Research Article Variational Principles for Bending and...

Research ArticleVariational Principles for Bending and Vibration ofPartially Composite Timoshenko Beams

Halil Oumlzer

Division of Mechanics Mechanical Engineering Department Yıldız Technical University (YTU) Yıldız Besiktas34349 Istanbul Turkey

Correspondence should be addressed to Halil Ozer hozeryildizedutr

Received 19 April 2014 Accepted 29 June 2014 Published 13 July 2014

Academic Editor Robertt A Valente

Copyright copy 2014 Halil Ozer This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Variational principles are established for the partially composite Timoshenko beam using the semi-inverse methodThe principlesare derived directly from governing differential equations for bending and vibration of the beam considered It is concluded that thesemi-inverse method is a powerful tool for searching for variational principles directly from the governing equations Comparisonbetween our results and the results reported in literature is given

1 Introduction

Composite beams composed of different elastic materialshave been widely used in many engineering applicationsThe individual beam components of the composite beamare combined by using the shear connectors Therefore theoverall behavior of the composite beam depends on thestiffness of connectors Connector having infinite stiffnesseliminates any interlayer shear slip between the individualbeam components which leads to the full interaction con-nection However the stiffness of connector has a finite valueand the interlayer slip between the individual componentsoccurs This type of connection is called partial-interactionconnection Therefore analysis of the partial-interactioncomposite beams requires the consideration of the interlayerslip between the beam components The Euler-Bernoullibeam theory has been extensively used in bending vibrationand buckling analyses Ecsedi and Baksa [1] analyzed thestatic behavior of elastic two-layer beams with interlayer slipand developed closed-form solutions for displacements andinterlayer slips Girhammar and Pan [2] presented generalsolutions for the deflection and internal actions for partiallycomposite Euler-Bernoulli beams and beam-columns Ranziet al [3] presented an analytical formulation for the analysis

of two-layered composite beams with longitudinal and ver-tical partial-interaction Their formulation is based on theprinciple of virtual work expressed in terms of the verticaland axial displacements of the two layers The model waspresented in both its weak and its strong forms Xu andWu [4] developed a new plane stress model of compositebeamswith interlayer slips using the one-dimensional theoryThey concluded that the shear force produced by the shearconnectors increases with the increase in rigidity of shearconnectors

However the effect of transverse shear deformation wasneglected in the Euler-Bernoulli beam theory When thebeam is thick the effect of shear deformation becomessignificant and cannot be neglected for a valid analysis Themost widely used and fundamentally simpler theory wasdeveloped by Timoshenko [5] Sousa and da Silva [6] studiedthe behavior of the general case of multilayered compositebeams with interlayer slip under Euler-Bernoulli as well asTimoshenko beam theory (TBT) assumptions Xu andWang[7] formulated the principle of virtual work and recipro-cal theorem of work for the partial-interaction compositebeams using the kinematic assumptions of Timoshenkorsquosbeam theory The variational principles for the frequencyof free vibration and critical load of buckling were also

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 435613 5 pageshttpdxdoiorg1011552014435613

2 Mathematical Problems in Engineering

Shear connector

Centroid of beam 1

Centroid of whole sectionCentroid of beam 2

0

E1 G1 A1 I1 1205881

E2 G2 A2 I2 1205882

L

H

z w

x u

q

m

h2

h1

h = h1 + h2

Figure 1 A two-layer composite beam

deduced Xu and Wang [8] derived the relationships of solu-tions between single-span Euler-Bernoulli and Timoshenkopartial-interaction composite beams

Variational formulations provide the basis for a numberof approximate and numerical methods Recently two sig-nificant variational methods are proposed by He one is thesemi-inverse method [9 10] and the other method is thevariational iteration method [11] The semi-inverse methodis used to establish variational principles directly from thegoverning differential equations His second method thevariational iteration method depends on constructing acorrection functional by a general Lagrange multiplierThenthe optimal value of the Lagrange multiplier is identified byusing the stationary conditions [12 13] However in the semi-inverse method the term involving the Lagrange multiplieris replaced by an unknown function 119865 The semi-inversemethod eliminates two important variational crises one isthat the Lagrange multiplier is equal to zero and the othercrisis is that making the Lagrangian stationary leads to onlysome parts of Euler equations In this study we will applythe semi-inverse method to establish variational principlesdirectly from the governing differential equations definingthe bending and vibration of Timoshenko composite beamwith partial-interaction The variational formulations wereobtained by following the rules of the calculus of variations

2 Timoshenko Composite Beam withPartial-Interaction

Before applying the semi-inverse method the problemis briefly discussed in Figure 1 Figure 1 shows a partial-interaction composite beam that is composed of two-layerbeams with different materials

In Figure 1 119864119894 119866119894 119860119894 119868119894 and 120588

119894(119894 = 1 2) denote the

elasticity modulus shear modulus cross-sectional area andmoment of inertia of two beam components respectively119871 is the beam length 119867 is the beam height and ℎ is thedistance between the centroids of two beam sections 119902and 119898 denote the distributed load and distributed bendingmoment respectively As seen in Figure 1 shear connectorsare used to connect the beam members of the compositebeam Figure 2 shows geometrical relationship among the

interlayer slip rotary angle (the rotation of the cross section)and longitudinal displacements

In Figure 2 120595 is the rotary angle and 119906119904is the interlayer

slip between two beam layers 1199061and 119906

2are the longitudinal

displacements at the centroids of beams 1 and 2 respectivelyFrom Figure 2 the kinematic relationship among the inter-layer slip rotary angle and longitudinal displacements canbe written as follows

119906119904= 1199062minus 1199061+ 120595ℎ (1)

The bending moment shear force and interlayer shear forceare given respectively as [7 8]

119872 = minus119863119889120595

119889119909+ 119864119860ℎ

119889119906119904

119889119909 (2a)

119876 = 119862(119889119908

119889119909minus 120595) (2b)

119876119904= 119896119904119906119904 (2c)

where 119908 denotes the deflection of the composite beam in the119911-direction (see Figure 1) and 119896

119904denotes the rigidity of the

shear connectors The other quantities used in (2a) (2b) and(2c) are defined as follows

119863 = 119863 + ℎ2119864119860 (3a)

119863 = 11986411198681+ 11986421198682 (3b)

119864119860 =1198641119860111986421198602

11986411198601+ 11986421198602

(3c)

119862 = 119896111986611198601+ 119896211986621198602 (3d)

in which 1198961and 119896

2are the shear correction factors of the

Timoshenko beam119863 is the flexural stiffness of the compositebeam in full interaction 119863 is the flexural stiffness of thecomposite beamwithout shear connection119864119860 is the effectiveaxial stiffness and 119862 is the shear rigidity of the whole crosssections Deflection of the composite beam is then obtainedusing the relation below

119908 = 1199080+ 119908slip + 119908shear (4)

Mathematical Problems in Engineering 3

u2 minus u1

h2

h1

h

us

120595h1

120595h2

120595

Figure 2 Kinematic model of a two-layer partially compositeTimoshenko beam [7]

where 1199080is the deflection of the full interaction composite

beam119908slip and119908shear are the additional deflections due to theinterlayer slip and transverse shear deformation respectively

In the next sections using the semi-inverse method wewill illustrate how to establish variational principles directlyfrom the governing differential equations for bending andvibration of the partially composite Timoshenko beams

3 Derivation of Variational Principle forBending of the Composite Beam

Consider the governing differential equations for bending ofthe partial-interaction composite Timoshenko beam underuniformly distributed load and bending moment [7]

minus1198631198892120595

1198891199092+ 119864119860ℎ

1198892

1198891199092(119906119904minus 120595ℎ) minus 119862(

119889119908

119889119909minus 120595) minus 119898 = 0

(5a)

minus119862(1198892119908

1198891199092minus119889120595

119889119909) minus 119902 = 0 (5b)

minus1198641198601198892

1198891199092(119906119904minus 120595ℎ) + 119896

119904119906119904= 0 (5c)

Using the semi-inverse method a trial variational principlecan be constructed as follows [9 10]

119869 (120595 119908 119906119904) = int119871119889119909 (6)

where 119871 is a trial Lagrangian There are many approaches forconstructing the trial Lagrangian see [14ndash17] We search forsuch a trial Lagrangian so that its trial Euler equation gives

one of the governing equations say (5a) Referring to (5a) anenergy-like trial Lagrangian can be constructed as follows

119871 =1

2119863(

119889120595

119889119909)

2

+1

2119864119860[

119889

119889119909(119906119904minus 120595ℎ)]

2

minus 119862120595119889119908

119889119909+1

21205952minus 119898120595 + 119865

1 (119908)

(7)

where 1198651is an unknown function of 119908 andor its derivatives

The advantage of the above trial Lagrangian lies in the factthat the stationary condition with respect to 120595 results in (5a)Now by making (7) stationary with respect to 119908 one can getthe following trial Euler equation for 120575119908

120575119908 119862119889120595

119889119909+1205751198651

120575119908= 0 (8)

where the operator 120575 is called a variational operator and 120575119908 isthe first order variation of119908 120575119865

1120575119908 is called Hersquos variational

derivative with respect to 119908 which is defined as

1205751198651

120575119908=1205971198651

120597119908minus120597

120597119909(1205971198651

1205971199081015840) 119908

1015840=119889119908

119889119909 (9)

We search for such an1198651so that (8) is equivalent to (5b) From

(9) the unknown function 1198651can be determined as

1198651 (119908) =

1

2119862(

119889119908

119889119909)

2

minus 119902119908 (10)

By adding the above relation the trial Lagrangian can berenewed as follows

119871 =1

2119863(

119889120595

119889119909)

2

+1

2119864119860[

119889

119889119909(119906119904minus 120595ℎ)]

2

minus 119862120595119889119908

119889119909+1

21205952

+1

2119862(

119889119908

119889119909)

2

minus 119902119908 minus 119898120595 + 1198652(119906119904)

(11)

It can be easily proved that the stationary condition of theabove Lagrangian with respect to 119908 satisfies (5b) In (11) 119865

2

is a newly introduced undetermined function of 119906119904andor its

derivatives and is free from the variables120595 and119908 Making thenew trial Lagrangian (11) stationary with respect to 119906

119904results

in the relation below

120575119906119904 minus119864119860

1198892

1198891199092(119906119904minus 120595ℎ) +

1205751198652

120575119906119904

= 0 (12)

which is the last trial Euler equationThe second term on theleft is the variational derivative with respect to 119906

119904and reads

1205751198652

120575119906119904

=1205971198652

120597119906119904

minus120597

120597119909(1205971198652

1205971199061015840119904

) 1199061015840

119904=119889119906119904

119889119909 (13)

from which the unknown 1198652can be determined as

1198652(119906119904) =

1

21198961199041199062

119904 (14)


Substituting1198652into (11) and rearranging lead to the necessary

variational principle as

119869 (120595 119908 119906119904)

= int

119871

0

1

2119863(

119889120595

119889119909)

2

+1

2119864119860[

119889

119889119909(119906119904minus 120595ℎ)]

2

+1

2119862(

119889119908

119889119909minus 120595)

2

+1

21198961199041199062

119904minus 119902119908 minus 119898120595119889119909

(15)

which is the total potential energy of partial-interaction com-posite Timoshenko beam subjected to uniformly distributedload and bending moment (see [7]) and yields the minimumpotential energy principle by letting 120575119869 = 0

Proof Making the above functional (15) stationary withrespect to 120595 119908 and 119906

119904 the Euler equations turn out to be

(5a)ndash(5c) respectively

The Ritz method can be used to obtain an approximateanalytical solution of the problemWe canwrite the one-termtrial functions which satisfy the boundary conditions as

119908 = 11990801198911 (119909) 120595 = 120595

01198912 (119909) 119906

119904= 11990601198913 (119909) (16)

where 1199080 1205950 and 119906

0are unknown constants which can be

determined from the following stationary conditions

120597119869

1205971199080

= 0120597119869

1205971205950

= 0120597119869

1205971199060

= 0 (17)

By solving the system of (17) simultaneously the unknownconstants can then be obtained

4 Derivation of Variational Principle for FreeVibration of the Composite Beam

Differential equations of motion for partial-interaction com-posite members under uniformly distributed load and bend-ing moment can be written as [7]

minus 1198631205972120595

1205971199092+ 119864119860ℎ

1205972

1205971199092(119906119904minus 120595ℎ)

minus 119862(120597119908

120597119909minus 120595) + 120588119868

0

1205972120595

1205971199052minus 119898 = 0

(18a)

minus119862(1205972119908

1205971199092minus120597120595

120597119909) + 120588119860

0

1205972119908

1205971199052minus 119902 = 0 (18b)

minus1198641198601205972

1205971199092(119906119904minus 120595ℎ) + 119896

119904119906119904= 0 (18c)

where 119905 denotes time 1205881198600= 12058811198601+ 12058821198602 and 120588119868

0= 12058811198681+

12058821198682We can construct the following trial variational principle

using the semi-inverse method [9 10]

119869 (120595 119908 119906119904) = ∬119871119889119909119889119905 (19)

Similarly referring to (18a) and making some modificationsso that the stationary condition with respect to120595 can identify(18a) lead us to the following trial Lagrangian

119871 =1

2119863(

120597120595

120597119909)

2

+1

2119864119860[

120597

120597119909(119906119904minus 120595ℎ)]

2

minus 119862120595120597119908

120597119909

+1

21205952minus1

21205881198680(120597120595

120597119905)

2

minus 119898120595 + 1198653 (119908)

(20)

with 1198653being an unknown function of 119908 andor its deriva-

tives As can be seen easily the stationary condition of theabove Lagrangian with respect to 120595 results in (18a) Nowmaking (20) stationary with respect to 119908 we obtain thefollowing trial Euler equation

120575119908 119862120597120595

120597119909+1205751198653

120575119908= 0 (21)

where 1205751198653120575119908 is defined as

1205751198653

120575119908=1205971198653

120597119908minus120597

120597119909(1205971198653

1205971199081015840) minus

120597

120597119905(1205971198653

120597) +

1205972

1205971199092(1205971198653

12059711990810158401015840)

+1205972

1205971199052(1205971198653

120597) + sdot sdot sdot =

120597119908

120597119905 1199081015840=120597119908

120597119909

(22)

From the above relation we can identify 1198653in the form

1198653 (119908) =

1

2119862(

120597119908

120597119909)

2

minus1

21205881198600(120597119908

120597119905)

2

minus 119902119908 (23)

Then the Lagrangian (20) is further updated as follows

119871 =1

2119863(

120597120595

120597119909)

2

+1

2119864119860[

120597

120597119909(119906119904minus 120595ℎ)]

2

minus 119862120595120597119908

120597119909+1

21205952

minus 119898120595 minus1

21205881198680(120597120595

120597119905)

2

+1

2119862(

120597119908

120597119909)

2

minus1

21205881198600(120597119908

120597119905)

2

minus 119902119908 + 1198654(119906119904)

(24)

It is obvious that making the renewed trial functional sta-tionary with respect to 119908 satisfies (18b) In (24) 119865

4is a new

undetermined function of 119906119904andor its derivatives It must

be noted that (18c) has the same form as (5c) Therefore byfollowing the same steps as before (see (12)-(13)) it is easilyseen that 119865

4= 1198652 Finally we can easily arrive at the required

variational principle

119869 (120595 119908 119906119904)

= ∬1

2119863(

120597120595

120597119909)

2

+1

2119864119860[

120597

120597119909(119906119904minus 120595ℎ)]

2

+1

2119862(

120597119908

120597119909minus 120595)

2

+1

21198961199041199062

119904

minus1

2[1205881198680(120597120595

120597119905)

2

+1

21205881198600(120597119908

120597119905)

2

]minus119902119908minus119898120595119889119909119889119905

(25)


The above functional is the same as that reported in [7]and yields Hamiltonrsquos principle by letting 120575119869 = 0 The fifthterm inside the braces is the kinetic energyof the beamcomponentsand reads

119879 =1

2int

119871

0

[1205881198680(120597120595

120597119905)

2

+1

21205881198600(120597119908

120597119905)

2

]119889119909 (26)


119904 the Euler equations correspond to


By following the same procedures performed for beambending the approximate solutions are obtained conve-niently for beam vibrating by the Ritz method

5 Conclusion

We used the semi-inverse method to establish a set ofvariational principles directly from governing differentialequations By following the rules of the calculus of variationswe obtained necessary variational principles for bending andvibration of the Timoshenko composite beam with partial-interaction The obtained variational principles have beencompared with those reported in literature and proved tobe correct It is concluded that the semi-inverse methodis a powerful tool for searching for variational principlesdirectly from the governing equations Moreover introduc-ing an unknown function instead of a Lagrange multiplieradditional variational principles can also be written byconstraining the trial Lagrangian with the different bound-ary conditions which may facilitate the implementationof complicated boundary conditions The direct variationalmethod such as the Ritz method can be used to obtain theapproximate solutions of the problem

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] I Ecsedi and A Baksa ldquoStatic analysis of composite beams withweak shear connectionrdquo Applied Mathematical Modelling vol35 no 4 pp 1739ndash1750 2011

[2] U A Girhammar and D H Pan ldquoExact static analysis ofpartially composite beams and beam-columnsrdquo InternationalJournal of Mechanical Sciences vol 49 no 2 pp 239ndash255 2007

[3] G Ranzi F Gara and P Ansourian ldquoGeneral method ofanalysis for composite beams with longitudinal and transversepartial interactionrdquo Computers amp Structures vol 84 no 31-32pp 2373ndash2384 2006

[4] R Xu and Y Wu ldquoTwo-dimensional analytical solutions ofsimply supported composite beams with interlayer slipsrdquo Inter-national Journal of Solids and Structures vol 44 no 1 pp 165ndash175 2007

[5] S P Timoshenko ldquoOn the correction for shear of the differentialequation for transverse vibrations of prismatic barsrdquo Philosoph-ical Magazine vol 41 pp 744ndash746 1921

[6] J B M Sousa and A R da Silva ldquoAnalytical and numericalanalysis of multilayered beams with interlayer sliprdquo EngineeringStructures vol 32 no 6 pp 1671ndash1680 2010

[7] R Xu andGWang ldquoVariational principle of partial-interactioncomposite beams using Timoshenkos beam theoryrdquo Interna-tional Journal of Mechanical Sciences vol 60 no 1 pp 72ndash832012

[8] R Xu and G Wang ldquoBending solutions of the Timoshenkopartial-interaction composite beams using Euler-Bernoullisolutionsrdquo Journal of Engineering Mechanics vol 139 no 12 pp1881ndash1885 2013

[9] J H He ldquoSemi-inverse method and generalized variationalprinciples withmulti-variables in elasticityrdquoAppliedMathemat-ics and Mechanics Yingyong Shuxue he Lixue vol 21 no 7 pp797ndash808 2000

[10] J H He ldquoDerivation of generalized variational principleswithout using lagrange multipliers part I applications to fluidmechanicsrdquo Mechanics Automatic Control and Robotics vol 3no 11 pp 5ndash12 2001

[11] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquoThe International Journalof Non-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[12] H Ozer ldquoApplication of the variational iteration method tothe boundary value problems with jump discontinuities arisingin solid mechanicsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 4 pp 513ndash518 2007

[13] H Ozer ldquoApplication of the variational iterationmethod to thincircular platesrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 9 no 1 pp 25ndash30 2008

[14] J H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[15] J He ldquoA Lagrangian for von Karman equations of largedeflection problem of thin circular platerdquo Applied Mathematicsand Computation vol 143 no 2-3 pp 543ndash549 2003

[16] J H He ldquoOn the semi-inverse method and variational princi-plerdquoThermal Science vol 17 no 5 pp 1565ndash1568 2013

[17] I Kucuk I S Sadek and S Adali ldquoVariational principles formultiwalled carbon nanotubes undergoing vibrations based onnonlocal timoshenko beam theoryrdquo Journal of Nanomaterialsvol 2010 Article ID 461252 7 pages 2010

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Stochastic AnalysisInternational Journal of


Shear connector

Centroid of beam 1

Centroid of whole sectionCentroid of beam 2

0

E1 G1 A1 I1 1205881

E2 G2 A2 I2 1205882

L

H

z w

x u

q

m

h2

h1

h = h1 + h2

Figure 1 A two-layer composite beam

deduced Xu and Wang [8] derived the relationships of solu-tions between single-span Euler-Bernoulli and Timoshenkopartial-interaction composite beams

Variational formulations provide the basis for a numberof approximate and numerical methods Recently two sig-nificant variational methods are proposed by He one is thesemi-inverse method [9 10] and the other method is thevariational iteration method [11] The semi-inverse methodis used to establish variational principles directly from thegoverning differential equations His second method thevariational iteration method depends on constructing acorrection functional by a general Lagrange multiplierThenthe optimal value of the Lagrange multiplier is identified byusing the stationary conditions [12 13] However in the semi-inverse method the term involving the Lagrange multiplieris replaced by an unknown function 119865 The semi-inversemethod eliminates two important variational crises one isthat the Lagrange multiplier is equal to zero and the othercrisis is that making the Lagrangian stationary leads to onlysome parts of Euler equations In this study we will applythe semi-inverse method to establish variational principlesdirectly from the governing differential equations definingthe bending and vibration of Timoshenko composite beamwith partial-interaction The variational formulations wereobtained by following the rules of the calculus of variations

2 Timoshenko Composite Beam withPartial-Interaction

Before applying the semi-inverse method the problemis briefly discussed in Figure 1 Figure 1 shows a partial-interaction composite beam that is composed of two-layerbeams with different materials

In Figure 1 119864119894 119866119894 119860119894 119868119894 and 120588

119894(119894 = 1 2) denote the

elasticity modulus shear modulus cross-sectional area andmoment of inertia of two beam components respectively119871 is the beam length 119867 is the beam height and ℎ is thedistance between the centroids of two beam sections 119902and 119898 denote the distributed load and distributed bendingmoment respectively As seen in Figure 1 shear connectorsare used to connect the beam members of the compositebeam Figure 2 shows geometrical relationship among the

interlayer slip rotary angle (the rotation of the cross section)and longitudinal displacements

In Figure 2 120595 is the rotary angle and 119906119904is the interlayer

slip between two beam layers 1199061and 119906

2are the longitudinal

displacements at the centroids of beams 1 and 2 respectivelyFrom Figure 2 the kinematic relationship among the inter-layer slip rotary angle and longitudinal displacements canbe written as follows

119906119904= 1199062minus 1199061+ 120595ℎ (1)

The bending moment shear force and interlayer shear forceare given respectively as [7 8]

119872 = minus119863119889120595

119889119909+ 119864119860ℎ

119889119906119904

119889119909 (2a)

119876 = 119862(119889119908

119889119909minus 120595) (2b)

119876119904= 119896119904119906119904 (2c)

where 119908 denotes the deflection of the composite beam in the119911-direction (see Figure 1) and 119896

119904denotes the rigidity of the

shear connectors The other quantities used in (2a) (2b) and(2c) are defined as follows

119863 = 119863 + ℎ2119864119860 (3a)

119863 = 11986411198681+ 11986421198682 (3b)

119864119860 =1198641119860111986421198602

11986411198601+ 11986421198602

(3c)

119862 = 119896111986611198601+ 119896211986621198602 (3d)

in which 1198961and 119896

2are the shear correction factors of the

Timoshenko beam119863 is the flexural stiffness of the compositebeam in full interaction 119863 is the flexural stiffness of thecomposite beamwithout shear connection119864119860 is the effectiveaxial stiffness and 119862 is the shear rigidity of the whole crosssections Deflection of the composite beam is then obtainedusing the relation below

119908 = 1199080+ 119908slip + 119908shear (4)


u2 minus u1

h2

h1

h

us

120595h1

120595h2

120595







minus1198631198892120595

1198891199092+ 119864119860ℎ

1198892

1198891199092(119906119904minus 120595ℎ) minus 119862(

119889119908

119889119909minus 120595) minus 119898 = 0

(5a)

minus119862(1198892119908

1198891199092minus119889120595

119889119909) minus 119902 = 0 (5b)

minus1198641198601198892

1198891199092(119906119904minus 120595ℎ) + 119896

119904119906119904= 0 (5c)


119869 (120595 119908 119906119904) = int119871119889119909 (6)



119871 =1

2119863(

119889120595

119889119909)

2

+1

2119864119860[

119889

119889119909(119906119904minus 120595ℎ)]

2

minus 119862120595119889119908

119889119909+1

21205952minus 119898120595 + 119865

1 (119908)

(7)



120575119908 119862119889120595

119889119909+1205751198651

120575119908= 0 (8)




1205751198651

120575119908=1205971198651

120597119908minus120597

120597119909(1205971198651

1205971199081015840) 119908

1015840=119889119908

119889119909 (9)



1198651 (119908) =

1

2119862(

119889119908

119889119909)

2

minus 119902119908 (10)


119871 =1

2119863(

119889120595

119889119909)

2

+1

2119864119860[

119889

119889119909(119906119904minus 120595ℎ)]

2

minus 119862120595119889119908

119889119909+1

21205952

+1

2119862(

119889119908

119889119909)

2

minus 119902119908 minus 119898120595 + 1198652(119906119904)

(11)


2



119904results


120575119906119904 minus119864119860

1198892

1198891199092(119906119904minus 120595ℎ) +

1205751198652

120575119906119904

= 0 (12)


119904and reads

1205751198652

120575119906119904

=1205971198652

120597119906119904

minus120597

120597119909(1205971198652

1205971199061015840119904

) 1199061015840

119904=119889119906119904

119889119909 (13)


1198652(119906119904) =

1

21198961199041199062

119904 (14)




119869 (120595 119908 119906119904)

= int

119871

0

1

2119863(

119889120595

119889119909)

2

+1

2119864119860[

119889

119889119909(119906119904minus 120595ℎ)]

2

+1

2119862(

119889119908

119889119909minus 120595)

2

+1

21198961199041199062

119904minus 119902119908 minus 119898120595119889119909

(15)






119908 = 11990801198911 (119909) 120595 = 120595

01198912 (119909) 119906

119904= 11990601198913 (119909) (16)

where 1199080 1205950 and 119906



120597119869

1205971199080

= 0120597119869

1205971205950

= 0120597119869

1205971199060

= 0 (17)




minus 1198631205972120595

1205971199092+ 119864119860ℎ

1205972

1205971199092(119906119904minus 120595ℎ)

minus 119862(120597119908

120597119909minus 120595) + 120588119868

0

1205972120595

1205971199052minus 119898 = 0

(18a)

minus119862(1205972119908

1205971199092minus120597120595

120597119909) + 120588119860

0

1205972119908

1205971199052minus 119902 = 0 (18b)

minus1198641198601205972

1205971199092(119906119904minus 120595ℎ) + 119896

119904119906119904= 0 (18c)


0= 12058811198681+



119869 (120595 119908 119906119904) = ∬119871119889119909119889119905 (19)


119871 =1

2119863(

120597120595

120597119909)

2

+1

2119864119860[

120597

120597119909(119906119904minus 120595ℎ)]

2

minus 119862120595120597119908

120597119909

+1

21205952minus1

21205881198680(120597120595

120597119905)

2

minus 119898120595 + 1198653 (119908)

(20)



120575119908 119862120597120595

120597119909+1205751198653

120575119908= 0 (21)


1205751198653

120575119908=1205971198653

120597119908minus120597

120597119909(1205971198653

1205971199081015840) minus

120597

120597119905(1205971198653

120597) +

1205972

1205971199092(1205971198653

12059711990810158401015840)

+1205972

1205971199052(1205971198653


120597119908

120597119905 1199081015840=120597119908

120597119909

(22)


1198653 (119908) =

1

2119862(

120597119908

120597119909)

2

minus1

21205881198600(120597119908

120597119905)

2

minus 119902119908 (23)


119871 =1

2119863(

120597120595

120597119909)

2

+1

2119864119860[

120597

120597119909(119906119904minus 120595ℎ)]

2

minus 119862120595120597119908

120597119909+1

21205952


21205881198680(120597120595

120597119905)

2

+1

2119862(

120597119908

120597119909)

2

minus1

21205881198600(120597119908

120597119905)

2

minus 119902119908 + 1198654(119906119904)

(24)


4is a new





119869 (120595 119908 119906119904)

= ∬1

2119863(

120597120595

120597119909)

2

+1

2119864119860[

120597

120597119909(119906119904minus 120595ℎ)]

2

+1

2119862(

120597119908

120597119909minus 120595)

2

+1

21198961199041199062

119904

minus1

2[1205881198680(120597120595

120597119905)

2

+1

21205881198600(120597119908

120597119905)

2

]minus119902119908minus119898120595119889119909119889119905

(25)



119879 =1

2int

119871

0

[1205881198680(120597120595

120597119905)

2

+1

21205881198600(120597119908

120597119905)

2

]119889119909 (26)





5 Conclusion




References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










u2 minus u1

h2

h1

h

us

120595h1

120595h2

120595







minus1198631198892120595

1198891199092+ 119864119860ℎ

1198892

1198891199092(119906119904minus 120595ℎ) minus 119862(

119889119908

119889119909minus 120595) minus 119898 = 0

(5a)

minus119862(1198892119908

1198891199092minus119889120595

119889119909) minus 119902 = 0 (5b)

minus1198641198601198892

1198891199092(119906119904minus 120595ℎ) + 119896

119904119906119904= 0 (5c)


119869 (120595 119908 119906119904) = int119871119889119909 (6)



119871 =1

2119863(

119889120595

119889119909)

2

+1

2119864119860[

119889

119889119909(119906119904minus 120595ℎ)]

2

minus 119862120595119889119908

119889119909+1

21205952minus 119898120595 + 119865

1 (119908)

(7)



120575119908 119862119889120595

119889119909+1205751198651

120575119908= 0 (8)




1205751198651

120575119908=1205971198651

120597119908minus120597

120597119909(1205971198651

1205971199081015840) 119908

1015840=119889119908

119889119909 (9)



1198651 (119908) =

1

2119862(

119889119908

119889119909)

2

minus 119902119908 (10)


119871 =1

2119863(

119889120595

119889119909)

2

+1

2119864119860[

119889

119889119909(119906119904minus 120595ℎ)]

2

minus 119862120595119889119908

119889119909+1

21205952

+1

2119862(

119889119908

119889119909)

2

minus 119902119908 minus 119898120595 + 1198652(119906119904)

(11)


2



119904results


120575119906119904 minus119864119860

1198892

1198891199092(119906119904minus 120595ℎ) +

1205751198652

120575119906119904

= 0 (12)


119904and reads

1205751198652

120575119906119904

=1205971198652

120597119906119904

minus120597

120597119909(1205971198652

1205971199061015840119904

) 1199061015840

119904=119889119906119904

119889119909 (13)


1198652(119906119904) =

1

21198961199041199062

119904 (14)




119869 (120595 119908 119906119904)

= int

119871

0

1

2119863(

119889120595

119889119909)

2

+1

2119864119860[

119889

119889119909(119906119904minus 120595ℎ)]

2

+1

2119862(

119889119908

119889119909minus 120595)

2

+1

21198961199041199062

119904minus 119902119908 minus 119898120595119889119909

(15)






119908 = 11990801198911 (119909) 120595 = 120595

01198912 (119909) 119906

119904= 11990601198913 (119909) (16)

where 1199080 1205950 and 119906



120597119869

1205971199080

= 0120597119869

1205971205950

= 0120597119869

1205971199060

= 0 (17)




minus 1198631205972120595

1205971199092+ 119864119860ℎ

1205972

1205971199092(119906119904minus 120595ℎ)

minus 119862(120597119908

120597119909minus 120595) + 120588119868

0

1205972120595

1205971199052minus 119898 = 0

(18a)

minus119862(1205972119908

1205971199092minus120597120595

120597119909) + 120588119860

0

1205972119908

1205971199052minus 119902 = 0 (18b)

minus1198641198601205972

1205971199092(119906119904minus 120595ℎ) + 119896

119904119906119904= 0 (18c)


0= 12058811198681+



119869 (120595 119908 119906119904) = ∬119871119889119909119889119905 (19)


119871 =1

2119863(

120597120595

120597119909)

2

+1

2119864119860[

120597

120597119909(119906119904minus 120595ℎ)]

2

minus 119862120595120597119908

120597119909

+1

21205952minus1

21205881198680(120597120595

120597119905)

2

minus 119898120595 + 1198653 (119908)

(20)



120575119908 119862120597120595

120597119909+1205751198653

120575119908= 0 (21)


1205751198653

120575119908=1205971198653

120597119908minus120597

120597119909(1205971198653

1205971199081015840) minus

120597

120597119905(1205971198653

120597) +

1205972

1205971199092(1205971198653

12059711990810158401015840)

+1205972

1205971199052(1205971198653


120597119908

120597119905 1199081015840=120597119908

120597119909

(22)


1198653 (119908) =

1

2119862(

120597119908

120597119909)

2

minus1

21205881198600(120597119908

120597119905)

2

minus 119902119908 (23)


119871 =1

2119863(

120597120595

120597119909)

2

+1

2119864119860[

120597

120597119909(119906119904minus 120595ℎ)]

2

minus 119862120595120597119908

120597119909+1

21205952


21205881198680(120597120595

120597119905)

2

+1

2119862(

120597119908

120597119909)

2

minus1

21205881198600(120597119908

120597119905)

2

minus 119902119908 + 1198654(119906119904)

(24)


4is a new





119869 (120595 119908 119906119904)

= ∬1

2119863(

120597120595

120597119909)

2

+1

2119864119860[

120597

120597119909(119906119904minus 120595ℎ)]

2

+1

2119862(

120597119908

120597119909minus 120595)

2

+1

21198961199041199062

119904

minus1

2[1205881198680(120597120595

120597119905)

2

+1

21205881198600(120597119908

120597119905)

2

]minus119902119908minus119898120595119889119909119889119905

(25)



119879 =1

2int

119871

0

[1205881198680(120597120595

120597119905)

2

+1

21205881198600(120597119908

120597119905)

2

]119889119909 (26)





5 Conclusion




References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119869 (120595 119908 119906119904)

= int

119871

0

1

2119863(

119889120595

119889119909)

2

+1

2119864119860[

119889

119889119909(119906119904minus 120595ℎ)]

2

+1

2119862(

119889119908

119889119909minus 120595)

2

+1

21198961199041199062

119904minus 119902119908 minus 119898120595119889119909

(15)






119908 = 11990801198911 (119909) 120595 = 120595

01198912 (119909) 119906

119904= 11990601198913 (119909) (16)

where 1199080 1205950 and 119906



120597119869

1205971199080

= 0120597119869

1205971205950

= 0120597119869

1205971199060

= 0 (17)




minus 1198631205972120595

1205971199092+ 119864119860ℎ

1205972

1205971199092(119906119904minus 120595ℎ)

minus 119862(120597119908

120597119909minus 120595) + 120588119868

0

1205972120595

1205971199052minus 119898 = 0

(18a)

minus119862(1205972119908

1205971199092minus120597120595

120597119909) + 120588119860

0

1205972119908

1205971199052minus 119902 = 0 (18b)

minus1198641198601205972

1205971199092(119906119904minus 120595ℎ) + 119896

119904119906119904= 0 (18c)


0= 12058811198681+



119869 (120595 119908 119906119904) = ∬119871119889119909119889119905 (19)


119871 =1

2119863(

120597120595

120597119909)

2

+1

2119864119860[

120597

120597119909(119906119904minus 120595ℎ)]

2

minus 119862120595120597119908

120597119909

+1

21205952minus1

21205881198680(120597120595

120597119905)

2

minus 119898120595 + 1198653 (119908)

(20)



120575119908 119862120597120595

120597119909+1205751198653

120575119908= 0 (21)


1205751198653

120575119908=1205971198653

120597119908minus120597

120597119909(1205971198653

1205971199081015840) minus

120597

120597119905(1205971198653

120597) +

1205972

1205971199092(1205971198653

12059711990810158401015840)

+1205972

1205971199052(1205971198653


120597119908

120597119905 1199081015840=120597119908

120597119909

(22)


1198653 (119908) =

1

2119862(

120597119908

120597119909)

2

minus1

21205881198600(120597119908

120597119905)

2

minus 119902119908 (23)


119871 =1

2119863(

120597120595

120597119909)

2

+1

2119864119860[

120597

120597119909(119906119904minus 120595ℎ)]

2

minus 119862120595120597119908

120597119909+1

21205952


21205881198680(120597120595

120597119905)

2

+1

2119862(

120597119908

120597119909)

2

minus1

21205881198600(120597119908

120597119905)

2

minus 119902119908 + 1198654(119906119904)

(24)


4is a new





119869 (120595 119908 119906119904)

= ∬1

2119863(

120597120595

120597119909)

2

+1

2119864119860[

120597

120597119909(119906119904minus 120595ℎ)]

2

+1

2119862(

120597119908

120597119909minus 120595)

2

+1

21198961199041199062

119904

minus1

2[1205881198680(120597120595

120597119905)

2

+1

21205881198600(120597119908

120597119905)

2

]minus119902119908minus119898120595119889119909119889119905

(25)



119879 =1

2int

119871

0

[1205881198680(120597120595

120597119905)

2

+1

21205881198600(120597119908

120597119905)

2

]119889119909 (26)





5 Conclusion




References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879 =1

2int

119871

0

[1205881198680(120597120595

120597119905)

2

+1

21205881198600(120597119908

120597119905)

2

]119889119909 (26)





5 Conclusion




References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Variational Principles for Bending and...

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