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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 136358 10 pageshttpdxdoiorg1011552013136358
Research ArticleA Numerical Approach to Static Deflection Analysisof an Infinite Beam on a Nonlinear Elastic FoundationOne-Way Spring Model
Jinsoo Park Hyeree Bai and T S Jang
Department of Naval Architecture and Ocean Engineering Pusan National UniversityBusan 609-735 Republic of Korea
Correspondence should be addressed to T S Jang taekpusanackr
Received 28 February 2013 Accepted 26 March 2013
Academic Editor Giuseppe Marino
Copyright copy 2013 Jinsoo Park et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A numerical procedure proposed by Jang et al (2011) is applied for the numerical analyzing of static deflection of an infinitebeam on a nonlinear elastic foundation And one-way spring model is used for the modeling of fully nonlinear elastic foundationThe nonlinear procedure involves Greenrsquos function technique and an iterative method using the pseudo spring coefficient Theworkability of the numerical procedure is demonstrated through showing the validity of the solution and the convergence test withsome external loads
1 Introduction
Accurate modeling of nonlinear deflection of an infinitebeam on a nonlinear elastic foundation is crucial for materialand structural engineering The research can be applied tostrength analysis and practical engineering design applica-tion say to curved plate manufacturing Therefore manytheoretical and experimental studies have been carried outon the nonlinear modeling of an infinite beam on a nonlinearelastic foundation
The closed-form solutions for the static and dynamicresponse of a uniform beam resting on a linear elastic foun-dation can be found in several references [1ndash3] Timoshenko[4] Kenney [5] Saito and Murakami [6] and Fryba [7]formulated a closed-form solution using Greenrsquos functionapproach based on a linear assumption Beaufait andHoadley[8] Massalas [9] Lakshmanan [10] and Hui [11] proposedthe static dynamic and elastic stability analysis of a beamresting on a nonlinear elastic foundation And there aremanyresearches concerning the linear elastic foundation [12ndash15]Among the references Beaufait and Hoadley [8] approxi-mated the relationship of the stress-strain curve to be hyper-bolic but they approximated the bilinear curve to handlethe nonlinear problem The applied nonlinear foundation is
active only when the beam is pressing against the foundationand it is assumed to be inactive in the regions where the beamhas been displaced away from the foundation Soldatos andSelvadurai [16] also applied the hyperbolic-type nonlinearelastic foundation to analyze finite or infinite beams Leeet al [17ndash19] developed the exact and semiexact analysisof a nonuniform beam with general elastic end-restraintsKuo and Lee [20] derived the static deflection of a generalelastically end-restrained nonuniform beam on a nonlinearelastic foundation under axial and transverse forces
Recently Jang et al [21] proposed a new method forassessing the nonlinear deflection of an infinite beam ona nonlinear elastic foundation They approach the highnonlinear problems using Greenrsquos function technique withan iterative method Jang and Sung [22] proposed a newfunctional iterative method for static beam deflection whichhas a variable cross-section Choi and Jang [23] proved theexistence and uniqueness of the nonlinear deflections of aninfinite beam resting on a nonlinear elastic foundation usingthe Banach fixed point theorem Jang [24] also proposed anew iterative method for the large deflection of an infinitebeam resting on an elastic foundation based on the v Karmanapproximation of geometrical nonlinearity From the existing
2 Journal of Applied Mathematics
0Deflection 119906
Non
linea
r spr
ing
forc
e119891(119906)
One wayTwo way
Figure 1 Nonlinear elastic foundation model a nonlinear spring model (one way) and a conventional one (two way)
119891(119906)
119906
119909
Euler-Bernoulli beam
119906(119909)
119908(119909)
Separable
Figure 2 An infinite beam on a nonlinear elastic foundation a nonlinear spring model
literature a number of studies have analyzed a beam onan elastic foundation however they just use linear plus anonlinear term of spring force that is linear-cubic modelAnd they are related to the static analysis of nonuniformbeams which is resting on a nonlinear elastic foundation andthe recovered solution is not accurate or hasmany limits Fewstudies have fully adopted the nonlinear elastic foundationmodel whose spring force is based on one-way springmodelas shown in Figure 1 In the real world at the steady state thesoil or foundation would not be raised or they are separable(in Figures 1 and 2) A nonlinear spring force exists when theinfinite beam deflects downward but does not exist in case ofthe other cases
Although there are many researches fully nonlinearelastic foundation was not considered Beaufait and Hoadley[8] and Soldatos and Selvadurai [16] approximated the stress-strain relationship as a bilinear curve In this paper one-wayspring model is successfully used to examine the real non-linear elastic foundation and the nonlinear iterative methodproposed by Jang et al [21] is applied Some numerical exper-iments are carried out to report the accuracy of the methodand the convergence of the solution is investigated accordingto several physical properties of the system
2 Mathematical Modeling
21 Euler-Bernoullirsquos Beam on a Nonlinear Elastic FoundationANonlinear SpringModel In this paper the nonlinear springforce is fully analyzed by the one-way spring model instead ofthe conventional mathematical form of the two-way springmodel [25]
The well-known classical Euler-Bernoullirsquos beam the-ory is considered for the solution procedure which is a
simplification of elasticitywhich provides ameans of calculat-ing the load-carrying and deflection characteristics of beamsThe governing equation for the linear deflection of an infinitebeam on an elastic foundation that satisfies the fourth-orderdifferential equation is as follows (the weight of the beam isneglected)
1198641198681198894119906
1198891199094+ 119891 (119906) = 119908 (119909) (1)
And the reaction force 119891(119906)119891 (119906) = 119896 sdot 119906 + 119873 (119906) (2)
and 119864 119868 119896 119873(119906) and 119908(119909) are Youngrsquos modulus the massmoment of inertia a linear spring coefficient a nonlinear partof spring force and external load respectively
The boundary
119906119889119906
1198891199091198892119906
1198891199092 and 119889
3119906
1198891199093997888rarr 0 as |119909| 997888rarr infin (3)
Therefore (1) and (3) together form a well-defined boundaryvalue problem Timoshenko [4] Kenney [5] Saito andMurakami [6] and Fryba [7] derived the general linearsolutions neglecting the nonlinear part 119873(119906) in (2) asfollows
119906 (119909) = int
infin
minusinfin
119866 (119909 120585 119896) sdot 119908 (120585) 119889120585 (4)
where Greenrsquos function 119866 can be defined as
119866 (119909 120585 119896) =120572
2119896119890minus120572|120585minus119909|radic2
times sin(1205721003816100381610038161003816120585 minus 119909
1003816100381610038161003816
radic2
+120587
4) 120572 =
4radic119896119864119868
(5)
Journal of Applied Mathematics 3
minus10 minus5 0 5 10
minus05
0
05
1
119909119906(119909)
Case aCase bCase c
Figure 3 Exact solutions in Table 1
minus20
minus10
0
10
20
119908(119909)
minus10 minus5 0 5 10119909
Case aCase bCase c
Figure 4 Applied loading conditions corresponding to the exact solution in Table 1
The loading condition is assumed to be localized so 119906(119909)
in (4) satisfies the boundary conditions in (3) Deriving thelinear solution in (4) a uniformly upward nonlinear springforce depends on the beam deflection 119906
In this study the main idea of the present study is pro-posed by Jang et al [21] The nonlinear spring model is usedfor the formulation of a realistic nonlinear elastic foundationA pseudo linear spring coefficient 119896
119901and a (real) spring
force 119891(119906) are used Therefore the fourth-order differentialequation in (1) is equivalent to the following equation
1198641198681198894119906
1198891199094+ 119896119901119906 + 119891 (119906) = 119908 (119909) + 119896
119901119906 (6)
where the nonlinear spring force119891 depends on the deflection119906
119891 (119906) = 119896 sdot 119906 + 119873 (119906) for 119906 ge 00 for 119906 lt 0
(7)
or
1198641198681198894119906
1198891199094+ 119896119901119906 = 119908 (119909) + 119896
119901119906 minus 119891 (119906) equiv 120601 (8)
Table 1 Three cases of the exact solution
Case Exact solution 119906(119909)a 119890
minus1199092
b Sin119909 sdot 119890minus1199092
c Sin 119909 sdot 119890minus11990924
In (8) 119896119901119906 is the pseudolinear spring force term and is finally
compensated so it does not affect the nonlinear solutionTherefore (4) shows that (8) must be equivalent to the
following equation
119906 (119909) = int
infin
minusinfin
119866(119909 120585 119896119901) sdot 120601 (120585) 119889120585 (9)
where Greenrsquos function with pseudo linear spring coefficientcan be expressed as
119866(119909 120585 119896119901) =
120573
2119896119890minus120573|120585minus119909|radic2
times sin(1205731003816100381610038161003816120585 minus 119909
1003816100381610038161003816
radic2
+120587
4) 120573 =
4radic119896119901119864119868
(10)
4 Journal of Applied Mathematics
minus10 minus5 0 5 10
0
02
04
06
08
1
119909
119906(119909)
ExactNumerical
(a)
ExactNumerical
minus10 minus5 0 5 10119909
119906(119909)
minus04
minus02
0
02
04
(b)
ExactNumerical
minus10 minus5 0 5 10119909
119906(119909)
minus05
0
05
(c)
Figure 5 Numerical solutions compared to the exact solutions
Regarding (8) the nonlinear relation for 119906 can be derived asfollows
119906 (119909) = int
infin
minusinfin
119866(119909 120585 119896119901) sdot 119908 (120585) 119889120585
+ int
infin
minusinfin
119866(119909 120585 119896119901) sdot 119870 [119906 (120585)] 119889120585
(11)
where the function119870 can be written as
119870 (119906) = 119896119901sdot 119906 minus 119891 (119906) (12)
Equation (11) is a nonlinear Fredholm integral equation of thesecond kind for 119906
22 Iterative Procedure From (11) the nonlinear iterativeprocedure is applied [21ndash24]
119906119899+1
(119909) = int
infin
minusinfin
119866(119909 120585 119896119901) sdot 119908 (120585) 119889120585
+ int
infin
minusinfin
119866(119909 120585 119896119901) sdot 119870 [119906
119899(120585)] 119889120585
(13)
where119870(119906) satisfies (7) and (12)
To examine the nonlinear iterative procedure in (13) bysimulation it should be discretized as follows
119906119899+1
(119909) =
Nsum
119895=1
119882119895119866 (119909 120585
119895 119896119901) sdot 119908 (120585
119895)
+ 119866 (119909 120585 119896119901) sdot 119870 [119906
119899(120585119895)]
119895 = 0 1 2 119873
(14)
where 119882119895denotes the weights for the integration rule The
number 119873 in the summation of (14) denotes the totalsegments of the interval (minus119877 119877) and 119877 is a sufficiently largevalue satisfying (3)
3 Numerical Experiments
In this section numerical experiments are performed todetermine the validity of the iterative method This studyassumes a nonlinear spring force 119891(119906) in (7) and examineshow accurate the solution converges to an exact solutionTheconvergence of themethod is investigated with some externalloading conditions
Journal of Applied Mathematics 5
0
05
1
119899 = 1
119899 = 2
119899 = 3
119899 = 300
Exact solution
minus10 minus5 0 5 10119909
119906(119909)
(a)
119899 = 1
119899 = 5
119899 = 10
119899 = 300
Exact solution
minus10 minus5 0 5 10119909
119906(119909)
minus04
minus02
0
02
04
(b)
119899 = 1
119899 = 5
119899 = 10
119899 = 300
Exact solution
minus10 minus5 0 5 10119909
119906(119909)
5 50minus
05
0
minus05
(c)
Figure 6 Convergence behaviors of the iterative solutions
31 Nonlinear Spring Model For simplicity the present studyconsiders an infinite beam on a nonlinear elastic foundationwhose spring force is derived as follows
119891 (119906) = 119896 sdot 119906 + 120574 sdot 119906
3 for 119906 ge 0
0 for 119906 lt 0(15)
where119873(119906) in (7) is chosen immediately as a cubic form
119873(119906) = 120574 sdot 1199063 (16)
32 Comparison with Exact Solution To determine if theiterative method converges to an exact solution the 3 casesof exact solutions listed in Table 1 are first assumed and areillustrated in Figure 3 These cases are chosen to be infinitelydifferentiable and satisfy the conditions in (3) The followingare assumed 119864 = 119868 = 119896 = 1 119896
119901= 3 and 120574 = 02 and the
initial guess of the deflection is 1199060= 0 Simpsonrsquos integration
rule is applied to the numerical integration Three cases ofexternal loads shown in Figure 4 are obtained by substitut-ing the exact solutions in Table 1 to (6) Under the load-ing condition the nonlinear iterative method in Section 2
is applied to obtain the solution Figure 5 compares the exactand numerical solutions and Figure 6 shows the convergencebehavior of the solutions in Figure 5 The errors of the solu-tions at the 119899th iteration are defined as follows
Error (119899) =1003817100381710038171003817119906exact minus 119906119899
100381710038171003817100381721003817100381710038171003817119906exact
10038171003817100381710038172
where 1199112equiv (
119873
sum
119894=1
10038161003816100381610038161199111198941003816100381610038161003816
2
)
12
(17)Three cases of Error (119899) are plotted as a function of theiteration number in Figure 7 The solutions converge at 200ndash300 iterations
33 Convergence of the Procedure The accuracy of theapplied iterative method for the nonlinear spring model isproven in Section 32 In this section 2 cases of loadingconditions are investigated to show the convergence of thesolutions
The locally distributed rectangular-type loadings inFigure 8 are taken
119908normal (single) (119909) = 1 |119909| le 1
0 otherwise(18)
6 Journal of Applied Mathematics
100
101
102
103
0
02
04
06
The number of iterations (119899)
Erro
r (119899
)
(a)
100
101
102
103
The number of iterations (119899)
Erro
r (119899
)
minus01
0
01
02
03
04
05
(b)
100
101
102
103
0
02
04
06
The number of iterations (119899)
Erro
r (119899
)
(c)
Figure 7 Errors between the exact solutions and iterative solutions
minus10 minus5 0 5 10119909
minus1
0
1
2
3
119908no
rmal
(single)(119909)
Figure 8 Applied loading
Figure 9 presents the convergence behavior of the solutionsaccording to the 3 cases of 119896
119901 119864 = 119868 = 119896 = 1
and 120574 = 02 As specified in Section 21 119896119901does not
affect the converged iterative solutions but the convergencecharacteristics changed The solution converged to a steadystate faster when 119896
119901is small
In Figure 10 two cases of converged solutions are com-pared the numerical results using the one-way and two-wayspring models The loading condition is 119908normal (single)(119909) in(18)They are different only when the beamdeflected upwardthat is the deflections near 119909 = plusmn5 are larger when the beamis separable (or free) from the foundation whereas the one-way spring model is not Of course at 119909 = 0 the solutions areequivalent because the physical system is the same in Figure 1
for 119906 gt 0 Therefore Figure 10 shows the validity of theapplied iterative method for the nonlinear spring model
Another loading condition in Figure 11 is taken as follows
119908normal (119909) = 1 4 le |119909| le 5 1 le |119909| le 2
0 otherwise(19)
Other numerical experiments are also conducted accordingto the 3 cases of a linear spring coefficient 119896 whereas theproperties (119864 = 119868 = 119896 = 1 and 120574 = 02) are fixed andthe solution converged (Figure 12) Figure 13 compares thesolutions from the one-way and two-way spring models 119864 =
119868 = 119896 = 1 120574 = 02 and 119896119901= 3
Journal of Applied Mathematics 7
minus02
0
02
04
06
08
119899 = 1
119899 = 2
119899 = 5
119899 = 10
119899 = 30
minus20 minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
(a)
minus02
0
02
04
06
08
119899 = 1
119899 = 3
119899 = 10
119899 = 30
119899 = 60
minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
minus20
(b)
minus02
0
02
04
06
08
119899 = 1
119899 = 3
119899 = 10
119899 = 40
119899 = 90
minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
minus20
(c)
Figure 9 Convergence behavior of the applied iterative method 119864 = 119868 = 119896 = 1 120574 = 02 (a) 119896119901= 3 (b) 119896
119901= 6 and (c) 119896
119901= 10
minus20 minus10 0 10 20119909
119906(119909)
One wayTwo way
0
02
04
06
08
minus02
(a)
One wayTwo way
minus5 0 5
minus004
minus002
0
002
004
006
119909
119906(119909)
(b)
Figure 10 Validity of the solution
8 Journal of Applied Mathematics
minus10 minus5 0 5 10119909
minus1
0
1
2
3
119908no
rmal(119909)
Figure 11 Applied loading conditions
119899 = 1
119899 = 2
119899 = 4
119899 = 20
119899 = 40
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(a)
119899 = 1
119899 = 2
119899 = 3
119899 = 10
119899 = 20
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(b)
119899 = 1
119899 = 2
119899 = 3
119899 = 5
119899 = 10
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(c)
Figure 12 Convergence behavior of the solutions 119864 = 119868 = 1 120574 = 02 and 119896119901= 3 (a) 119896 = 1 (b) 119896 = 15 and (c) 119896 = 2
Finally the validity and accuracy of the applied iterativeprocedure are investigated The nonlinear spring force isconsidered and the iterative procedure is applied successfullyfor the solution Convergence of the deflections according tothe external loading conditions is observed for the effects ofthe pseudo 119896
119901and real 119896
4 Conclusion
In this work we succeeded in applying the numerical methodproposed by Jang et al [21] to find the static deflection ofan infinite beam on a full nonlinear elastic foundation Forthat one-way spring model is considered for the formulation
Journal of Applied Mathematics 9
minus20 minus10 0 10 20119909
One wayTwo way
119906(119909)
0
02
04
06
(a)
One wayTwo way
119909
119906(119909)
minus15 minus10 minus5 0 5 10 15minus003
minus002
minus001
0
001
002
(b)
Figure 13 Validity of the solution in Figure 12(a)
of the nonlinear elastic one Since the problem concernsone-way spring force the governing equation for the staticbeam deflection is transformed as in (11) For the solution aniterative procedure is applied for the calculation of the highnonlinearity Some numerical experiments are carried out forshowing the validity and the fast convergence of the appliednumerical method And we can also find that the resultsconverge to the solutions fast for certain external loads
Acknowledgment
This work was supported by a 2-year research grant of PusanNational University
References
[1] M Hetenyi Beams on Elastic Foundation The University ofMichigan Press Ann Arbor Mich USA 1946
[2] C Miranda and K Nair ldquoFinite beams on elastic foundationrdquoJournal of the Structural Division vol 92 pp 131ndash142 1966
[3] B Y Ting ldquoFinite beams on elastic foundation with restraintsrdquoJournal of the Structural Division vol 108 no 3 pp 611ndash6211982
[4] S Timoshenko ldquoMethod of analysis of statistical and dynamicalstress in railrdquo in Proceeding of the International Congress ofApplied Mechanics pp 407ndash418 Zurich Switzerland 1926
[5] J T Kenney ldquoSteady-state vibrations of beam on elastic founda-tion for moving loadrdquo Journal of Applied Mechanics vol 21 pp359ndash364 1954
[6] H Saito and TMurakami ldquoVibrations of an infinite beam on anelastic foundation with consideration of mass of a foundationrdquoThe Japan Society of Mechanical Engineers vol 12 pp 200ndash2051969
[7] L Fryba ldquoInfinitely long beam on elastic foundation undermoving loadrdquo Aplikace Matematiky vol 2 pp 105ndash132 1957
[8] F W Beaufait and P W Hoadley ldquoAnalysis of elastic beams onnonlinear foundationsrdquo Computers amp Structures vol 12 no 5pp 669ndash676 1980
[9] C Massalas ldquoFundamental frequency of vibration of a beam ona non-linear elastic foundationrdquo Journal of Sound andVibrationvol 54 no 4 pp 613ndash615 1977
[10] M Lakshmanan ldquoComment on the fundamental frequency ofvibration of a beam on a non-linear elastic foundationrdquo Journalof Sound and Vibration vol 58 no 3 pp 455ndash457 1978
[11] D Hui ldquoPostbuckling behavior of infinite beams on elasticfoundations using Koiterrsquos improved theoryrdquo International Jour-nal of Non-Linear Mechanics vol 23 no 2 pp 113ndash123 1988
[12] D Z Yankelevsky and M Eisenberger ldquoAnalysis of a beam col-umn on elastic foundationrdquoComputers amp Structures vol 23 no3 pp 351ndash356 1986
[13] M Eisenberger and J Clastornik ldquoBeams on variable two-parameter elastic foundationrdquo Journal of EngineeringMechanicsvol 113 no 10 pp l454ndash1466 1987
[14] D Karamanlidis and V Prakash ldquoExact transfer and stiffnessmatrices for a beamcolumn resting on a two-parameter foun-dationrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 72 no 1 pp 77ndash89 1989
[15] S N Sirosh A Ghali and A G Razaqpur ldquoA general finite ele-ment for beams or beam-columns with or without an elasticfoundationrdquo International Journal for Numerical Methods inEngineering vol 28 no 5 pp 1061ndash1076 1989
[16] K P Soldatos and A P S Selvadurai ldquoFlexure of beams rest-ing on hyperbolic elastic foundationsrdquo International Journal ofSolids amp Structures vol 21 no 4 pp 373ndash388 1985
[17] S Y Lee H Y Ke and Y H Kuo ldquoExact static deflection ofa non-uniform bernoulli-euler beam with general elastic endrestraintsrdquoComputersampStructures vol 36 no 1 pp 91ndash97 1990
[18] S Y Lee and H Y Ke ldquoFree vibrations of a non-uniformbeam with general elastically restrained boundary conditionsrdquoJournal of Sound andVibration vol 136 no 3 pp 425ndash437 1990
[19] S Y Lee H Y Ke and Y H Kuo ldquoAnalysis of non-uniformbeam vibrationrdquo Journal of Sound and Vibration vol 142 no1 pp 15ndash29 1990
[20] Y H Kuo and S Y Lee ldquoDeflection of nonuniform beams rest-ing on a nonlinear elastic foundationrdquo Computers amp Structuresvol 51 no 5 pp 513ndash519 1994
[21] T S Jang H S Baek and J K Paik ldquoA newmethod for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundationrdquo International Journal of Non-LinearMechanics vol 46 no 1 pp 339ndash346 2011
[22] T S Jang and H G Sung ldquoA new semi-analytical method forthe non-linear static analysis of an infinite beam on a non-linear
10 Journal of Applied Mathematics
elastic foundation a general approach to a variable beam cross-sectionrdquo International Journal of Non-Linear Mechanics vol 47pp 132ndash139 2012
[23] S W Choi and T S Jang ldquoExistence and uniqueness of non-linear deflections of an infinite beam resting on a non-uniformnonlinear elastic foundationrdquo Boundary Value Problems vol2012 article 5 2012
[24] T S Jang ldquoA new semi-analytical approach to large deflectionsof BernoullindashEuler-v Karman beams on a linear elastic founda-tion nonlinear analysis of infinite beamsrdquo International Journalof Mechanical Sciences vol 66 pp 22ndash32 2013
[25] M D Greenberg Foundations of Applied Mathematics Pren-tice-Hall New Jersey NJ USA 1978
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Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
0Deflection 119906
Non
linea
r spr
ing
forc
e119891(119906)
One wayTwo way
Figure 1 Nonlinear elastic foundation model a nonlinear spring model (one way) and a conventional one (two way)
119891(119906)
119906
119909
Euler-Bernoulli beam
119906(119909)
119908(119909)
Separable
Figure 2 An infinite beam on a nonlinear elastic foundation a nonlinear spring model
literature a number of studies have analyzed a beam onan elastic foundation however they just use linear plus anonlinear term of spring force that is linear-cubic modelAnd they are related to the static analysis of nonuniformbeams which is resting on a nonlinear elastic foundation andthe recovered solution is not accurate or hasmany limits Fewstudies have fully adopted the nonlinear elastic foundationmodel whose spring force is based on one-way springmodelas shown in Figure 1 In the real world at the steady state thesoil or foundation would not be raised or they are separable(in Figures 1 and 2) A nonlinear spring force exists when theinfinite beam deflects downward but does not exist in case ofthe other cases
Although there are many researches fully nonlinearelastic foundation was not considered Beaufait and Hoadley[8] and Soldatos and Selvadurai [16] approximated the stress-strain relationship as a bilinear curve In this paper one-wayspring model is successfully used to examine the real non-linear elastic foundation and the nonlinear iterative methodproposed by Jang et al [21] is applied Some numerical exper-iments are carried out to report the accuracy of the methodand the convergence of the solution is investigated accordingto several physical properties of the system
2 Mathematical Modeling
21 Euler-Bernoullirsquos Beam on a Nonlinear Elastic FoundationANonlinear SpringModel In this paper the nonlinear springforce is fully analyzed by the one-way spring model instead ofthe conventional mathematical form of the two-way springmodel [25]
The well-known classical Euler-Bernoullirsquos beam the-ory is considered for the solution procedure which is a
simplification of elasticitywhich provides ameans of calculat-ing the load-carrying and deflection characteristics of beamsThe governing equation for the linear deflection of an infinitebeam on an elastic foundation that satisfies the fourth-orderdifferential equation is as follows (the weight of the beam isneglected)
1198641198681198894119906
1198891199094+ 119891 (119906) = 119908 (119909) (1)
And the reaction force 119891(119906)119891 (119906) = 119896 sdot 119906 + 119873 (119906) (2)
and 119864 119868 119896 119873(119906) and 119908(119909) are Youngrsquos modulus the massmoment of inertia a linear spring coefficient a nonlinear partof spring force and external load respectively
The boundary
119906119889119906
1198891199091198892119906
1198891199092 and 119889
3119906
1198891199093997888rarr 0 as |119909| 997888rarr infin (3)
Therefore (1) and (3) together form a well-defined boundaryvalue problem Timoshenko [4] Kenney [5] Saito andMurakami [6] and Fryba [7] derived the general linearsolutions neglecting the nonlinear part 119873(119906) in (2) asfollows
119906 (119909) = int
infin
minusinfin
119866 (119909 120585 119896) sdot 119908 (120585) 119889120585 (4)
where Greenrsquos function 119866 can be defined as
119866 (119909 120585 119896) =120572
2119896119890minus120572|120585minus119909|radic2
times sin(1205721003816100381610038161003816120585 minus 119909
1003816100381610038161003816
radic2
+120587
4) 120572 =
4radic119896119864119868
(5)
Journal of Applied Mathematics 3
minus10 minus5 0 5 10
minus05
0
05
1
119909119906(119909)
Case aCase bCase c
Figure 3 Exact solutions in Table 1
minus20
minus10
0
10
20
119908(119909)
minus10 minus5 0 5 10119909
Case aCase bCase c
Figure 4 Applied loading conditions corresponding to the exact solution in Table 1
The loading condition is assumed to be localized so 119906(119909)
in (4) satisfies the boundary conditions in (3) Deriving thelinear solution in (4) a uniformly upward nonlinear springforce depends on the beam deflection 119906
In this study the main idea of the present study is pro-posed by Jang et al [21] The nonlinear spring model is usedfor the formulation of a realistic nonlinear elastic foundationA pseudo linear spring coefficient 119896
119901and a (real) spring
force 119891(119906) are used Therefore the fourth-order differentialequation in (1) is equivalent to the following equation
1198641198681198894119906
1198891199094+ 119896119901119906 + 119891 (119906) = 119908 (119909) + 119896
119901119906 (6)
where the nonlinear spring force119891 depends on the deflection119906
119891 (119906) = 119896 sdot 119906 + 119873 (119906) for 119906 ge 00 for 119906 lt 0
(7)
or
1198641198681198894119906
1198891199094+ 119896119901119906 = 119908 (119909) + 119896
119901119906 minus 119891 (119906) equiv 120601 (8)
Table 1 Three cases of the exact solution
Case Exact solution 119906(119909)a 119890
minus1199092
b Sin119909 sdot 119890minus1199092
c Sin 119909 sdot 119890minus11990924
In (8) 119896119901119906 is the pseudolinear spring force term and is finally
compensated so it does not affect the nonlinear solutionTherefore (4) shows that (8) must be equivalent to the
following equation
119906 (119909) = int
infin
minusinfin
119866(119909 120585 119896119901) sdot 120601 (120585) 119889120585 (9)
where Greenrsquos function with pseudo linear spring coefficientcan be expressed as
119866(119909 120585 119896119901) =
120573
2119896119890minus120573|120585minus119909|radic2
times sin(1205731003816100381610038161003816120585 minus 119909
1003816100381610038161003816
radic2
+120587
4) 120573 =
4radic119896119901119864119868
(10)
4 Journal of Applied Mathematics
minus10 minus5 0 5 10
0
02
04
06
08
1
119909
119906(119909)
ExactNumerical
(a)
ExactNumerical
minus10 minus5 0 5 10119909
119906(119909)
minus04
minus02
0
02
04
(b)
ExactNumerical
minus10 minus5 0 5 10119909
119906(119909)
minus05
0
05
(c)
Figure 5 Numerical solutions compared to the exact solutions
Regarding (8) the nonlinear relation for 119906 can be derived asfollows
119906 (119909) = int
infin
minusinfin
119866(119909 120585 119896119901) sdot 119908 (120585) 119889120585
+ int
infin
minusinfin
119866(119909 120585 119896119901) sdot 119870 [119906 (120585)] 119889120585
(11)
where the function119870 can be written as
119870 (119906) = 119896119901sdot 119906 minus 119891 (119906) (12)
Equation (11) is a nonlinear Fredholm integral equation of thesecond kind for 119906
22 Iterative Procedure From (11) the nonlinear iterativeprocedure is applied [21ndash24]
119906119899+1
(119909) = int
infin
minusinfin
119866(119909 120585 119896119901) sdot 119908 (120585) 119889120585
+ int
infin
minusinfin
119866(119909 120585 119896119901) sdot 119870 [119906
119899(120585)] 119889120585
(13)
where119870(119906) satisfies (7) and (12)
To examine the nonlinear iterative procedure in (13) bysimulation it should be discretized as follows
119906119899+1
(119909) =
Nsum
119895=1
119882119895119866 (119909 120585
119895 119896119901) sdot 119908 (120585
119895)
+ 119866 (119909 120585 119896119901) sdot 119870 [119906
119899(120585119895)]
119895 = 0 1 2 119873
(14)
where 119882119895denotes the weights for the integration rule The
number 119873 in the summation of (14) denotes the totalsegments of the interval (minus119877 119877) and 119877 is a sufficiently largevalue satisfying (3)
3 Numerical Experiments
In this section numerical experiments are performed todetermine the validity of the iterative method This studyassumes a nonlinear spring force 119891(119906) in (7) and examineshow accurate the solution converges to an exact solutionTheconvergence of themethod is investigated with some externalloading conditions
Journal of Applied Mathematics 5
0
05
1
119899 = 1
119899 = 2
119899 = 3
119899 = 300
Exact solution
minus10 minus5 0 5 10119909
119906(119909)
(a)
119899 = 1
119899 = 5
119899 = 10
119899 = 300
Exact solution
minus10 minus5 0 5 10119909
119906(119909)
minus04
minus02
0
02
04
(b)
119899 = 1
119899 = 5
119899 = 10
119899 = 300
Exact solution
minus10 minus5 0 5 10119909
119906(119909)
5 50minus
05
0
minus05
(c)
Figure 6 Convergence behaviors of the iterative solutions
31 Nonlinear Spring Model For simplicity the present studyconsiders an infinite beam on a nonlinear elastic foundationwhose spring force is derived as follows
119891 (119906) = 119896 sdot 119906 + 120574 sdot 119906
3 for 119906 ge 0
0 for 119906 lt 0(15)
where119873(119906) in (7) is chosen immediately as a cubic form
119873(119906) = 120574 sdot 1199063 (16)
32 Comparison with Exact Solution To determine if theiterative method converges to an exact solution the 3 casesof exact solutions listed in Table 1 are first assumed and areillustrated in Figure 3 These cases are chosen to be infinitelydifferentiable and satisfy the conditions in (3) The followingare assumed 119864 = 119868 = 119896 = 1 119896
119901= 3 and 120574 = 02 and the
initial guess of the deflection is 1199060= 0 Simpsonrsquos integration
rule is applied to the numerical integration Three cases ofexternal loads shown in Figure 4 are obtained by substitut-ing the exact solutions in Table 1 to (6) Under the load-ing condition the nonlinear iterative method in Section 2
is applied to obtain the solution Figure 5 compares the exactand numerical solutions and Figure 6 shows the convergencebehavior of the solutions in Figure 5 The errors of the solu-tions at the 119899th iteration are defined as follows
Error (119899) =1003817100381710038171003817119906exact minus 119906119899
100381710038171003817100381721003817100381710038171003817119906exact
10038171003817100381710038172
where 1199112equiv (
119873
sum
119894=1
10038161003816100381610038161199111198941003816100381610038161003816
2
)
12
(17)Three cases of Error (119899) are plotted as a function of theiteration number in Figure 7 The solutions converge at 200ndash300 iterations
33 Convergence of the Procedure The accuracy of theapplied iterative method for the nonlinear spring model isproven in Section 32 In this section 2 cases of loadingconditions are investigated to show the convergence of thesolutions
The locally distributed rectangular-type loadings inFigure 8 are taken
119908normal (single) (119909) = 1 |119909| le 1
0 otherwise(18)
6 Journal of Applied Mathematics
100
101
102
103
0
02
04
06
The number of iterations (119899)
Erro
r (119899
)
(a)
100
101
102
103
The number of iterations (119899)
Erro
r (119899
)
minus01
0
01
02
03
04
05
(b)
100
101
102
103
0
02
04
06
The number of iterations (119899)
Erro
r (119899
)
(c)
Figure 7 Errors between the exact solutions and iterative solutions
minus10 minus5 0 5 10119909
minus1
0
1
2
3
119908no
rmal
(single)(119909)
Figure 8 Applied loading
Figure 9 presents the convergence behavior of the solutionsaccording to the 3 cases of 119896
119901 119864 = 119868 = 119896 = 1
and 120574 = 02 As specified in Section 21 119896119901does not
affect the converged iterative solutions but the convergencecharacteristics changed The solution converged to a steadystate faster when 119896
119901is small
In Figure 10 two cases of converged solutions are com-pared the numerical results using the one-way and two-wayspring models The loading condition is 119908normal (single)(119909) in(18)They are different only when the beamdeflected upwardthat is the deflections near 119909 = plusmn5 are larger when the beamis separable (or free) from the foundation whereas the one-way spring model is not Of course at 119909 = 0 the solutions areequivalent because the physical system is the same in Figure 1
for 119906 gt 0 Therefore Figure 10 shows the validity of theapplied iterative method for the nonlinear spring model
Another loading condition in Figure 11 is taken as follows
119908normal (119909) = 1 4 le |119909| le 5 1 le |119909| le 2
0 otherwise(19)
Other numerical experiments are also conducted accordingto the 3 cases of a linear spring coefficient 119896 whereas theproperties (119864 = 119868 = 119896 = 1 and 120574 = 02) are fixed andthe solution converged (Figure 12) Figure 13 compares thesolutions from the one-way and two-way spring models 119864 =
119868 = 119896 = 1 120574 = 02 and 119896119901= 3
Journal of Applied Mathematics 7
minus02
0
02
04
06
08
119899 = 1
119899 = 2
119899 = 5
119899 = 10
119899 = 30
minus20 minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
(a)
minus02
0
02
04
06
08
119899 = 1
119899 = 3
119899 = 10
119899 = 30
119899 = 60
minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
minus20
(b)
minus02
0
02
04
06
08
119899 = 1
119899 = 3
119899 = 10
119899 = 40
119899 = 90
minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
minus20
(c)
Figure 9 Convergence behavior of the applied iterative method 119864 = 119868 = 119896 = 1 120574 = 02 (a) 119896119901= 3 (b) 119896
119901= 6 and (c) 119896
119901= 10
minus20 minus10 0 10 20119909
119906(119909)
One wayTwo way
0
02
04
06
08
minus02
(a)
One wayTwo way
minus5 0 5
minus004
minus002
0
002
004
006
119909
119906(119909)
(b)
Figure 10 Validity of the solution
8 Journal of Applied Mathematics
minus10 minus5 0 5 10119909
minus1
0
1
2
3
119908no
rmal(119909)
Figure 11 Applied loading conditions
119899 = 1
119899 = 2
119899 = 4
119899 = 20
119899 = 40
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(a)
119899 = 1
119899 = 2
119899 = 3
119899 = 10
119899 = 20
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(b)
119899 = 1
119899 = 2
119899 = 3
119899 = 5
119899 = 10
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(c)
Figure 12 Convergence behavior of the solutions 119864 = 119868 = 1 120574 = 02 and 119896119901= 3 (a) 119896 = 1 (b) 119896 = 15 and (c) 119896 = 2
Finally the validity and accuracy of the applied iterativeprocedure are investigated The nonlinear spring force isconsidered and the iterative procedure is applied successfullyfor the solution Convergence of the deflections according tothe external loading conditions is observed for the effects ofthe pseudo 119896
119901and real 119896
4 Conclusion
In this work we succeeded in applying the numerical methodproposed by Jang et al [21] to find the static deflection ofan infinite beam on a full nonlinear elastic foundation Forthat one-way spring model is considered for the formulation
Journal of Applied Mathematics 9
minus20 minus10 0 10 20119909
One wayTwo way
119906(119909)
0
02
04
06
(a)
One wayTwo way
119909
119906(119909)
minus15 minus10 minus5 0 5 10 15minus003
minus002
minus001
0
001
002
(b)
Figure 13 Validity of the solution in Figure 12(a)
of the nonlinear elastic one Since the problem concernsone-way spring force the governing equation for the staticbeam deflection is transformed as in (11) For the solution aniterative procedure is applied for the calculation of the highnonlinearity Some numerical experiments are carried out forshowing the validity and the fast convergence of the appliednumerical method And we can also find that the resultsconverge to the solutions fast for certain external loads
Acknowledgment
This work was supported by a 2-year research grant of PusanNational University
References
[1] M Hetenyi Beams on Elastic Foundation The University ofMichigan Press Ann Arbor Mich USA 1946
[2] C Miranda and K Nair ldquoFinite beams on elastic foundationrdquoJournal of the Structural Division vol 92 pp 131ndash142 1966
[3] B Y Ting ldquoFinite beams on elastic foundation with restraintsrdquoJournal of the Structural Division vol 108 no 3 pp 611ndash6211982
[4] S Timoshenko ldquoMethod of analysis of statistical and dynamicalstress in railrdquo in Proceeding of the International Congress ofApplied Mechanics pp 407ndash418 Zurich Switzerland 1926
[5] J T Kenney ldquoSteady-state vibrations of beam on elastic founda-tion for moving loadrdquo Journal of Applied Mechanics vol 21 pp359ndash364 1954
[6] H Saito and TMurakami ldquoVibrations of an infinite beam on anelastic foundation with consideration of mass of a foundationrdquoThe Japan Society of Mechanical Engineers vol 12 pp 200ndash2051969
[7] L Fryba ldquoInfinitely long beam on elastic foundation undermoving loadrdquo Aplikace Matematiky vol 2 pp 105ndash132 1957
[8] F W Beaufait and P W Hoadley ldquoAnalysis of elastic beams onnonlinear foundationsrdquo Computers amp Structures vol 12 no 5pp 669ndash676 1980
[9] C Massalas ldquoFundamental frequency of vibration of a beam ona non-linear elastic foundationrdquo Journal of Sound andVibrationvol 54 no 4 pp 613ndash615 1977
[10] M Lakshmanan ldquoComment on the fundamental frequency ofvibration of a beam on a non-linear elastic foundationrdquo Journalof Sound and Vibration vol 58 no 3 pp 455ndash457 1978
[11] D Hui ldquoPostbuckling behavior of infinite beams on elasticfoundations using Koiterrsquos improved theoryrdquo International Jour-nal of Non-Linear Mechanics vol 23 no 2 pp 113ndash123 1988
[12] D Z Yankelevsky and M Eisenberger ldquoAnalysis of a beam col-umn on elastic foundationrdquoComputers amp Structures vol 23 no3 pp 351ndash356 1986
[13] M Eisenberger and J Clastornik ldquoBeams on variable two-parameter elastic foundationrdquo Journal of EngineeringMechanicsvol 113 no 10 pp l454ndash1466 1987
[14] D Karamanlidis and V Prakash ldquoExact transfer and stiffnessmatrices for a beamcolumn resting on a two-parameter foun-dationrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 72 no 1 pp 77ndash89 1989
[15] S N Sirosh A Ghali and A G Razaqpur ldquoA general finite ele-ment for beams or beam-columns with or without an elasticfoundationrdquo International Journal for Numerical Methods inEngineering vol 28 no 5 pp 1061ndash1076 1989
[16] K P Soldatos and A P S Selvadurai ldquoFlexure of beams rest-ing on hyperbolic elastic foundationsrdquo International Journal ofSolids amp Structures vol 21 no 4 pp 373ndash388 1985
[17] S Y Lee H Y Ke and Y H Kuo ldquoExact static deflection ofa non-uniform bernoulli-euler beam with general elastic endrestraintsrdquoComputersampStructures vol 36 no 1 pp 91ndash97 1990
[18] S Y Lee and H Y Ke ldquoFree vibrations of a non-uniformbeam with general elastically restrained boundary conditionsrdquoJournal of Sound andVibration vol 136 no 3 pp 425ndash437 1990
[19] S Y Lee H Y Ke and Y H Kuo ldquoAnalysis of non-uniformbeam vibrationrdquo Journal of Sound and Vibration vol 142 no1 pp 15ndash29 1990
[20] Y H Kuo and S Y Lee ldquoDeflection of nonuniform beams rest-ing on a nonlinear elastic foundationrdquo Computers amp Structuresvol 51 no 5 pp 513ndash519 1994
[21] T S Jang H S Baek and J K Paik ldquoA newmethod for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundationrdquo International Journal of Non-LinearMechanics vol 46 no 1 pp 339ndash346 2011
[22] T S Jang and H G Sung ldquoA new semi-analytical method forthe non-linear static analysis of an infinite beam on a non-linear
10 Journal of Applied Mathematics
elastic foundation a general approach to a variable beam cross-sectionrdquo International Journal of Non-Linear Mechanics vol 47pp 132ndash139 2012
[23] S W Choi and T S Jang ldquoExistence and uniqueness of non-linear deflections of an infinite beam resting on a non-uniformnonlinear elastic foundationrdquo Boundary Value Problems vol2012 article 5 2012
[24] T S Jang ldquoA new semi-analytical approach to large deflectionsof BernoullindashEuler-v Karman beams on a linear elastic founda-tion nonlinear analysis of infinite beamsrdquo International Journalof Mechanical Sciences vol 66 pp 22ndash32 2013
[25] M D Greenberg Foundations of Applied Mathematics Pren-tice-Hall New Jersey NJ USA 1978
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
minus10 minus5 0 5 10
minus05
0
05
1
119909119906(119909)
Case aCase bCase c
Figure 3 Exact solutions in Table 1
minus20
minus10
0
10
20
119908(119909)
minus10 minus5 0 5 10119909
Case aCase bCase c
Figure 4 Applied loading conditions corresponding to the exact solution in Table 1
The loading condition is assumed to be localized so 119906(119909)
in (4) satisfies the boundary conditions in (3) Deriving thelinear solution in (4) a uniformly upward nonlinear springforce depends on the beam deflection 119906
In this study the main idea of the present study is pro-posed by Jang et al [21] The nonlinear spring model is usedfor the formulation of a realistic nonlinear elastic foundationA pseudo linear spring coefficient 119896
119901and a (real) spring
force 119891(119906) are used Therefore the fourth-order differentialequation in (1) is equivalent to the following equation
1198641198681198894119906
1198891199094+ 119896119901119906 + 119891 (119906) = 119908 (119909) + 119896
119901119906 (6)
where the nonlinear spring force119891 depends on the deflection119906
119891 (119906) = 119896 sdot 119906 + 119873 (119906) for 119906 ge 00 for 119906 lt 0
(7)
or
1198641198681198894119906
1198891199094+ 119896119901119906 = 119908 (119909) + 119896
119901119906 minus 119891 (119906) equiv 120601 (8)
Table 1 Three cases of the exact solution
Case Exact solution 119906(119909)a 119890
minus1199092
b Sin119909 sdot 119890minus1199092
c Sin 119909 sdot 119890minus11990924
In (8) 119896119901119906 is the pseudolinear spring force term and is finally
compensated so it does not affect the nonlinear solutionTherefore (4) shows that (8) must be equivalent to the
following equation
119906 (119909) = int
infin
minusinfin
119866(119909 120585 119896119901) sdot 120601 (120585) 119889120585 (9)
where Greenrsquos function with pseudo linear spring coefficientcan be expressed as
119866(119909 120585 119896119901) =
120573
2119896119890minus120573|120585minus119909|radic2
times sin(1205731003816100381610038161003816120585 minus 119909
1003816100381610038161003816
radic2
+120587
4) 120573 =
4radic119896119901119864119868
(10)
4 Journal of Applied Mathematics
minus10 minus5 0 5 10
0
02
04
06
08
1
119909
119906(119909)
ExactNumerical
(a)
ExactNumerical
minus10 minus5 0 5 10119909
119906(119909)
minus04
minus02
0
02
04
(b)
ExactNumerical
minus10 minus5 0 5 10119909
119906(119909)
minus05
0
05
(c)
Figure 5 Numerical solutions compared to the exact solutions
Regarding (8) the nonlinear relation for 119906 can be derived asfollows
119906 (119909) = int
infin
minusinfin
119866(119909 120585 119896119901) sdot 119908 (120585) 119889120585
+ int
infin
minusinfin
119866(119909 120585 119896119901) sdot 119870 [119906 (120585)] 119889120585
(11)
where the function119870 can be written as
119870 (119906) = 119896119901sdot 119906 minus 119891 (119906) (12)
Equation (11) is a nonlinear Fredholm integral equation of thesecond kind for 119906
22 Iterative Procedure From (11) the nonlinear iterativeprocedure is applied [21ndash24]
119906119899+1
(119909) = int
infin
minusinfin
119866(119909 120585 119896119901) sdot 119908 (120585) 119889120585
+ int
infin
minusinfin
119866(119909 120585 119896119901) sdot 119870 [119906
119899(120585)] 119889120585
(13)
where119870(119906) satisfies (7) and (12)
To examine the nonlinear iterative procedure in (13) bysimulation it should be discretized as follows
119906119899+1
(119909) =
Nsum
119895=1
119882119895119866 (119909 120585
119895 119896119901) sdot 119908 (120585
119895)
+ 119866 (119909 120585 119896119901) sdot 119870 [119906
119899(120585119895)]
119895 = 0 1 2 119873
(14)
where 119882119895denotes the weights for the integration rule The
number 119873 in the summation of (14) denotes the totalsegments of the interval (minus119877 119877) and 119877 is a sufficiently largevalue satisfying (3)
3 Numerical Experiments
In this section numerical experiments are performed todetermine the validity of the iterative method This studyassumes a nonlinear spring force 119891(119906) in (7) and examineshow accurate the solution converges to an exact solutionTheconvergence of themethod is investigated with some externalloading conditions
Journal of Applied Mathematics 5
0
05
1
119899 = 1
119899 = 2
119899 = 3
119899 = 300
Exact solution
minus10 minus5 0 5 10119909
119906(119909)
(a)
119899 = 1
119899 = 5
119899 = 10
119899 = 300
Exact solution
minus10 minus5 0 5 10119909
119906(119909)
minus04
minus02
0
02
04
(b)
119899 = 1
119899 = 5
119899 = 10
119899 = 300
Exact solution
minus10 minus5 0 5 10119909
119906(119909)
5 50minus
05
0
minus05
(c)
Figure 6 Convergence behaviors of the iterative solutions
31 Nonlinear Spring Model For simplicity the present studyconsiders an infinite beam on a nonlinear elastic foundationwhose spring force is derived as follows
119891 (119906) = 119896 sdot 119906 + 120574 sdot 119906
3 for 119906 ge 0
0 for 119906 lt 0(15)
where119873(119906) in (7) is chosen immediately as a cubic form
119873(119906) = 120574 sdot 1199063 (16)
32 Comparison with Exact Solution To determine if theiterative method converges to an exact solution the 3 casesof exact solutions listed in Table 1 are first assumed and areillustrated in Figure 3 These cases are chosen to be infinitelydifferentiable and satisfy the conditions in (3) The followingare assumed 119864 = 119868 = 119896 = 1 119896
119901= 3 and 120574 = 02 and the
initial guess of the deflection is 1199060= 0 Simpsonrsquos integration
rule is applied to the numerical integration Three cases ofexternal loads shown in Figure 4 are obtained by substitut-ing the exact solutions in Table 1 to (6) Under the load-ing condition the nonlinear iterative method in Section 2
is applied to obtain the solution Figure 5 compares the exactand numerical solutions and Figure 6 shows the convergencebehavior of the solutions in Figure 5 The errors of the solu-tions at the 119899th iteration are defined as follows
Error (119899) =1003817100381710038171003817119906exact minus 119906119899
100381710038171003817100381721003817100381710038171003817119906exact
10038171003817100381710038172
where 1199112equiv (
119873
sum
119894=1
10038161003816100381610038161199111198941003816100381610038161003816
2
)
12
(17)Three cases of Error (119899) are plotted as a function of theiteration number in Figure 7 The solutions converge at 200ndash300 iterations
33 Convergence of the Procedure The accuracy of theapplied iterative method for the nonlinear spring model isproven in Section 32 In this section 2 cases of loadingconditions are investigated to show the convergence of thesolutions
The locally distributed rectangular-type loadings inFigure 8 are taken
119908normal (single) (119909) = 1 |119909| le 1
0 otherwise(18)
6 Journal of Applied Mathematics
100
101
102
103
0
02
04
06
The number of iterations (119899)
Erro
r (119899
)
(a)
100
101
102
103
The number of iterations (119899)
Erro
r (119899
)
minus01
0
01
02
03
04
05
(b)
100
101
102
103
0
02
04
06
The number of iterations (119899)
Erro
r (119899
)
(c)
Figure 7 Errors between the exact solutions and iterative solutions
minus10 minus5 0 5 10119909
minus1
0
1
2
3
119908no
rmal
(single)(119909)
Figure 8 Applied loading
Figure 9 presents the convergence behavior of the solutionsaccording to the 3 cases of 119896
119901 119864 = 119868 = 119896 = 1
and 120574 = 02 As specified in Section 21 119896119901does not
affect the converged iterative solutions but the convergencecharacteristics changed The solution converged to a steadystate faster when 119896
119901is small
In Figure 10 two cases of converged solutions are com-pared the numerical results using the one-way and two-wayspring models The loading condition is 119908normal (single)(119909) in(18)They are different only when the beamdeflected upwardthat is the deflections near 119909 = plusmn5 are larger when the beamis separable (or free) from the foundation whereas the one-way spring model is not Of course at 119909 = 0 the solutions areequivalent because the physical system is the same in Figure 1
for 119906 gt 0 Therefore Figure 10 shows the validity of theapplied iterative method for the nonlinear spring model
Another loading condition in Figure 11 is taken as follows
119908normal (119909) = 1 4 le |119909| le 5 1 le |119909| le 2
0 otherwise(19)
Other numerical experiments are also conducted accordingto the 3 cases of a linear spring coefficient 119896 whereas theproperties (119864 = 119868 = 119896 = 1 and 120574 = 02) are fixed andthe solution converged (Figure 12) Figure 13 compares thesolutions from the one-way and two-way spring models 119864 =
119868 = 119896 = 1 120574 = 02 and 119896119901= 3
Journal of Applied Mathematics 7
minus02
0
02
04
06
08
119899 = 1
119899 = 2
119899 = 5
119899 = 10
119899 = 30
minus20 minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
(a)
minus02
0
02
04
06
08
119899 = 1
119899 = 3
119899 = 10
119899 = 30
119899 = 60
minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
minus20
(b)
minus02
0
02
04
06
08
119899 = 1
119899 = 3
119899 = 10
119899 = 40
119899 = 90
minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
minus20
(c)
Figure 9 Convergence behavior of the applied iterative method 119864 = 119868 = 119896 = 1 120574 = 02 (a) 119896119901= 3 (b) 119896
119901= 6 and (c) 119896
119901= 10
minus20 minus10 0 10 20119909
119906(119909)
One wayTwo way
0
02
04
06
08
minus02
(a)
One wayTwo way
minus5 0 5
minus004
minus002
0
002
004
006
119909
119906(119909)
(b)
Figure 10 Validity of the solution
8 Journal of Applied Mathematics
minus10 minus5 0 5 10119909
minus1
0
1
2
3
119908no
rmal(119909)
Figure 11 Applied loading conditions
119899 = 1
119899 = 2
119899 = 4
119899 = 20
119899 = 40
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(a)
119899 = 1
119899 = 2
119899 = 3
119899 = 10
119899 = 20
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(b)
119899 = 1
119899 = 2
119899 = 3
119899 = 5
119899 = 10
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(c)
Figure 12 Convergence behavior of the solutions 119864 = 119868 = 1 120574 = 02 and 119896119901= 3 (a) 119896 = 1 (b) 119896 = 15 and (c) 119896 = 2
Finally the validity and accuracy of the applied iterativeprocedure are investigated The nonlinear spring force isconsidered and the iterative procedure is applied successfullyfor the solution Convergence of the deflections according tothe external loading conditions is observed for the effects ofthe pseudo 119896
119901and real 119896
4 Conclusion
In this work we succeeded in applying the numerical methodproposed by Jang et al [21] to find the static deflection ofan infinite beam on a full nonlinear elastic foundation Forthat one-way spring model is considered for the formulation
Journal of Applied Mathematics 9
minus20 minus10 0 10 20119909
One wayTwo way
119906(119909)
0
02
04
06
(a)
One wayTwo way
119909
119906(119909)
minus15 minus10 minus5 0 5 10 15minus003
minus002
minus001
0
001
002
(b)
Figure 13 Validity of the solution in Figure 12(a)
of the nonlinear elastic one Since the problem concernsone-way spring force the governing equation for the staticbeam deflection is transformed as in (11) For the solution aniterative procedure is applied for the calculation of the highnonlinearity Some numerical experiments are carried out forshowing the validity and the fast convergence of the appliednumerical method And we can also find that the resultsconverge to the solutions fast for certain external loads
Acknowledgment
This work was supported by a 2-year research grant of PusanNational University
References
[1] M Hetenyi Beams on Elastic Foundation The University ofMichigan Press Ann Arbor Mich USA 1946
[2] C Miranda and K Nair ldquoFinite beams on elastic foundationrdquoJournal of the Structural Division vol 92 pp 131ndash142 1966
[3] B Y Ting ldquoFinite beams on elastic foundation with restraintsrdquoJournal of the Structural Division vol 108 no 3 pp 611ndash6211982
[4] S Timoshenko ldquoMethod of analysis of statistical and dynamicalstress in railrdquo in Proceeding of the International Congress ofApplied Mechanics pp 407ndash418 Zurich Switzerland 1926
[5] J T Kenney ldquoSteady-state vibrations of beam on elastic founda-tion for moving loadrdquo Journal of Applied Mechanics vol 21 pp359ndash364 1954
[6] H Saito and TMurakami ldquoVibrations of an infinite beam on anelastic foundation with consideration of mass of a foundationrdquoThe Japan Society of Mechanical Engineers vol 12 pp 200ndash2051969
[7] L Fryba ldquoInfinitely long beam on elastic foundation undermoving loadrdquo Aplikace Matematiky vol 2 pp 105ndash132 1957
[8] F W Beaufait and P W Hoadley ldquoAnalysis of elastic beams onnonlinear foundationsrdquo Computers amp Structures vol 12 no 5pp 669ndash676 1980
[9] C Massalas ldquoFundamental frequency of vibration of a beam ona non-linear elastic foundationrdquo Journal of Sound andVibrationvol 54 no 4 pp 613ndash615 1977
[10] M Lakshmanan ldquoComment on the fundamental frequency ofvibration of a beam on a non-linear elastic foundationrdquo Journalof Sound and Vibration vol 58 no 3 pp 455ndash457 1978
[11] D Hui ldquoPostbuckling behavior of infinite beams on elasticfoundations using Koiterrsquos improved theoryrdquo International Jour-nal of Non-Linear Mechanics vol 23 no 2 pp 113ndash123 1988
[12] D Z Yankelevsky and M Eisenberger ldquoAnalysis of a beam col-umn on elastic foundationrdquoComputers amp Structures vol 23 no3 pp 351ndash356 1986
[13] M Eisenberger and J Clastornik ldquoBeams on variable two-parameter elastic foundationrdquo Journal of EngineeringMechanicsvol 113 no 10 pp l454ndash1466 1987
[14] D Karamanlidis and V Prakash ldquoExact transfer and stiffnessmatrices for a beamcolumn resting on a two-parameter foun-dationrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 72 no 1 pp 77ndash89 1989
[15] S N Sirosh A Ghali and A G Razaqpur ldquoA general finite ele-ment for beams or beam-columns with or without an elasticfoundationrdquo International Journal for Numerical Methods inEngineering vol 28 no 5 pp 1061ndash1076 1989
[16] K P Soldatos and A P S Selvadurai ldquoFlexure of beams rest-ing on hyperbolic elastic foundationsrdquo International Journal ofSolids amp Structures vol 21 no 4 pp 373ndash388 1985
[17] S Y Lee H Y Ke and Y H Kuo ldquoExact static deflection ofa non-uniform bernoulli-euler beam with general elastic endrestraintsrdquoComputersampStructures vol 36 no 1 pp 91ndash97 1990
[18] S Y Lee and H Y Ke ldquoFree vibrations of a non-uniformbeam with general elastically restrained boundary conditionsrdquoJournal of Sound andVibration vol 136 no 3 pp 425ndash437 1990
[19] S Y Lee H Y Ke and Y H Kuo ldquoAnalysis of non-uniformbeam vibrationrdquo Journal of Sound and Vibration vol 142 no1 pp 15ndash29 1990
[20] Y H Kuo and S Y Lee ldquoDeflection of nonuniform beams rest-ing on a nonlinear elastic foundationrdquo Computers amp Structuresvol 51 no 5 pp 513ndash519 1994
[21] T S Jang H S Baek and J K Paik ldquoA newmethod for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundationrdquo International Journal of Non-LinearMechanics vol 46 no 1 pp 339ndash346 2011
[22] T S Jang and H G Sung ldquoA new semi-analytical method forthe non-linear static analysis of an infinite beam on a non-linear
10 Journal of Applied Mathematics
elastic foundation a general approach to a variable beam cross-sectionrdquo International Journal of Non-Linear Mechanics vol 47pp 132ndash139 2012
[23] S W Choi and T S Jang ldquoExistence and uniqueness of non-linear deflections of an infinite beam resting on a non-uniformnonlinear elastic foundationrdquo Boundary Value Problems vol2012 article 5 2012
[24] T S Jang ldquoA new semi-analytical approach to large deflectionsof BernoullindashEuler-v Karman beams on a linear elastic founda-tion nonlinear analysis of infinite beamsrdquo International Journalof Mechanical Sciences vol 66 pp 22ndash32 2013
[25] M D Greenberg Foundations of Applied Mathematics Pren-tice-Hall New Jersey NJ USA 1978
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
minus10 minus5 0 5 10
0
02
04
06
08
1
119909
119906(119909)
ExactNumerical
(a)
ExactNumerical
minus10 minus5 0 5 10119909
119906(119909)
minus04
minus02
0
02
04
(b)
ExactNumerical
minus10 minus5 0 5 10119909
119906(119909)
minus05
0
05
(c)
Figure 5 Numerical solutions compared to the exact solutions
Regarding (8) the nonlinear relation for 119906 can be derived asfollows
119906 (119909) = int
infin
minusinfin
119866(119909 120585 119896119901) sdot 119908 (120585) 119889120585
+ int
infin
minusinfin
119866(119909 120585 119896119901) sdot 119870 [119906 (120585)] 119889120585
(11)
where the function119870 can be written as
119870 (119906) = 119896119901sdot 119906 minus 119891 (119906) (12)
Equation (11) is a nonlinear Fredholm integral equation of thesecond kind for 119906
22 Iterative Procedure From (11) the nonlinear iterativeprocedure is applied [21ndash24]
119906119899+1
(119909) = int
infin
minusinfin
119866(119909 120585 119896119901) sdot 119908 (120585) 119889120585
+ int
infin
minusinfin
119866(119909 120585 119896119901) sdot 119870 [119906
119899(120585)] 119889120585
(13)
where119870(119906) satisfies (7) and (12)
To examine the nonlinear iterative procedure in (13) bysimulation it should be discretized as follows
119906119899+1
(119909) =
Nsum
119895=1
119882119895119866 (119909 120585
119895 119896119901) sdot 119908 (120585
119895)
+ 119866 (119909 120585 119896119901) sdot 119870 [119906
119899(120585119895)]
119895 = 0 1 2 119873
(14)
where 119882119895denotes the weights for the integration rule The
number 119873 in the summation of (14) denotes the totalsegments of the interval (minus119877 119877) and 119877 is a sufficiently largevalue satisfying (3)
3 Numerical Experiments
In this section numerical experiments are performed todetermine the validity of the iterative method This studyassumes a nonlinear spring force 119891(119906) in (7) and examineshow accurate the solution converges to an exact solutionTheconvergence of themethod is investigated with some externalloading conditions
Journal of Applied Mathematics 5
0
05
1
119899 = 1
119899 = 2
119899 = 3
119899 = 300
Exact solution
minus10 minus5 0 5 10119909
119906(119909)
(a)
119899 = 1
119899 = 5
119899 = 10
119899 = 300
Exact solution
minus10 minus5 0 5 10119909
119906(119909)
minus04
minus02
0
02
04
(b)
119899 = 1
119899 = 5
119899 = 10
119899 = 300
Exact solution
minus10 minus5 0 5 10119909
119906(119909)
5 50minus
05
0
minus05
(c)
Figure 6 Convergence behaviors of the iterative solutions
31 Nonlinear Spring Model For simplicity the present studyconsiders an infinite beam on a nonlinear elastic foundationwhose spring force is derived as follows
119891 (119906) = 119896 sdot 119906 + 120574 sdot 119906
3 for 119906 ge 0
0 for 119906 lt 0(15)
where119873(119906) in (7) is chosen immediately as a cubic form
119873(119906) = 120574 sdot 1199063 (16)
32 Comparison with Exact Solution To determine if theiterative method converges to an exact solution the 3 casesof exact solutions listed in Table 1 are first assumed and areillustrated in Figure 3 These cases are chosen to be infinitelydifferentiable and satisfy the conditions in (3) The followingare assumed 119864 = 119868 = 119896 = 1 119896
119901= 3 and 120574 = 02 and the
initial guess of the deflection is 1199060= 0 Simpsonrsquos integration
rule is applied to the numerical integration Three cases ofexternal loads shown in Figure 4 are obtained by substitut-ing the exact solutions in Table 1 to (6) Under the load-ing condition the nonlinear iterative method in Section 2
is applied to obtain the solution Figure 5 compares the exactand numerical solutions and Figure 6 shows the convergencebehavior of the solutions in Figure 5 The errors of the solu-tions at the 119899th iteration are defined as follows
Error (119899) =1003817100381710038171003817119906exact minus 119906119899
100381710038171003817100381721003817100381710038171003817119906exact
10038171003817100381710038172
where 1199112equiv (
119873
sum
119894=1
10038161003816100381610038161199111198941003816100381610038161003816
2
)
12
(17)Three cases of Error (119899) are plotted as a function of theiteration number in Figure 7 The solutions converge at 200ndash300 iterations
33 Convergence of the Procedure The accuracy of theapplied iterative method for the nonlinear spring model isproven in Section 32 In this section 2 cases of loadingconditions are investigated to show the convergence of thesolutions
The locally distributed rectangular-type loadings inFigure 8 are taken
119908normal (single) (119909) = 1 |119909| le 1
0 otherwise(18)
6 Journal of Applied Mathematics
100
101
102
103
0
02
04
06
The number of iterations (119899)
Erro
r (119899
)
(a)
100
101
102
103
The number of iterations (119899)
Erro
r (119899
)
minus01
0
01
02
03
04
05
(b)
100
101
102
103
0
02
04
06
The number of iterations (119899)
Erro
r (119899
)
(c)
Figure 7 Errors between the exact solutions and iterative solutions
minus10 minus5 0 5 10119909
minus1
0
1
2
3
119908no
rmal
(single)(119909)
Figure 8 Applied loading
Figure 9 presents the convergence behavior of the solutionsaccording to the 3 cases of 119896
119901 119864 = 119868 = 119896 = 1
and 120574 = 02 As specified in Section 21 119896119901does not
affect the converged iterative solutions but the convergencecharacteristics changed The solution converged to a steadystate faster when 119896
119901is small
In Figure 10 two cases of converged solutions are com-pared the numerical results using the one-way and two-wayspring models The loading condition is 119908normal (single)(119909) in(18)They are different only when the beamdeflected upwardthat is the deflections near 119909 = plusmn5 are larger when the beamis separable (or free) from the foundation whereas the one-way spring model is not Of course at 119909 = 0 the solutions areequivalent because the physical system is the same in Figure 1
for 119906 gt 0 Therefore Figure 10 shows the validity of theapplied iterative method for the nonlinear spring model
Another loading condition in Figure 11 is taken as follows
119908normal (119909) = 1 4 le |119909| le 5 1 le |119909| le 2
0 otherwise(19)
Other numerical experiments are also conducted accordingto the 3 cases of a linear spring coefficient 119896 whereas theproperties (119864 = 119868 = 119896 = 1 and 120574 = 02) are fixed andthe solution converged (Figure 12) Figure 13 compares thesolutions from the one-way and two-way spring models 119864 =
119868 = 119896 = 1 120574 = 02 and 119896119901= 3
Journal of Applied Mathematics 7
minus02
0
02
04
06
08
119899 = 1
119899 = 2
119899 = 5
119899 = 10
119899 = 30
minus20 minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
(a)
minus02
0
02
04
06
08
119899 = 1
119899 = 3
119899 = 10
119899 = 30
119899 = 60
minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
minus20
(b)
minus02
0
02
04
06
08
119899 = 1
119899 = 3
119899 = 10
119899 = 40
119899 = 90
minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
minus20
(c)
Figure 9 Convergence behavior of the applied iterative method 119864 = 119868 = 119896 = 1 120574 = 02 (a) 119896119901= 3 (b) 119896
119901= 6 and (c) 119896
119901= 10
minus20 minus10 0 10 20119909
119906(119909)
One wayTwo way
0
02
04
06
08
minus02
(a)
One wayTwo way
minus5 0 5
minus004
minus002
0
002
004
006
119909
119906(119909)
(b)
Figure 10 Validity of the solution
8 Journal of Applied Mathematics
minus10 minus5 0 5 10119909
minus1
0
1
2
3
119908no
rmal(119909)
Figure 11 Applied loading conditions
119899 = 1
119899 = 2
119899 = 4
119899 = 20
119899 = 40
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(a)
119899 = 1
119899 = 2
119899 = 3
119899 = 10
119899 = 20
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(b)
119899 = 1
119899 = 2
119899 = 3
119899 = 5
119899 = 10
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(c)
Figure 12 Convergence behavior of the solutions 119864 = 119868 = 1 120574 = 02 and 119896119901= 3 (a) 119896 = 1 (b) 119896 = 15 and (c) 119896 = 2
Finally the validity and accuracy of the applied iterativeprocedure are investigated The nonlinear spring force isconsidered and the iterative procedure is applied successfullyfor the solution Convergence of the deflections according tothe external loading conditions is observed for the effects ofthe pseudo 119896
119901and real 119896
4 Conclusion
In this work we succeeded in applying the numerical methodproposed by Jang et al [21] to find the static deflection ofan infinite beam on a full nonlinear elastic foundation Forthat one-way spring model is considered for the formulation
Journal of Applied Mathematics 9
minus20 minus10 0 10 20119909
One wayTwo way
119906(119909)
0
02
04
06
(a)
One wayTwo way
119909
119906(119909)
minus15 minus10 minus5 0 5 10 15minus003
minus002
minus001
0
001
002
(b)
Figure 13 Validity of the solution in Figure 12(a)
of the nonlinear elastic one Since the problem concernsone-way spring force the governing equation for the staticbeam deflection is transformed as in (11) For the solution aniterative procedure is applied for the calculation of the highnonlinearity Some numerical experiments are carried out forshowing the validity and the fast convergence of the appliednumerical method And we can also find that the resultsconverge to the solutions fast for certain external loads
Acknowledgment
This work was supported by a 2-year research grant of PusanNational University
References
[1] M Hetenyi Beams on Elastic Foundation The University ofMichigan Press Ann Arbor Mich USA 1946
[2] C Miranda and K Nair ldquoFinite beams on elastic foundationrdquoJournal of the Structural Division vol 92 pp 131ndash142 1966
[3] B Y Ting ldquoFinite beams on elastic foundation with restraintsrdquoJournal of the Structural Division vol 108 no 3 pp 611ndash6211982
[4] S Timoshenko ldquoMethod of analysis of statistical and dynamicalstress in railrdquo in Proceeding of the International Congress ofApplied Mechanics pp 407ndash418 Zurich Switzerland 1926
[5] J T Kenney ldquoSteady-state vibrations of beam on elastic founda-tion for moving loadrdquo Journal of Applied Mechanics vol 21 pp359ndash364 1954
[6] H Saito and TMurakami ldquoVibrations of an infinite beam on anelastic foundation with consideration of mass of a foundationrdquoThe Japan Society of Mechanical Engineers vol 12 pp 200ndash2051969
[7] L Fryba ldquoInfinitely long beam on elastic foundation undermoving loadrdquo Aplikace Matematiky vol 2 pp 105ndash132 1957
[8] F W Beaufait and P W Hoadley ldquoAnalysis of elastic beams onnonlinear foundationsrdquo Computers amp Structures vol 12 no 5pp 669ndash676 1980
[9] C Massalas ldquoFundamental frequency of vibration of a beam ona non-linear elastic foundationrdquo Journal of Sound andVibrationvol 54 no 4 pp 613ndash615 1977
[10] M Lakshmanan ldquoComment on the fundamental frequency ofvibration of a beam on a non-linear elastic foundationrdquo Journalof Sound and Vibration vol 58 no 3 pp 455ndash457 1978
[11] D Hui ldquoPostbuckling behavior of infinite beams on elasticfoundations using Koiterrsquos improved theoryrdquo International Jour-nal of Non-Linear Mechanics vol 23 no 2 pp 113ndash123 1988
[12] D Z Yankelevsky and M Eisenberger ldquoAnalysis of a beam col-umn on elastic foundationrdquoComputers amp Structures vol 23 no3 pp 351ndash356 1986
[13] M Eisenberger and J Clastornik ldquoBeams on variable two-parameter elastic foundationrdquo Journal of EngineeringMechanicsvol 113 no 10 pp l454ndash1466 1987
[14] D Karamanlidis and V Prakash ldquoExact transfer and stiffnessmatrices for a beamcolumn resting on a two-parameter foun-dationrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 72 no 1 pp 77ndash89 1989
[15] S N Sirosh A Ghali and A G Razaqpur ldquoA general finite ele-ment for beams or beam-columns with or without an elasticfoundationrdquo International Journal for Numerical Methods inEngineering vol 28 no 5 pp 1061ndash1076 1989
[16] K P Soldatos and A P S Selvadurai ldquoFlexure of beams rest-ing on hyperbolic elastic foundationsrdquo International Journal ofSolids amp Structures vol 21 no 4 pp 373ndash388 1985
[17] S Y Lee H Y Ke and Y H Kuo ldquoExact static deflection ofa non-uniform bernoulli-euler beam with general elastic endrestraintsrdquoComputersampStructures vol 36 no 1 pp 91ndash97 1990
[18] S Y Lee and H Y Ke ldquoFree vibrations of a non-uniformbeam with general elastically restrained boundary conditionsrdquoJournal of Sound andVibration vol 136 no 3 pp 425ndash437 1990
[19] S Y Lee H Y Ke and Y H Kuo ldquoAnalysis of non-uniformbeam vibrationrdquo Journal of Sound and Vibration vol 142 no1 pp 15ndash29 1990
[20] Y H Kuo and S Y Lee ldquoDeflection of nonuniform beams rest-ing on a nonlinear elastic foundationrdquo Computers amp Structuresvol 51 no 5 pp 513ndash519 1994
[21] T S Jang H S Baek and J K Paik ldquoA newmethod for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundationrdquo International Journal of Non-LinearMechanics vol 46 no 1 pp 339ndash346 2011
[22] T S Jang and H G Sung ldquoA new semi-analytical method forthe non-linear static analysis of an infinite beam on a non-linear
10 Journal of Applied Mathematics
elastic foundation a general approach to a variable beam cross-sectionrdquo International Journal of Non-Linear Mechanics vol 47pp 132ndash139 2012
[23] S W Choi and T S Jang ldquoExistence and uniqueness of non-linear deflections of an infinite beam resting on a non-uniformnonlinear elastic foundationrdquo Boundary Value Problems vol2012 article 5 2012
[24] T S Jang ldquoA new semi-analytical approach to large deflectionsof BernoullindashEuler-v Karman beams on a linear elastic founda-tion nonlinear analysis of infinite beamsrdquo International Journalof Mechanical Sciences vol 66 pp 22ndash32 2013
[25] M D Greenberg Foundations of Applied Mathematics Pren-tice-Hall New Jersey NJ USA 1978
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
1
119899 = 1
119899 = 2
119899 = 3
119899 = 300
Exact solution
minus10 minus5 0 5 10119909
119906(119909)
(a)
119899 = 1
119899 = 5
119899 = 10
119899 = 300
Exact solution
minus10 minus5 0 5 10119909
119906(119909)
minus04
minus02
0
02
04
(b)
119899 = 1
119899 = 5
119899 = 10
119899 = 300
Exact solution
minus10 minus5 0 5 10119909
119906(119909)
5 50minus
05
0
minus05
(c)
Figure 6 Convergence behaviors of the iterative solutions
31 Nonlinear Spring Model For simplicity the present studyconsiders an infinite beam on a nonlinear elastic foundationwhose spring force is derived as follows
119891 (119906) = 119896 sdot 119906 + 120574 sdot 119906
3 for 119906 ge 0
0 for 119906 lt 0(15)
where119873(119906) in (7) is chosen immediately as a cubic form
119873(119906) = 120574 sdot 1199063 (16)
32 Comparison with Exact Solution To determine if theiterative method converges to an exact solution the 3 casesof exact solutions listed in Table 1 are first assumed and areillustrated in Figure 3 These cases are chosen to be infinitelydifferentiable and satisfy the conditions in (3) The followingare assumed 119864 = 119868 = 119896 = 1 119896
119901= 3 and 120574 = 02 and the
initial guess of the deflection is 1199060= 0 Simpsonrsquos integration
rule is applied to the numerical integration Three cases ofexternal loads shown in Figure 4 are obtained by substitut-ing the exact solutions in Table 1 to (6) Under the load-ing condition the nonlinear iterative method in Section 2
is applied to obtain the solution Figure 5 compares the exactand numerical solutions and Figure 6 shows the convergencebehavior of the solutions in Figure 5 The errors of the solu-tions at the 119899th iteration are defined as follows
Error (119899) =1003817100381710038171003817119906exact minus 119906119899
100381710038171003817100381721003817100381710038171003817119906exact
10038171003817100381710038172
where 1199112equiv (
119873
sum
119894=1
10038161003816100381610038161199111198941003816100381610038161003816
2
)
12
(17)Three cases of Error (119899) are plotted as a function of theiteration number in Figure 7 The solutions converge at 200ndash300 iterations
33 Convergence of the Procedure The accuracy of theapplied iterative method for the nonlinear spring model isproven in Section 32 In this section 2 cases of loadingconditions are investigated to show the convergence of thesolutions
The locally distributed rectangular-type loadings inFigure 8 are taken
119908normal (single) (119909) = 1 |119909| le 1
0 otherwise(18)
6 Journal of Applied Mathematics
100
101
102
103
0
02
04
06
The number of iterations (119899)
Erro
r (119899
)
(a)
100
101
102
103
The number of iterations (119899)
Erro
r (119899
)
minus01
0
01
02
03
04
05
(b)
100
101
102
103
0
02
04
06
The number of iterations (119899)
Erro
r (119899
)
(c)
Figure 7 Errors between the exact solutions and iterative solutions
minus10 minus5 0 5 10119909
minus1
0
1
2
3
119908no
rmal
(single)(119909)
Figure 8 Applied loading
Figure 9 presents the convergence behavior of the solutionsaccording to the 3 cases of 119896
119901 119864 = 119868 = 119896 = 1
and 120574 = 02 As specified in Section 21 119896119901does not
affect the converged iterative solutions but the convergencecharacteristics changed The solution converged to a steadystate faster when 119896
119901is small
In Figure 10 two cases of converged solutions are com-pared the numerical results using the one-way and two-wayspring models The loading condition is 119908normal (single)(119909) in(18)They are different only when the beamdeflected upwardthat is the deflections near 119909 = plusmn5 are larger when the beamis separable (or free) from the foundation whereas the one-way spring model is not Of course at 119909 = 0 the solutions areequivalent because the physical system is the same in Figure 1
for 119906 gt 0 Therefore Figure 10 shows the validity of theapplied iterative method for the nonlinear spring model
Another loading condition in Figure 11 is taken as follows
119908normal (119909) = 1 4 le |119909| le 5 1 le |119909| le 2
0 otherwise(19)
Other numerical experiments are also conducted accordingto the 3 cases of a linear spring coefficient 119896 whereas theproperties (119864 = 119868 = 119896 = 1 and 120574 = 02) are fixed andthe solution converged (Figure 12) Figure 13 compares thesolutions from the one-way and two-way spring models 119864 =
119868 = 119896 = 1 120574 = 02 and 119896119901= 3
Journal of Applied Mathematics 7
minus02
0
02
04
06
08
119899 = 1
119899 = 2
119899 = 5
119899 = 10
119899 = 30
minus20 minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
(a)
minus02
0
02
04
06
08
119899 = 1
119899 = 3
119899 = 10
119899 = 30
119899 = 60
minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
minus20
(b)
minus02
0
02
04
06
08
119899 = 1
119899 = 3
119899 = 10
119899 = 40
119899 = 90
minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
minus20
(c)
Figure 9 Convergence behavior of the applied iterative method 119864 = 119868 = 119896 = 1 120574 = 02 (a) 119896119901= 3 (b) 119896
119901= 6 and (c) 119896
119901= 10
minus20 minus10 0 10 20119909
119906(119909)
One wayTwo way
0
02
04
06
08
minus02
(a)
One wayTwo way
minus5 0 5
minus004
minus002
0
002
004
006
119909
119906(119909)
(b)
Figure 10 Validity of the solution
8 Journal of Applied Mathematics
minus10 minus5 0 5 10119909
minus1
0
1
2
3
119908no
rmal(119909)
Figure 11 Applied loading conditions
119899 = 1
119899 = 2
119899 = 4
119899 = 20
119899 = 40
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(a)
119899 = 1
119899 = 2
119899 = 3
119899 = 10
119899 = 20
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(b)
119899 = 1
119899 = 2
119899 = 3
119899 = 5
119899 = 10
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(c)
Figure 12 Convergence behavior of the solutions 119864 = 119868 = 1 120574 = 02 and 119896119901= 3 (a) 119896 = 1 (b) 119896 = 15 and (c) 119896 = 2
Finally the validity and accuracy of the applied iterativeprocedure are investigated The nonlinear spring force isconsidered and the iterative procedure is applied successfullyfor the solution Convergence of the deflections according tothe external loading conditions is observed for the effects ofthe pseudo 119896
119901and real 119896
4 Conclusion
In this work we succeeded in applying the numerical methodproposed by Jang et al [21] to find the static deflection ofan infinite beam on a full nonlinear elastic foundation Forthat one-way spring model is considered for the formulation
Journal of Applied Mathematics 9
minus20 minus10 0 10 20119909
One wayTwo way
119906(119909)
0
02
04
06
(a)
One wayTwo way
119909
119906(119909)
minus15 minus10 minus5 0 5 10 15minus003
minus002
minus001
0
001
002
(b)
Figure 13 Validity of the solution in Figure 12(a)
of the nonlinear elastic one Since the problem concernsone-way spring force the governing equation for the staticbeam deflection is transformed as in (11) For the solution aniterative procedure is applied for the calculation of the highnonlinearity Some numerical experiments are carried out forshowing the validity and the fast convergence of the appliednumerical method And we can also find that the resultsconverge to the solutions fast for certain external loads
Acknowledgment
This work was supported by a 2-year research grant of PusanNational University
References
[1] M Hetenyi Beams on Elastic Foundation The University ofMichigan Press Ann Arbor Mich USA 1946
[2] C Miranda and K Nair ldquoFinite beams on elastic foundationrdquoJournal of the Structural Division vol 92 pp 131ndash142 1966
[3] B Y Ting ldquoFinite beams on elastic foundation with restraintsrdquoJournal of the Structural Division vol 108 no 3 pp 611ndash6211982
[4] S Timoshenko ldquoMethod of analysis of statistical and dynamicalstress in railrdquo in Proceeding of the International Congress ofApplied Mechanics pp 407ndash418 Zurich Switzerland 1926
[5] J T Kenney ldquoSteady-state vibrations of beam on elastic founda-tion for moving loadrdquo Journal of Applied Mechanics vol 21 pp359ndash364 1954
[6] H Saito and TMurakami ldquoVibrations of an infinite beam on anelastic foundation with consideration of mass of a foundationrdquoThe Japan Society of Mechanical Engineers vol 12 pp 200ndash2051969
[7] L Fryba ldquoInfinitely long beam on elastic foundation undermoving loadrdquo Aplikace Matematiky vol 2 pp 105ndash132 1957
[8] F W Beaufait and P W Hoadley ldquoAnalysis of elastic beams onnonlinear foundationsrdquo Computers amp Structures vol 12 no 5pp 669ndash676 1980
[9] C Massalas ldquoFundamental frequency of vibration of a beam ona non-linear elastic foundationrdquo Journal of Sound andVibrationvol 54 no 4 pp 613ndash615 1977
[10] M Lakshmanan ldquoComment on the fundamental frequency ofvibration of a beam on a non-linear elastic foundationrdquo Journalof Sound and Vibration vol 58 no 3 pp 455ndash457 1978
[11] D Hui ldquoPostbuckling behavior of infinite beams on elasticfoundations using Koiterrsquos improved theoryrdquo International Jour-nal of Non-Linear Mechanics vol 23 no 2 pp 113ndash123 1988
[12] D Z Yankelevsky and M Eisenberger ldquoAnalysis of a beam col-umn on elastic foundationrdquoComputers amp Structures vol 23 no3 pp 351ndash356 1986
[13] M Eisenberger and J Clastornik ldquoBeams on variable two-parameter elastic foundationrdquo Journal of EngineeringMechanicsvol 113 no 10 pp l454ndash1466 1987
[14] D Karamanlidis and V Prakash ldquoExact transfer and stiffnessmatrices for a beamcolumn resting on a two-parameter foun-dationrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 72 no 1 pp 77ndash89 1989
[15] S N Sirosh A Ghali and A G Razaqpur ldquoA general finite ele-ment for beams or beam-columns with or without an elasticfoundationrdquo International Journal for Numerical Methods inEngineering vol 28 no 5 pp 1061ndash1076 1989
[16] K P Soldatos and A P S Selvadurai ldquoFlexure of beams rest-ing on hyperbolic elastic foundationsrdquo International Journal ofSolids amp Structures vol 21 no 4 pp 373ndash388 1985
[17] S Y Lee H Y Ke and Y H Kuo ldquoExact static deflection ofa non-uniform bernoulli-euler beam with general elastic endrestraintsrdquoComputersampStructures vol 36 no 1 pp 91ndash97 1990
[18] S Y Lee and H Y Ke ldquoFree vibrations of a non-uniformbeam with general elastically restrained boundary conditionsrdquoJournal of Sound andVibration vol 136 no 3 pp 425ndash437 1990
[19] S Y Lee H Y Ke and Y H Kuo ldquoAnalysis of non-uniformbeam vibrationrdquo Journal of Sound and Vibration vol 142 no1 pp 15ndash29 1990
[20] Y H Kuo and S Y Lee ldquoDeflection of nonuniform beams rest-ing on a nonlinear elastic foundationrdquo Computers amp Structuresvol 51 no 5 pp 513ndash519 1994
[21] T S Jang H S Baek and J K Paik ldquoA newmethod for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundationrdquo International Journal of Non-LinearMechanics vol 46 no 1 pp 339ndash346 2011
[22] T S Jang and H G Sung ldquoA new semi-analytical method forthe non-linear static analysis of an infinite beam on a non-linear
10 Journal of Applied Mathematics
elastic foundation a general approach to a variable beam cross-sectionrdquo International Journal of Non-Linear Mechanics vol 47pp 132ndash139 2012
[23] S W Choi and T S Jang ldquoExistence and uniqueness of non-linear deflections of an infinite beam resting on a non-uniformnonlinear elastic foundationrdquo Boundary Value Problems vol2012 article 5 2012
[24] T S Jang ldquoA new semi-analytical approach to large deflectionsof BernoullindashEuler-v Karman beams on a linear elastic founda-tion nonlinear analysis of infinite beamsrdquo International Journalof Mechanical Sciences vol 66 pp 22ndash32 2013
[25] M D Greenberg Foundations of Applied Mathematics Pren-tice-Hall New Jersey NJ USA 1978
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
100
101
102
103
0
02
04
06
The number of iterations (119899)
Erro
r (119899
)
(a)
100
101
102
103
The number of iterations (119899)
Erro
r (119899
)
minus01
0
01
02
03
04
05
(b)
100
101
102
103
0
02
04
06
The number of iterations (119899)
Erro
r (119899
)
(c)
Figure 7 Errors between the exact solutions and iterative solutions
minus10 minus5 0 5 10119909
minus1
0
1
2
3
119908no
rmal
(single)(119909)
Figure 8 Applied loading
Figure 9 presents the convergence behavior of the solutionsaccording to the 3 cases of 119896
119901 119864 = 119868 = 119896 = 1
and 120574 = 02 As specified in Section 21 119896119901does not
affect the converged iterative solutions but the convergencecharacteristics changed The solution converged to a steadystate faster when 119896
119901is small
In Figure 10 two cases of converged solutions are com-pared the numerical results using the one-way and two-wayspring models The loading condition is 119908normal (single)(119909) in(18)They are different only when the beamdeflected upwardthat is the deflections near 119909 = plusmn5 are larger when the beamis separable (or free) from the foundation whereas the one-way spring model is not Of course at 119909 = 0 the solutions areequivalent because the physical system is the same in Figure 1
for 119906 gt 0 Therefore Figure 10 shows the validity of theapplied iterative method for the nonlinear spring model
Another loading condition in Figure 11 is taken as follows
119908normal (119909) = 1 4 le |119909| le 5 1 le |119909| le 2
0 otherwise(19)
Other numerical experiments are also conducted accordingto the 3 cases of a linear spring coefficient 119896 whereas theproperties (119864 = 119868 = 119896 = 1 and 120574 = 02) are fixed andthe solution converged (Figure 12) Figure 13 compares thesolutions from the one-way and two-way spring models 119864 =
119868 = 119896 = 1 120574 = 02 and 119896119901= 3
Journal of Applied Mathematics 7
minus02
0
02
04
06
08
119899 = 1
119899 = 2
119899 = 5
119899 = 10
119899 = 30
minus20 minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
(a)
minus02
0
02
04
06
08
119899 = 1
119899 = 3
119899 = 10
119899 = 30
119899 = 60
minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
minus20
(b)
minus02
0
02
04
06
08
119899 = 1
119899 = 3
119899 = 10
119899 = 40
119899 = 90
minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
minus20
(c)
Figure 9 Convergence behavior of the applied iterative method 119864 = 119868 = 119896 = 1 120574 = 02 (a) 119896119901= 3 (b) 119896
119901= 6 and (c) 119896
119901= 10
minus20 minus10 0 10 20119909
119906(119909)
One wayTwo way
0
02
04
06
08
minus02
(a)
One wayTwo way
minus5 0 5
minus004
minus002
0
002
004
006
119909
119906(119909)
(b)
Figure 10 Validity of the solution
8 Journal of Applied Mathematics
minus10 minus5 0 5 10119909
minus1
0
1
2
3
119908no
rmal(119909)
Figure 11 Applied loading conditions
119899 = 1
119899 = 2
119899 = 4
119899 = 20
119899 = 40
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(a)
119899 = 1
119899 = 2
119899 = 3
119899 = 10
119899 = 20
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(b)
119899 = 1
119899 = 2
119899 = 3
119899 = 5
119899 = 10
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(c)
Figure 12 Convergence behavior of the solutions 119864 = 119868 = 1 120574 = 02 and 119896119901= 3 (a) 119896 = 1 (b) 119896 = 15 and (c) 119896 = 2
Finally the validity and accuracy of the applied iterativeprocedure are investigated The nonlinear spring force isconsidered and the iterative procedure is applied successfullyfor the solution Convergence of the deflections according tothe external loading conditions is observed for the effects ofthe pseudo 119896
119901and real 119896
4 Conclusion
In this work we succeeded in applying the numerical methodproposed by Jang et al [21] to find the static deflection ofan infinite beam on a full nonlinear elastic foundation Forthat one-way spring model is considered for the formulation
Journal of Applied Mathematics 9
minus20 minus10 0 10 20119909
One wayTwo way
119906(119909)
0
02
04
06
(a)
One wayTwo way
119909
119906(119909)
minus15 minus10 minus5 0 5 10 15minus003
minus002
minus001
0
001
002
(b)
Figure 13 Validity of the solution in Figure 12(a)
of the nonlinear elastic one Since the problem concernsone-way spring force the governing equation for the staticbeam deflection is transformed as in (11) For the solution aniterative procedure is applied for the calculation of the highnonlinearity Some numerical experiments are carried out forshowing the validity and the fast convergence of the appliednumerical method And we can also find that the resultsconverge to the solutions fast for certain external loads
Acknowledgment
This work was supported by a 2-year research grant of PusanNational University
References
[1] M Hetenyi Beams on Elastic Foundation The University ofMichigan Press Ann Arbor Mich USA 1946
[2] C Miranda and K Nair ldquoFinite beams on elastic foundationrdquoJournal of the Structural Division vol 92 pp 131ndash142 1966
[3] B Y Ting ldquoFinite beams on elastic foundation with restraintsrdquoJournal of the Structural Division vol 108 no 3 pp 611ndash6211982
[4] S Timoshenko ldquoMethod of analysis of statistical and dynamicalstress in railrdquo in Proceeding of the International Congress ofApplied Mechanics pp 407ndash418 Zurich Switzerland 1926
[5] J T Kenney ldquoSteady-state vibrations of beam on elastic founda-tion for moving loadrdquo Journal of Applied Mechanics vol 21 pp359ndash364 1954
[6] H Saito and TMurakami ldquoVibrations of an infinite beam on anelastic foundation with consideration of mass of a foundationrdquoThe Japan Society of Mechanical Engineers vol 12 pp 200ndash2051969
[7] L Fryba ldquoInfinitely long beam on elastic foundation undermoving loadrdquo Aplikace Matematiky vol 2 pp 105ndash132 1957
[8] F W Beaufait and P W Hoadley ldquoAnalysis of elastic beams onnonlinear foundationsrdquo Computers amp Structures vol 12 no 5pp 669ndash676 1980
[9] C Massalas ldquoFundamental frequency of vibration of a beam ona non-linear elastic foundationrdquo Journal of Sound andVibrationvol 54 no 4 pp 613ndash615 1977
[10] M Lakshmanan ldquoComment on the fundamental frequency ofvibration of a beam on a non-linear elastic foundationrdquo Journalof Sound and Vibration vol 58 no 3 pp 455ndash457 1978
[11] D Hui ldquoPostbuckling behavior of infinite beams on elasticfoundations using Koiterrsquos improved theoryrdquo International Jour-nal of Non-Linear Mechanics vol 23 no 2 pp 113ndash123 1988
[12] D Z Yankelevsky and M Eisenberger ldquoAnalysis of a beam col-umn on elastic foundationrdquoComputers amp Structures vol 23 no3 pp 351ndash356 1986
[13] M Eisenberger and J Clastornik ldquoBeams on variable two-parameter elastic foundationrdquo Journal of EngineeringMechanicsvol 113 no 10 pp l454ndash1466 1987
[14] D Karamanlidis and V Prakash ldquoExact transfer and stiffnessmatrices for a beamcolumn resting on a two-parameter foun-dationrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 72 no 1 pp 77ndash89 1989
[15] S N Sirosh A Ghali and A G Razaqpur ldquoA general finite ele-ment for beams or beam-columns with or without an elasticfoundationrdquo International Journal for Numerical Methods inEngineering vol 28 no 5 pp 1061ndash1076 1989
[16] K P Soldatos and A P S Selvadurai ldquoFlexure of beams rest-ing on hyperbolic elastic foundationsrdquo International Journal ofSolids amp Structures vol 21 no 4 pp 373ndash388 1985
[17] S Y Lee H Y Ke and Y H Kuo ldquoExact static deflection ofa non-uniform bernoulli-euler beam with general elastic endrestraintsrdquoComputersampStructures vol 36 no 1 pp 91ndash97 1990
[18] S Y Lee and H Y Ke ldquoFree vibrations of a non-uniformbeam with general elastically restrained boundary conditionsrdquoJournal of Sound andVibration vol 136 no 3 pp 425ndash437 1990
[19] S Y Lee H Y Ke and Y H Kuo ldquoAnalysis of non-uniformbeam vibrationrdquo Journal of Sound and Vibration vol 142 no1 pp 15ndash29 1990
[20] Y H Kuo and S Y Lee ldquoDeflection of nonuniform beams rest-ing on a nonlinear elastic foundationrdquo Computers amp Structuresvol 51 no 5 pp 513ndash519 1994
[21] T S Jang H S Baek and J K Paik ldquoA newmethod for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundationrdquo International Journal of Non-LinearMechanics vol 46 no 1 pp 339ndash346 2011
[22] T S Jang and H G Sung ldquoA new semi-analytical method forthe non-linear static analysis of an infinite beam on a non-linear
10 Journal of Applied Mathematics
elastic foundation a general approach to a variable beam cross-sectionrdquo International Journal of Non-Linear Mechanics vol 47pp 132ndash139 2012
[23] S W Choi and T S Jang ldquoExistence and uniqueness of non-linear deflections of an infinite beam resting on a non-uniformnonlinear elastic foundationrdquo Boundary Value Problems vol2012 article 5 2012
[24] T S Jang ldquoA new semi-analytical approach to large deflectionsof BernoullindashEuler-v Karman beams on a linear elastic founda-tion nonlinear analysis of infinite beamsrdquo International Journalof Mechanical Sciences vol 66 pp 22ndash32 2013
[25] M D Greenberg Foundations of Applied Mathematics Pren-tice-Hall New Jersey NJ USA 1978
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
minus02
0
02
04
06
08
119899 = 1
119899 = 2
119899 = 5
119899 = 10
119899 = 30
minus20 minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
(a)
minus02
0
02
04
06
08
119899 = 1
119899 = 3
119899 = 10
119899 = 30
119899 = 60
minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
minus20
(b)
minus02
0
02
04
06
08
119899 = 1
119899 = 3
119899 = 10
119899 = 40
119899 = 90
minus10 0 10 20119909
minus15 minus5 5 15
119906(119909)
minus20
(c)
Figure 9 Convergence behavior of the applied iterative method 119864 = 119868 = 119896 = 1 120574 = 02 (a) 119896119901= 3 (b) 119896
119901= 6 and (c) 119896
119901= 10
minus20 minus10 0 10 20119909
119906(119909)
One wayTwo way
0
02
04
06
08
minus02
(a)
One wayTwo way
minus5 0 5
minus004
minus002
0
002
004
006
119909
119906(119909)
(b)
Figure 10 Validity of the solution
8 Journal of Applied Mathematics
minus10 minus5 0 5 10119909
minus1
0
1
2
3
119908no
rmal(119909)
Figure 11 Applied loading conditions
119899 = 1
119899 = 2
119899 = 4
119899 = 20
119899 = 40
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(a)
119899 = 1
119899 = 2
119899 = 3
119899 = 10
119899 = 20
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(b)
119899 = 1
119899 = 2
119899 = 3
119899 = 5
119899 = 10
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(c)
Figure 12 Convergence behavior of the solutions 119864 = 119868 = 1 120574 = 02 and 119896119901= 3 (a) 119896 = 1 (b) 119896 = 15 and (c) 119896 = 2
Finally the validity and accuracy of the applied iterativeprocedure are investigated The nonlinear spring force isconsidered and the iterative procedure is applied successfullyfor the solution Convergence of the deflections according tothe external loading conditions is observed for the effects ofthe pseudo 119896
119901and real 119896
4 Conclusion
In this work we succeeded in applying the numerical methodproposed by Jang et al [21] to find the static deflection ofan infinite beam on a full nonlinear elastic foundation Forthat one-way spring model is considered for the formulation
Journal of Applied Mathematics 9
minus20 minus10 0 10 20119909
One wayTwo way
119906(119909)
0
02
04
06
(a)
One wayTwo way
119909
119906(119909)
minus15 minus10 minus5 0 5 10 15minus003
minus002
minus001
0
001
002
(b)
Figure 13 Validity of the solution in Figure 12(a)
of the nonlinear elastic one Since the problem concernsone-way spring force the governing equation for the staticbeam deflection is transformed as in (11) For the solution aniterative procedure is applied for the calculation of the highnonlinearity Some numerical experiments are carried out forshowing the validity and the fast convergence of the appliednumerical method And we can also find that the resultsconverge to the solutions fast for certain external loads
Acknowledgment
This work was supported by a 2-year research grant of PusanNational University
References
[1] M Hetenyi Beams on Elastic Foundation The University ofMichigan Press Ann Arbor Mich USA 1946
[2] C Miranda and K Nair ldquoFinite beams on elastic foundationrdquoJournal of the Structural Division vol 92 pp 131ndash142 1966
[3] B Y Ting ldquoFinite beams on elastic foundation with restraintsrdquoJournal of the Structural Division vol 108 no 3 pp 611ndash6211982
[4] S Timoshenko ldquoMethod of analysis of statistical and dynamicalstress in railrdquo in Proceeding of the International Congress ofApplied Mechanics pp 407ndash418 Zurich Switzerland 1926
[5] J T Kenney ldquoSteady-state vibrations of beam on elastic founda-tion for moving loadrdquo Journal of Applied Mechanics vol 21 pp359ndash364 1954
[6] H Saito and TMurakami ldquoVibrations of an infinite beam on anelastic foundation with consideration of mass of a foundationrdquoThe Japan Society of Mechanical Engineers vol 12 pp 200ndash2051969
[7] L Fryba ldquoInfinitely long beam on elastic foundation undermoving loadrdquo Aplikace Matematiky vol 2 pp 105ndash132 1957
[8] F W Beaufait and P W Hoadley ldquoAnalysis of elastic beams onnonlinear foundationsrdquo Computers amp Structures vol 12 no 5pp 669ndash676 1980
[9] C Massalas ldquoFundamental frequency of vibration of a beam ona non-linear elastic foundationrdquo Journal of Sound andVibrationvol 54 no 4 pp 613ndash615 1977
[10] M Lakshmanan ldquoComment on the fundamental frequency ofvibration of a beam on a non-linear elastic foundationrdquo Journalof Sound and Vibration vol 58 no 3 pp 455ndash457 1978
[11] D Hui ldquoPostbuckling behavior of infinite beams on elasticfoundations using Koiterrsquos improved theoryrdquo International Jour-nal of Non-Linear Mechanics vol 23 no 2 pp 113ndash123 1988
[12] D Z Yankelevsky and M Eisenberger ldquoAnalysis of a beam col-umn on elastic foundationrdquoComputers amp Structures vol 23 no3 pp 351ndash356 1986
[13] M Eisenberger and J Clastornik ldquoBeams on variable two-parameter elastic foundationrdquo Journal of EngineeringMechanicsvol 113 no 10 pp l454ndash1466 1987
[14] D Karamanlidis and V Prakash ldquoExact transfer and stiffnessmatrices for a beamcolumn resting on a two-parameter foun-dationrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 72 no 1 pp 77ndash89 1989
[15] S N Sirosh A Ghali and A G Razaqpur ldquoA general finite ele-ment for beams or beam-columns with or without an elasticfoundationrdquo International Journal for Numerical Methods inEngineering vol 28 no 5 pp 1061ndash1076 1989
[16] K P Soldatos and A P S Selvadurai ldquoFlexure of beams rest-ing on hyperbolic elastic foundationsrdquo International Journal ofSolids amp Structures vol 21 no 4 pp 373ndash388 1985
[17] S Y Lee H Y Ke and Y H Kuo ldquoExact static deflection ofa non-uniform bernoulli-euler beam with general elastic endrestraintsrdquoComputersampStructures vol 36 no 1 pp 91ndash97 1990
[18] S Y Lee and H Y Ke ldquoFree vibrations of a non-uniformbeam with general elastically restrained boundary conditionsrdquoJournal of Sound andVibration vol 136 no 3 pp 425ndash437 1990
[19] S Y Lee H Y Ke and Y H Kuo ldquoAnalysis of non-uniformbeam vibrationrdquo Journal of Sound and Vibration vol 142 no1 pp 15ndash29 1990
[20] Y H Kuo and S Y Lee ldquoDeflection of nonuniform beams rest-ing on a nonlinear elastic foundationrdquo Computers amp Structuresvol 51 no 5 pp 513ndash519 1994
[21] T S Jang H S Baek and J K Paik ldquoA newmethod for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundationrdquo International Journal of Non-LinearMechanics vol 46 no 1 pp 339ndash346 2011
[22] T S Jang and H G Sung ldquoA new semi-analytical method forthe non-linear static analysis of an infinite beam on a non-linear
10 Journal of Applied Mathematics
elastic foundation a general approach to a variable beam cross-sectionrdquo International Journal of Non-Linear Mechanics vol 47pp 132ndash139 2012
[23] S W Choi and T S Jang ldquoExistence and uniqueness of non-linear deflections of an infinite beam resting on a non-uniformnonlinear elastic foundationrdquo Boundary Value Problems vol2012 article 5 2012
[24] T S Jang ldquoA new semi-analytical approach to large deflectionsof BernoullindashEuler-v Karman beams on a linear elastic founda-tion nonlinear analysis of infinite beamsrdquo International Journalof Mechanical Sciences vol 66 pp 22ndash32 2013
[25] M D Greenberg Foundations of Applied Mathematics Pren-tice-Hall New Jersey NJ USA 1978
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
minus10 minus5 0 5 10119909
minus1
0
1
2
3
119908no
rmal(119909)
Figure 11 Applied loading conditions
119899 = 1
119899 = 2
119899 = 4
119899 = 20
119899 = 40
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(a)
119899 = 1
119899 = 2
119899 = 3
119899 = 10
119899 = 20
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(b)
119899 = 1
119899 = 2
119899 = 3
119899 = 5
119899 = 10
minus10minus20 0 10 20119909
minus15 minus5 5 15
05
04
03
02
01
0
119906(119909)
(c)
Figure 12 Convergence behavior of the solutions 119864 = 119868 = 1 120574 = 02 and 119896119901= 3 (a) 119896 = 1 (b) 119896 = 15 and (c) 119896 = 2
Finally the validity and accuracy of the applied iterativeprocedure are investigated The nonlinear spring force isconsidered and the iterative procedure is applied successfullyfor the solution Convergence of the deflections according tothe external loading conditions is observed for the effects ofthe pseudo 119896
119901and real 119896
4 Conclusion
In this work we succeeded in applying the numerical methodproposed by Jang et al [21] to find the static deflection ofan infinite beam on a full nonlinear elastic foundation Forthat one-way spring model is considered for the formulation
Journal of Applied Mathematics 9
minus20 minus10 0 10 20119909
One wayTwo way
119906(119909)
0
02
04
06
(a)
One wayTwo way
119909
119906(119909)
minus15 minus10 minus5 0 5 10 15minus003
minus002
minus001
0
001
002
(b)
Figure 13 Validity of the solution in Figure 12(a)
of the nonlinear elastic one Since the problem concernsone-way spring force the governing equation for the staticbeam deflection is transformed as in (11) For the solution aniterative procedure is applied for the calculation of the highnonlinearity Some numerical experiments are carried out forshowing the validity and the fast convergence of the appliednumerical method And we can also find that the resultsconverge to the solutions fast for certain external loads
Acknowledgment
This work was supported by a 2-year research grant of PusanNational University
References
[1] M Hetenyi Beams on Elastic Foundation The University ofMichigan Press Ann Arbor Mich USA 1946
[2] C Miranda and K Nair ldquoFinite beams on elastic foundationrdquoJournal of the Structural Division vol 92 pp 131ndash142 1966
[3] B Y Ting ldquoFinite beams on elastic foundation with restraintsrdquoJournal of the Structural Division vol 108 no 3 pp 611ndash6211982
[4] S Timoshenko ldquoMethod of analysis of statistical and dynamicalstress in railrdquo in Proceeding of the International Congress ofApplied Mechanics pp 407ndash418 Zurich Switzerland 1926
[5] J T Kenney ldquoSteady-state vibrations of beam on elastic founda-tion for moving loadrdquo Journal of Applied Mechanics vol 21 pp359ndash364 1954
[6] H Saito and TMurakami ldquoVibrations of an infinite beam on anelastic foundation with consideration of mass of a foundationrdquoThe Japan Society of Mechanical Engineers vol 12 pp 200ndash2051969
[7] L Fryba ldquoInfinitely long beam on elastic foundation undermoving loadrdquo Aplikace Matematiky vol 2 pp 105ndash132 1957
[8] F W Beaufait and P W Hoadley ldquoAnalysis of elastic beams onnonlinear foundationsrdquo Computers amp Structures vol 12 no 5pp 669ndash676 1980
[9] C Massalas ldquoFundamental frequency of vibration of a beam ona non-linear elastic foundationrdquo Journal of Sound andVibrationvol 54 no 4 pp 613ndash615 1977
[10] M Lakshmanan ldquoComment on the fundamental frequency ofvibration of a beam on a non-linear elastic foundationrdquo Journalof Sound and Vibration vol 58 no 3 pp 455ndash457 1978
[11] D Hui ldquoPostbuckling behavior of infinite beams on elasticfoundations using Koiterrsquos improved theoryrdquo International Jour-nal of Non-Linear Mechanics vol 23 no 2 pp 113ndash123 1988
[12] D Z Yankelevsky and M Eisenberger ldquoAnalysis of a beam col-umn on elastic foundationrdquoComputers amp Structures vol 23 no3 pp 351ndash356 1986
[13] M Eisenberger and J Clastornik ldquoBeams on variable two-parameter elastic foundationrdquo Journal of EngineeringMechanicsvol 113 no 10 pp l454ndash1466 1987
[14] D Karamanlidis and V Prakash ldquoExact transfer and stiffnessmatrices for a beamcolumn resting on a two-parameter foun-dationrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 72 no 1 pp 77ndash89 1989
[15] S N Sirosh A Ghali and A G Razaqpur ldquoA general finite ele-ment for beams or beam-columns with or without an elasticfoundationrdquo International Journal for Numerical Methods inEngineering vol 28 no 5 pp 1061ndash1076 1989
[16] K P Soldatos and A P S Selvadurai ldquoFlexure of beams rest-ing on hyperbolic elastic foundationsrdquo International Journal ofSolids amp Structures vol 21 no 4 pp 373ndash388 1985
[17] S Y Lee H Y Ke and Y H Kuo ldquoExact static deflection ofa non-uniform bernoulli-euler beam with general elastic endrestraintsrdquoComputersampStructures vol 36 no 1 pp 91ndash97 1990
[18] S Y Lee and H Y Ke ldquoFree vibrations of a non-uniformbeam with general elastically restrained boundary conditionsrdquoJournal of Sound andVibration vol 136 no 3 pp 425ndash437 1990
[19] S Y Lee H Y Ke and Y H Kuo ldquoAnalysis of non-uniformbeam vibrationrdquo Journal of Sound and Vibration vol 142 no1 pp 15ndash29 1990
[20] Y H Kuo and S Y Lee ldquoDeflection of nonuniform beams rest-ing on a nonlinear elastic foundationrdquo Computers amp Structuresvol 51 no 5 pp 513ndash519 1994
[21] T S Jang H S Baek and J K Paik ldquoA newmethod for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundationrdquo International Journal of Non-LinearMechanics vol 46 no 1 pp 339ndash346 2011
[22] T S Jang and H G Sung ldquoA new semi-analytical method forthe non-linear static analysis of an infinite beam on a non-linear
10 Journal of Applied Mathematics
elastic foundation a general approach to a variable beam cross-sectionrdquo International Journal of Non-Linear Mechanics vol 47pp 132ndash139 2012
[23] S W Choi and T S Jang ldquoExistence and uniqueness of non-linear deflections of an infinite beam resting on a non-uniformnonlinear elastic foundationrdquo Boundary Value Problems vol2012 article 5 2012
[24] T S Jang ldquoA new semi-analytical approach to large deflectionsof BernoullindashEuler-v Karman beams on a linear elastic founda-tion nonlinear analysis of infinite beamsrdquo International Journalof Mechanical Sciences vol 66 pp 22ndash32 2013
[25] M D Greenberg Foundations of Applied Mathematics Pren-tice-Hall New Jersey NJ USA 1978
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 9
minus20 minus10 0 10 20119909
One wayTwo way
119906(119909)
0
02
04
06
(a)
One wayTwo way
119909
119906(119909)
minus15 minus10 minus5 0 5 10 15minus003
minus002
minus001
0
001
002
(b)
Figure 13 Validity of the solution in Figure 12(a)
of the nonlinear elastic one Since the problem concernsone-way spring force the governing equation for the staticbeam deflection is transformed as in (11) For the solution aniterative procedure is applied for the calculation of the highnonlinearity Some numerical experiments are carried out forshowing the validity and the fast convergence of the appliednumerical method And we can also find that the resultsconverge to the solutions fast for certain external loads
Acknowledgment
This work was supported by a 2-year research grant of PusanNational University
References
[1] M Hetenyi Beams on Elastic Foundation The University ofMichigan Press Ann Arbor Mich USA 1946
[2] C Miranda and K Nair ldquoFinite beams on elastic foundationrdquoJournal of the Structural Division vol 92 pp 131ndash142 1966
[3] B Y Ting ldquoFinite beams on elastic foundation with restraintsrdquoJournal of the Structural Division vol 108 no 3 pp 611ndash6211982
[4] S Timoshenko ldquoMethod of analysis of statistical and dynamicalstress in railrdquo in Proceeding of the International Congress ofApplied Mechanics pp 407ndash418 Zurich Switzerland 1926
[5] J T Kenney ldquoSteady-state vibrations of beam on elastic founda-tion for moving loadrdquo Journal of Applied Mechanics vol 21 pp359ndash364 1954
[6] H Saito and TMurakami ldquoVibrations of an infinite beam on anelastic foundation with consideration of mass of a foundationrdquoThe Japan Society of Mechanical Engineers vol 12 pp 200ndash2051969
[7] L Fryba ldquoInfinitely long beam on elastic foundation undermoving loadrdquo Aplikace Matematiky vol 2 pp 105ndash132 1957
[8] F W Beaufait and P W Hoadley ldquoAnalysis of elastic beams onnonlinear foundationsrdquo Computers amp Structures vol 12 no 5pp 669ndash676 1980
[9] C Massalas ldquoFundamental frequency of vibration of a beam ona non-linear elastic foundationrdquo Journal of Sound andVibrationvol 54 no 4 pp 613ndash615 1977
[10] M Lakshmanan ldquoComment on the fundamental frequency ofvibration of a beam on a non-linear elastic foundationrdquo Journalof Sound and Vibration vol 58 no 3 pp 455ndash457 1978
[11] D Hui ldquoPostbuckling behavior of infinite beams on elasticfoundations using Koiterrsquos improved theoryrdquo International Jour-nal of Non-Linear Mechanics vol 23 no 2 pp 113ndash123 1988
[12] D Z Yankelevsky and M Eisenberger ldquoAnalysis of a beam col-umn on elastic foundationrdquoComputers amp Structures vol 23 no3 pp 351ndash356 1986
[13] M Eisenberger and J Clastornik ldquoBeams on variable two-parameter elastic foundationrdquo Journal of EngineeringMechanicsvol 113 no 10 pp l454ndash1466 1987
[14] D Karamanlidis and V Prakash ldquoExact transfer and stiffnessmatrices for a beamcolumn resting on a two-parameter foun-dationrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 72 no 1 pp 77ndash89 1989
[15] S N Sirosh A Ghali and A G Razaqpur ldquoA general finite ele-ment for beams or beam-columns with or without an elasticfoundationrdquo International Journal for Numerical Methods inEngineering vol 28 no 5 pp 1061ndash1076 1989
[16] K P Soldatos and A P S Selvadurai ldquoFlexure of beams rest-ing on hyperbolic elastic foundationsrdquo International Journal ofSolids amp Structures vol 21 no 4 pp 373ndash388 1985
[17] S Y Lee H Y Ke and Y H Kuo ldquoExact static deflection ofa non-uniform bernoulli-euler beam with general elastic endrestraintsrdquoComputersampStructures vol 36 no 1 pp 91ndash97 1990
[18] S Y Lee and H Y Ke ldquoFree vibrations of a non-uniformbeam with general elastically restrained boundary conditionsrdquoJournal of Sound andVibration vol 136 no 3 pp 425ndash437 1990
[19] S Y Lee H Y Ke and Y H Kuo ldquoAnalysis of non-uniformbeam vibrationrdquo Journal of Sound and Vibration vol 142 no1 pp 15ndash29 1990
[20] Y H Kuo and S Y Lee ldquoDeflection of nonuniform beams rest-ing on a nonlinear elastic foundationrdquo Computers amp Structuresvol 51 no 5 pp 513ndash519 1994
[21] T S Jang H S Baek and J K Paik ldquoA newmethod for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundationrdquo International Journal of Non-LinearMechanics vol 46 no 1 pp 339ndash346 2011
[22] T S Jang and H G Sung ldquoA new semi-analytical method forthe non-linear static analysis of an infinite beam on a non-linear
10 Journal of Applied Mathematics
elastic foundation a general approach to a variable beam cross-sectionrdquo International Journal of Non-Linear Mechanics vol 47pp 132ndash139 2012
[23] S W Choi and T S Jang ldquoExistence and uniqueness of non-linear deflections of an infinite beam resting on a non-uniformnonlinear elastic foundationrdquo Boundary Value Problems vol2012 article 5 2012
[24] T S Jang ldquoA new semi-analytical approach to large deflectionsof BernoullindashEuler-v Karman beams on a linear elastic founda-tion nonlinear analysis of infinite beamsrdquo International Journalof Mechanical Sciences vol 66 pp 22ndash32 2013
[25] M D Greenberg Foundations of Applied Mathematics Pren-tice-Hall New Jersey NJ USA 1978
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
10 Journal of Applied Mathematics
elastic foundation a general approach to a variable beam cross-sectionrdquo International Journal of Non-Linear Mechanics vol 47pp 132ndash139 2012
[23] S W Choi and T S Jang ldquoExistence and uniqueness of non-linear deflections of an infinite beam resting on a non-uniformnonlinear elastic foundationrdquo Boundary Value Problems vol2012 article 5 2012
[24] T S Jang ldquoA new semi-analytical approach to large deflectionsof BernoullindashEuler-v Karman beams on a linear elastic founda-tion nonlinear analysis of infinite beamsrdquo International Journalof Mechanical Sciences vol 66 pp 22ndash32 2013
[25] M D Greenberg Foundations of Applied Mathematics Pren-tice-Hall New Jersey NJ USA 1978
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