger ldquoBuckling loads for variable cross-sectionmembers with variable axial forcesrdquo International Journal ofSolids and Structures vol 27 no 2 pp 135ndash143 1991

[13] J M Gere andW O Carter ldquoCritical buckling loads for taperedcolumnsrdquo Journal of Structural Engineering ASCE vol 88 no 1pp 1ndash11 1962

[14] F Arbabi and F Li ldquoBuckling of variable cross-section columnsintegral equation approachrdquo Journal of Structural EngineeringASCE vol 117 no 8 1991

[15] A Siginer ldquoBuckling of columns of variable flexural rigidityrdquoJournal of Engineering Mechanics ASCE vol 118 no 3 pp 543ndash640 1992

[16] Q Li H Cao and G Li ldquoStability analysis of a bar with multi-segments of varying cross-sectionrdquo Computers and Structuresvol 53 no 5 pp 1085ndash1089 1994

[17] L Qiusheng CHong and L Guiqing ldquoStability analysis of barswith varying cross-sectionrdquo International Journal of Solids andStructures vol 32 no 21 pp 3217ndash3228 1995

[18] Q Li H Cao and G Li ldquoStatic and dynamic analysis of straightbars with variable cross-sectionrdquoComputers and Structures vol59 no 6 pp 1185ndash1191 1996

[19] J H B Sampaio and J R Hundhausen ldquoA mathematical modeland analytical solution for buckling of inclined beam-columnsrdquoApplied Mathematical Modelling vol 22 no 6 pp 405ndash4211998

[20] J B Keller ldquoThe shape of the strongest columnrdquo Archive forRationalMechanics and Analysis vol 5 no 1 pp 275ndash285 1960

[21] I Tadjbakhsh and J B Keller ldquoStrongest columns and isoperi-metric inequalities for eigenvaluesrdquo Journal of Applied Mechan-ics ASME vol 29 pp 159ndash164 1962

[22] J E Taylor ldquoThe strongest columnmdashan energy approachrdquoJournal of Applied Mechanics ASME vol 34 pp 486ndash487 1967

[23] M T Atay and S B Coskun ldquoElastic stability of Euler columnswith a continuous elastic restraint using variational iterationmethodrdquoComputers andMathematics with Applications vol 58no 11-12 pp 2528ndash2534 2009

[24] S B Coskun andM T Atay ldquoDetermination of critical bucklingload for elastic columns of constant and variable cross-sectionsusing variational iteration methodrdquo Computers and Mathemat-ics with Applications vol 58 no 11-12 pp 2260ndash2266 2009

[25] M T Atay ldquoDetermination of buckling loads of tilted buckledcolumn with varying flexural rigidity using variational itera-tion methodrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 11 no 2 pp 97ndash103 2010

[26] F Okay M T Atay and S B Cockun ldquoDetermination of buck-ling loads and mode shapes of a heavy vertical column underits own weight using the variational iteration methodrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 11 no 10 pp 851ndash857 2010

8 Journal of Applied Mathematics

[27] S B Coskun ldquoDetermination of critical buckling loads for eulercolumns of variable flexural stiffness with a continuous elasticrestraint using homotopy perturbation methodrdquo InternationalJournal of Nonlinear Sciences and Numerical Simulation vol 10no 2 pp 191ndash197 2009

[28] MTAtay ldquoDetermination of critical buckling loads for variablestiffness euler columns using homotopy perturbation methodrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 10 no 2 pp 199ndash206 2009

[29] S B Coskun ldquoAnalysis of tilt-buckling of euler columns withvarying flexural stiffness using homotopy perturbation meth-odrdquoMathematicalModelling andAnalysis vol 15 no 3 pp 275ndash286 2010

[30] S Pinarbasi ldquoStability analysis of non-uniform rectangularbeams using homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2012 Article ID 197483 18 pages2012

[31] YHuang andQ Z Luo ldquoA simplemethod to determine the crit-ical buckling loads for axially inhomogeneous beams with elas-tic restraintrdquoComputers andMathematics withApplications vol61 no 9 pp 2510ndash2517 2011

[32] Z X Yuan and XWWang ldquoBuckling and post-buckling analy-sis of extensible beam-columns by using the differential quadra-ture methodrdquo Computers ampMathematics with Applications vol62 no 12 pp 4499ndash4513 2011

[33] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method CRC Press 2004

[34] S J Liao ldquoHomotopy analysis method a new analytic methodfor nonlinear problemsrdquo Applied Mathematics and Mechanicsvol 19 no 10 pp 957ndash962 1998

[35] S J Liao ldquoA uniformly valid analytic solution of two-dimen-sional viscous flow over a semi-infinite flat platerdquo Journal ofFluid Mechanics vol 385 pp 101ndash128 1999

[36] S J Liao ldquoAn explicit analytic solution to the Thomas-Fermiequationrdquo Applied Mathematics and Computation vol 144 no2-3 pp 495ndash506 2003

[37] T Hayat T Javed and M Sajid ldquoAnalytic solution for rotatingflow and heat transfer analysis of a third-grade fluidrdquo ActaMechanica vol 191 no 3-4 pp 219ndash229 2007

[38] T Hayat S B Khan M Sajid and S Asghar ldquoRotating flow of athird grade fluid in a porous space with Hall currentrdquoNonlinearDynamics vol 49 no 1-2 pp 83ndash91 2007

[39] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007

[40] S Abbasbandy and E Shivanian ldquoA new analytical technique tosolve Fredholmrsquos integral equationsrdquoNumerical Algorithms vol56 no 1 pp 27ndash43 2011

[41] S Abbasbandy and F S Zakaria ldquoSoliton solutions for the fifth-order KdV equation with the homotopy analysis methodrdquoNon-linear Dynamics vol 51 no 1-2 pp 83ndash87 2008

[42] Z Wang L Zou and H Zhang ldquoApplying homotopy analysismethod for solving differential-difference equationrdquo PhysicsLetters A vol 369 no 1-2 pp 77ndash84 2007

[43] A S Bataineh M S M Noorani and I Hashim ldquoSolutions oftime-dependent Emden-Fowler type equations by homotopyanalysis methodrdquo Physics Letters A vol 371 no 1-2 pp 72ndash822007

[44] M Inc ldquoOn exact solution of Laplace equation with Dirichletand Neumann boundary conditions by the homotopy analysismethodrdquo Physics Letters A vol 365 no 5-6 pp 412ndash415 2007

[45] F Awawdeh A Adawi and S Al-Shara ldquoAnalytic solution ofmulti pantograph equationrdquo Journal of Applied Mathematicsand Decision Sciences vol 2008 Article ID 605064 11 pages2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

Table 2 Comparison of exact and HAM solutions of the buckling loads for 120573 = 0 with ten iterations

Mode C-F column P-P column C-P column C-C column C-S columnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 24674 24674 98696 98696 201907 201907 394784 394784 98696 986962 222066 222066 394784 394779 596795 596813 807629 809336 394784 394784

Table 3 Comparison of exact and HAM solutions of the buckling loads for 120573 = 50 with ten iterations

Mode C-F Column P-P Column C-P Column C-C Column C-S ColumnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 88614 88614 149357 149357 242852 242852 432606 432606 235717 2357172 330879 330873 407449 407442 610966 611003 817943 820059 444104 444053

Table 4 Comparison of exact and HAM solutions of the buckling loads for 120573 = 100 with ten iterations

Mode C-F column P-P column C-P column C-C column C-S columnExact HAM Exact HAM Exact HAM Exact HAM Exact HAM

1 119964 119964 200017 200017 283066 283066 470066 470066 326690 3266902 452659 452574 420114 420105 625613 625678 828246 830859 528965 528105

Table 5 Absolute errors of HAM approximations for the first buckling mode

C-F column P-P column C-P column C-C column C-S column120573 = 0 130562119864 minus 13 103727119864 minus 12 000000 000000 539109119864 minus 11

120573 = 50 981515119864 minus 10 126009119864 minus 10 109165119864 minus 7 464191119864 minus 5 150249119864 minus 7

120573 = 100 304503119864 minus 8 165444119864 minus 8 737428119864 minus 7 59300119864 minus 7 181936119864 minus 5

Table 6 Percent relative errors of HAM approximations for the second buckling mode

C-F column P-P column C-P column C-C column C-S column120573 = 0 000000621 000124745 000303663 021128177 000008617120573 = 50 000163762 000173402 000599296 025870613 001135007120573 = 100 001874692 000227055 001038389 031551889 016247717

0

5

10

15

20

25

0 2 4 6 8 10

Nor

mal

ized

buc

klin

g lo

ad

Iteration number

Convergence for mode 1

β = 100

β = 50

β = 0

Figure 8 Convergence of solutions through the iterations for thefirst mode of P-P column

the following tables the buckling loads for the first twomodesare given and compared with exact results

From Tables 2ndash4 it is observed that normalized bucklingloads for the first mode which is the critical buckling load arein excellent agreement with analytical results Absolute errorsfor these cases are given in Table 5

The differences in the results of the second mode existdue to lack of iterations provided by the method Additionaliterations would improve the results for the second modeHowever the results for the second mode are still in goodagreement with the exact results Relative errors for this caseare provided in the following Table 6

These percent relative errors show that the presented solu-tion is in good agreement with analytical ones Convergenceof solutions for the PP column is simulated in Figures 8 and 9Solid lines show the exact solutions for different normalizedspring stiffnesses

Same convergence behaviors are observed for all casesconsidered in this study Furthermore Figure 8 shows that atleast 6 iterations are required for the first mode and Figure 9shows that at least 10 iterations are required for the secondmode to obtain satisfactory results from the analysis

5 Conclusions

In this work a reliable algorithm based on the HAM toobtain the normalized buckling loads of the Euler columns

Journal of Applied Mathematics 7

β = 100

β = 50

β = 0

36

38

40

42

44

0 2 4 6 8 10

Nor

mal

ized

buc

klin

g lo

ad

Iteration number

Convergence for mode 2

Figure 9 Convergence of solutions through the iterations for thesecond mode of P-P column

with constant flexural stiffness is presented Several cases aregiven to illustrate the validity and accuracy of this procedureThe series solutions of (3) by HAM contain the auxiliaryparameter ℎ In general by means of the so-called ℎ-curveit is straightforward to choose a proper value of ℎ whichensures that the series solution is convergent Figures 2 34 5 and 6 show the ℎ-curves obtained from the 119898th-orderHAM approximation solutions From these figures the validregions of ℎ correspond to the line segments nearly parallelto the horizontal axis By the use of the proper value withthis parameter two buckling loads for the first and secondmodes are obtained The method will provide the results forthe following modes if additional iterations are introducedin the analysis The buckling loads are positive real roots ofthe characteristic equationswhich are obtained consecutivelyThis is a huge advantage because it is still very difficult toobtain those roots consecutively even with a mathematicssoftware As a result HAM is an efficient powerful andaccurate tool for determining the buckling loads of Eulercolumns

References

[1] L Euler ldquoDie altitudine colomnarum sub proprio pondere cor-ruentiumrdquo Acta Academiae Scientiarum Imperialis Petropoli-tanae 1778 in Latin

[2] A G Greenhill ldquoDetermination of the greatest height consis-tent with stability that a vertical pole on mast can be made andof the greatest height to which a tree of given proportions cangrowrdquo in Proceedings of the Cambridge Philosophical Society IV1883

[3] A N Dinnik ldquoDesign of columns of varying cross-sectionrdquoTransactions of the ASME vol 51 1929

[4] T R Karman and M A Biot Mathematical Methods in Engi-neering McGraw Hill New York NY USA 1940

[5] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw Hill New York NY USA 1961

[6] C M Wang C Y Wang and J N Reddy Exact Solutions forBuckling of Structural Members CRC Press LLC Florida FlaUSA 2005

[7] H Bleich Buckling Strength of Metal Structures McGraw-HillNew York NY USA 1952

[8] A N Dinnik ldquoDesign of columns of varying cross-sectionrdquoTransactions of the ASME Applied Mechanics vol 51 1932

[9] I Elishakoff and F Pellegrini ldquoExact and effective approximatesolution solutions of some divergent type non-conservativeproblemsrdquo Journal of Sound and Vibration vol 114 pp 144ndash1481987

[10] I Elishakoff and F Pellegrini ldquoApplication of bessel and lommelfunctions and the undetermined multiplier galerkin methodversion for instability of non-uniform columnrdquo Journal ofSound and Vibration vol 115 no 1 pp 182ndash186 1987

[11] I Elishakoff and F Pellegrini ldquoExact solutions for buckling ofsome divergence-type nonconservative systems in terms of bes-sel and lommel functionsrdquo Computer Methods in AppliedMechanics and Engineering vol 66 no 1 pp 107ndash119 1988

[12] M Eisenberger ldquoBuckling loads for variable cross-sectionmembers with variable axial forcesrdquo International Journal ofSolids and Structures vol 27 no 2 pp 135ndash143 1991

[13] J M Gere andW O Carter ldquoCritical buckling loads for taperedcolumnsrdquo Journal of Structural Engineering ASCE vol 88 no 1pp 1ndash11 1962

[14] F Arbabi and F Li ldquoBuckling of variable cross-section columnsintegral equation approachrdquo Journal of Structural EngineeringASCE vol 117 no 8 1991

[15] A Siginer ldquoBuckling of columns of variable flexural rigidityrdquoJournal of Engineering Mechanics ASCE vol 118 no 3 pp 543ndash640 1992

[16] Q Li H Cao and G Li ldquoStability analysis of a bar with multi-segments of varying cross-sectionrdquo Computers and Structuresvol 53 no 5 pp 1085ndash1089 1994

[17] L Qiusheng CHong and L Guiqing ldquoStability analysis of barswith varying cross-sectionrdquo International Journal of Solids andStructures vol 32 no 21 pp 3217ndash3228 1995

[18] Q Li H Cao and G Li ldquoStatic and dynamic analysis of straightbars with variable cross-sectionrdquoComputers and Structures vol59 no 6 pp 1185ndash1191 1996

[19] J H B Sampaio and J R Hundhausen ldquoA mathematical modeland analytical solution for buckling of inclined beam-columnsrdquoApplied Mathematical Modelling vol 22 no 6 pp 405ndash4211998

[20] J B Keller ldquoThe shape of the strongest columnrdquo Archive forRationalMechanics and Analysis vol 5 no 1 pp 275ndash285 1960

[21] I Tadjbakhsh and J B Keller ldquoStrongest columns and isoperi-metric inequalities for eigenvaluesrdquo Journal of Applied Mechan-ics ASME vol 29 pp 159ndash164 1962

[22] J E Taylor ldquoThe strongest columnmdashan energy approachrdquoJournal of Applied Mechanics ASME vol 34 pp 486ndash487 1967

[23] M T Atay and S B Coskun ldquoElastic stability of Euler columnswith a continuous elastic restraint using variational iterationmethodrdquoComputers andMathematics with Applications vol 58no 11-12 pp 2528ndash2534 2009

[24] S B Coskun andM T Atay ldquoDetermination of critical bucklingload for elastic columns of constant and variable cross-sectionsusing variational iteration methodrdquo Computers and Mathemat-ics with Applications vol 58 no 11-12 pp 2260ndash2266 2009

[25] M T Atay ldquoDetermination of buckling loads of tilted buckledcolumn with varying flexural rigidity using variational itera-tion methodrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 11 no 2 pp 97ndash103 2010

[26] F Okay M T Atay and S B Cockun ldquoDetermination of buck-ling loads and mode shapes of a heavy vertical column underits own weight using the variational iteration methodrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 11 no 10 pp 851ndash857 2010

8 Journal of Applied Mathematics

[27] S B Coskun ldquoDetermination of critical buckling loads for eulercolumns of variable flexural stiffness with a continuous elasticrestraint using homotopy perturbation methodrdquo InternationalJournal of Nonlinear Sciences and Numerical Simulation vol 10no 2 pp 191ndash197 2009

[28] MTAtay ldquoDetermination of critical buckling loads for variablestiffness euler columns using homotopy perturbation methodrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 10 no 2 pp 199ndash206 2009

[29] S B Coskun ldquoAnalysis of tilt-buckling of euler columns withvarying flexural stiffness using homotopy perturbation meth-odrdquoMathematicalModelling andAnalysis vol 15 no 3 pp 275ndash286 2010

[30] S Pinarbasi ldquoStability analysis of non-uniform rectangularbeams using homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2012 Article ID 197483 18 pages2012

[31] YHuang andQ Z Luo ldquoA simplemethod to determine the crit-ical buckling loads for axially inhomogeneous beams with elas-tic restraintrdquoComputers andMathematics withApplications vol61 no 9 pp 2510ndash2517 2011

[32] Z X Yuan and XWWang ldquoBuckling and post-buckling analy-sis of extensible beam-columns by using the differential quadra-ture methodrdquo Computers ampMathematics with Applications vol62 no 12 pp 4499ndash4513 2011

[33] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method CRC Press 2004

[34] S J Liao ldquoHomotopy analysis method a new analytic methodfor nonlinear problemsrdquo Applied Mathematics and Mechanicsvol 19 no 10 pp 957ndash962 1998

[35] S J Liao ldquoA uniformly valid analytic solution of two-dimen-sional viscous flow over a semi-infinite flat platerdquo Journal ofFluid Mechanics vol 385 pp 101ndash128 1999

[36] S J Liao ldquoAn explicit analytic solution to the Thomas-Fermiequationrdquo Applied Mathematics and Computation vol 144 no2-3 pp 495ndash506 2003

[37] T Hayat T Javed and M Sajid ldquoAnalytic solution for rotatingflow and heat transfer analysis of a third-grade fluidrdquo ActaMechanica vol 191 no 3-4 pp 219ndash229 2007

[38] T Hayat S B Khan M Sajid and S Asghar ldquoRotating flow of athird grade fluid in a porous space with Hall currentrdquoNonlinearDynamics vol 49 no 1-2 pp 83ndash91 2007

[39] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007

[40] S Abbasbandy and E Shivanian ldquoA new analytical technique tosolve Fredholmrsquos integral equationsrdquoNumerical Algorithms vol56 no 1 pp 27ndash43 2011

[41] S Abbasbandy and F S Zakaria ldquoSoliton solutions for the fifth-order KdV equation with the homotopy analysis methodrdquoNon-linear Dynamics vol 51 no 1-2 pp 83ndash87 2008

[42] Z Wang L Zou and H Zhang ldquoApplying homotopy analysismethod for solving differential-difference equationrdquo PhysicsLetters A vol 369 no 1-2 pp 77ndash84 2007

[43] A S Bataineh M S M Noorani and I Hashim ldquoSolutions oftime-dependent Emden-Fowler type equations by homotopyanalysis methodrdquo Physics Letters A vol 371 no 1-2 pp 72ndash822007

[44] M Inc ldquoOn exact solution of Laplace equation with Dirichletand Neumann boundary conditions by the homotopy analysismethodrdquo Physics Letters A vol 365 no 5-6 pp 412ndash415 2007

[45] F Awawdeh A Adawi and S Al-Shara ldquoAnalytic solution ofmulti pantograph equationrdquo Journal of Applied Mathematicsand Decision Sciences vol 2008 Article ID 605064 11 pages2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

β = 100

β = 50

β = 0

36

38

40

42

44

0 2 4 6 8 10

Nor

mal

ized

buc

klin

g lo

ad

Iteration number

Convergence for mode 2

Figure 9 Convergence of solutions through the iterations for thesecond mode of P-P column

with constant flexural stiffness is presented Several cases aregiven to illustrate the validity and accuracy of this procedureThe series solutions of (3) by HAM contain the auxiliaryparameter ℎ In general by means of the so-called ℎ-curveit is straightforward to choose a proper value of ℎ whichensures that the series solution is convergent Figures 2 34 5 and 6 show the ℎ-curves obtained from the 119898th-orderHAM approximation solutions From these figures the validregions of ℎ correspond to the line segments nearly parallelto the horizontal axis By the use of the proper value withthis parameter two buckling loads for the first and secondmodes are obtained The method will provide the results forthe following modes if additional iterations are introducedin the analysis The buckling loads are positive real roots ofthe characteristic equationswhich are obtained consecutivelyThis is a huge advantage because it is still very difficult toobtain those roots consecutively even with a mathematicssoftware As a result HAM is an efficient powerful andaccurate tool for determining the buckling loads of Eulercolumns

References

[1] L Euler ldquoDie altitudine colomnarum sub proprio pondere cor-ruentiumrdquo Acta Academiae Scientiarum Imperialis Petropoli-tanae 1778 in Latin

[2] A G Greenhill ldquoDetermination of the greatest height consis-tent with stability that a vertical pole on mast can be made andof the greatest height to which a tree of given proportions cangrowrdquo in Proceedings of the Cambridge Philosophical Society IV1883

[3] A N Dinnik ldquoDesign of columns of varying cross-sectionrdquoTransactions of the ASME vol 51 1929

[4] T R Karman and M A Biot Mathematical Methods in Engi-neering McGraw Hill New York NY USA 1940

[5] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw Hill New York NY USA 1961

[6] C M Wang C Y Wang and J N Reddy Exact Solutions forBuckling of Structural Members CRC Press LLC Florida FlaUSA 2005

[7] H Bleich Buckling Strength of Metal Structures McGraw-HillNew York NY USA 1952

[8] A N Dinnik ldquoDesign of columns of varying cross-sectionrdquoTransactions of the ASME Applied Mechanics vol 51 1932

[9] I Elishakoff and F Pellegrini ldquoExact and effective approximatesolution solutions of some divergent type non-conservativeproblemsrdquo Journal of Sound and Vibration vol 114 pp 144ndash1481987

[10] I Elishakoff and F Pellegrini ldquoApplication of bessel and lommelfunctions and the undetermined multiplier galerkin methodversion for instability of non-uniform columnrdquo Journal ofSound and Vibration vol 115 no 1 pp 182ndash186 1987

[11] I Elishakoff and F Pellegrini ldquoExact solutions for buckling ofsome divergence-type nonconservative systems in terms of bes-sel and lommel functionsrdquo Computer Methods in AppliedMechanics and Engineering vol 66 no 1 pp 107ndash119 1988

[12] M Eisenberger ldquoBuckling loads for variable cross-sectionmembers with variable axial forcesrdquo International Journal ofSolids and Structures vol 27 no 2 pp 135ndash143 1991

[13] J M Gere andW O Carter ldquoCritical buckling loads for taperedcolumnsrdquo Journal of Structural Engineering ASCE vol 88 no 1pp 1ndash11 1962

[14] F Arbabi and F Li ldquoBuckling of variable cross-section columnsintegral equation approachrdquo Journal of Structural EngineeringASCE vol 117 no 8 1991

[15] A Siginer ldquoBuckling of columns of variable flexural rigidityrdquoJournal of Engineering Mechanics ASCE vol 118 no 3 pp 543ndash640 1992

[16] Q Li H Cao and G Li ldquoStability analysis of a bar with multi-segments of varying cross-sectionrdquo Computers and Structuresvol 53 no 5 pp 1085ndash1089 1994

[17] L Qiusheng CHong and L Guiqing ldquoStability analysis of barswith varying cross-sectionrdquo International Journal of Solids andStructures vol 32 no 21 pp 3217ndash3228 1995

[18] Q Li H Cao and G Li ldquoStatic and dynamic analysis of straightbars with variable cross-sectionrdquoComputers and Structures vol59 no 6 pp 1185ndash1191 1996

[19] J H B Sampaio and J R Hundhausen ldquoA mathematical modeland analytical solution for buckling of inclined beam-columnsrdquoApplied Mathematical Modelling vol 22 no 6 pp 405ndash4211998

[20] J B Keller ldquoThe shape of the strongest columnrdquo Archive forRationalMechanics and Analysis vol 5 no 1 pp 275ndash285 1960

[21] I Tadjbakhsh and J B Keller ldquoStrongest columns and isoperi-metric inequalities for eigenvaluesrdquo Journal of Applied Mechan-ics ASME vol 29 pp 159ndash164 1962

[22] J E Taylor ldquoThe strongest columnmdashan energy approachrdquoJournal of Applied Mechanics ASME vol 34 pp 486ndash487 1967

[23] M T Atay and S B Coskun ldquoElastic stability of Euler columnswith a continuous elastic restraint using variational iterationmethodrdquoComputers andMathematics with Applications vol 58no 11-12 pp 2528ndash2534 2009

[24] S B Coskun andM T Atay ldquoDetermination of critical bucklingload for elastic columns of constant and variable cross-sectionsusing variational iteration methodrdquo Computers and Mathemat-ics with Applications vol 58 no 11-12 pp 2260ndash2266 2009

[25] M T Atay ldquoDetermination of buckling loads of tilted buckledcolumn with varying flexural rigidity using variational itera-tion methodrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 11 no 2 pp 97ndash103 2010

[26] F Okay M T Atay and S B Cockun ldquoDetermination of buck-ling loads and mode shapes of a heavy vertical column underits own weight using the variational iteration methodrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 11 no 10 pp 851ndash857 2010

8 Journal of Applied Mathematics

[27] S B Coskun ldquoDetermination of critical buckling loads for eulercolumns of variable flexural stiffness with a continuous elasticrestraint using homotopy perturbation methodrdquo InternationalJournal of Nonlinear Sciences and Numerical Simulation vol 10no 2 pp 191ndash197 2009

[28] MTAtay ldquoDetermination of critical buckling loads for variablestiffness euler columns using homotopy perturbation methodrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 10 no 2 pp 199ndash206 2009

[29] S B Coskun ldquoAnalysis of tilt-buckling of euler columns withvarying flexural stiffness using homotopy perturbation meth-odrdquoMathematicalModelling andAnalysis vol 15 no 3 pp 275ndash286 2010

[30] S Pinarbasi ldquoStability analysis of non-uniform rectangularbeams using homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2012 Article ID 197483 18 pages2012

[31] YHuang andQ Z Luo ldquoA simplemethod to determine the crit-ical buckling loads for axially inhomogeneous beams with elas-tic restraintrdquoComputers andMathematics withApplications vol61 no 9 pp 2510ndash2517 2011

[32] Z X Yuan and XWWang ldquoBuckling and post-buckling analy-sis of extensible beam-columns by using the differential quadra-ture methodrdquo Computers ampMathematics with Applications vol62 no 12 pp 4499ndash4513 2011

[33] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method CRC Press 2004

[34] S J Liao ldquoHomotopy analysis method a new analytic methodfor nonlinear problemsrdquo Applied Mathematics and Mechanicsvol 19 no 10 pp 957ndash962 1998

[35] S J Liao ldquoA uniformly valid analytic solution of two-dimen-sional viscous flow over a semi-infinite flat platerdquo Journal ofFluid Mechanics vol 385 pp 101ndash128 1999

[36] S J Liao ldquoAn explicit analytic solution to the Thomas-Fermiequationrdquo Applied Mathematics and Computation vol 144 no2-3 pp 495ndash506 2003

[37] T Hayat T Javed and M Sajid ldquoAnalytic solution for rotatingflow and heat transfer analysis of a third-grade fluidrdquo ActaMechanica vol 191 no 3-4 pp 219ndash229 2007

[38] T Hayat S B Khan M Sajid and S Asghar ldquoRotating flow of athird grade fluid in a porous space with Hall currentrdquoNonlinearDynamics vol 49 no 1-2 pp 83ndash91 2007

[39] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007

[40] S Abbasbandy and E Shivanian ldquoA new analytical technique tosolve Fredholmrsquos integral equationsrdquoNumerical Algorithms vol56 no 1 pp 27ndash43 2011

[41] S Abbasbandy and F S Zakaria ldquoSoliton solutions for the fifth-order KdV equation with the homotopy analysis methodrdquoNon-linear Dynamics vol 51 no 1-2 pp 83ndash87 2008

[42] Z Wang L Zou and H Zhang ldquoApplying homotopy analysismethod for solving differential-difference equationrdquo PhysicsLetters A vol 369 no 1-2 pp 77ndash84 2007

[43] A S Bataineh M S M Noorani and I Hashim ldquoSolutions oftime-dependent Emden-Fowler type equations by homotopyanalysis methodrdquo Physics Letters A vol 371 no 1-2 pp 72ndash822007

[44] M Inc ldquoOn exact solution of Laplace equation with Dirichletand Neumann boundary conditions by the homotopy analysismethodrdquo Physics Letters A vol 365 no 5-6 pp 412ndash415 2007

[45] F Awawdeh A Adawi and S Al-Shara ldquoAnalytic solution ofmulti pantograph equationrdquo Journal of Applied Mathematicsand Decision Sciences vol 2008 Article ID 605064 11 pages2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

[27] S B Coskun ldquoDetermination of critical buckling loads for eulercolumns of variable flexural stiffness with a continuous elasticrestraint using homotopy perturbation methodrdquo InternationalJournal of Nonlinear Sciences and Numerical Simulation vol 10no 2 pp 191ndash197 2009

[28] MTAtay ldquoDetermination of critical buckling loads for variablestiffness euler columns using homotopy perturbation methodrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 10 no 2 pp 199ndash206 2009

[29] S B Coskun ldquoAnalysis of tilt-buckling of euler columns withvarying flexural stiffness using homotopy perturbation meth-odrdquoMathematicalModelling andAnalysis vol 15 no 3 pp 275ndash286 2010

[30] S Pinarbasi ldquoStability analysis of non-uniform rectangularbeams using homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2012 Article ID 197483 18 pages2012

[31] YHuang andQ Z Luo ldquoA simplemethod to determine the crit-ical buckling loads for axially inhomogeneous beams with elas-tic restraintrdquoComputers andMathematics withApplications vol61 no 9 pp 2510ndash2517 2011

[32] Z X Yuan and XWWang ldquoBuckling and post-buckling analy-sis of extensible beam-columns by using the differential quadra-ture methodrdquo Computers ampMathematics with Applications vol62 no 12 pp 4499ndash4513 2011

[33] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method CRC Press 2004

[34] S J Liao ldquoHomotopy analysis method a new analytic methodfor nonlinear problemsrdquo Applied Mathematics and Mechanicsvol 19 no 10 pp 957ndash962 1998

[35] S J Liao ldquoA uniformly valid analytic solution of two-dimen-sional viscous flow over a semi-infinite flat platerdquo Journal ofFluid Mechanics vol 385 pp 101ndash128 1999

[36] S J Liao ldquoAn explicit analytic solution to the Thomas-Fermiequationrdquo Applied Mathematics and Computation vol 144 no2-3 pp 495ndash506 2003

[37] T Hayat T Javed and M Sajid ldquoAnalytic solution for rotatingflow and heat transfer analysis of a third-grade fluidrdquo ActaMechanica vol 191 no 3-4 pp 219ndash229 2007

[38] T Hayat S B Khan M Sajid and S Asghar ldquoRotating flow of athird grade fluid in a porous space with Hall currentrdquoNonlinearDynamics vol 49 no 1-2 pp 83ndash91 2007

[39] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007

[40] S Abbasbandy and E Shivanian ldquoA new analytical technique tosolve Fredholmrsquos integral equationsrdquoNumerical Algorithms vol56 no 1 pp 27ndash43 2011

[41] S Abbasbandy and F S Zakaria ldquoSoliton solutions for the fifth-order KdV equation with the homotopy analysis methodrdquoNon-linear Dynamics vol 51 no 1-2 pp 83ndash87 2008

[42] Z Wang L Zou and H Zhang ldquoApplying homotopy analysismethod for solving differential-difference equationrdquo PhysicsLetters A vol 369 no 1-2 pp 77ndash84 2007

[43] A S Bataineh M S M Noorani and I Hashim ldquoSolutions oftime-dependent Emden-Fowler type equations by homotopyanalysis methodrdquo Physics Letters A vol 371 no 1-2 pp 72ndash822007

[44] M Inc ldquoOn exact solution of Laplace equation with Dirichletand Neumann boundary conditions by the homotopy analysismethodrdquo Physics Letters A vol 365 no 5-6 pp 412ndash415 2007

[45] F Awawdeh A Adawi and S Al-Shara ldquoAnalytic solution ofmulti pantograph equationrdquo Journal of Applied Mathematicsand Decision Sciences vol 2008 Article ID 605064 11 pages2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of