Research ArticleElement for Beam Dynamic Analysis Based onAnalytical Deflection Trial Function

Qiongqiong Cao1 Min Ding1 Xiugen Jiang1 Jinsan Ju1 Hongzhi Wang1 and Peng Zhang12

1College of Water Resources amp Civil Engineering China Agricultural University Beijing 100083 China2China Aerospace Construction Group Co Ltd Beijing 100071 China

Correspondence should be addressed to Xiugen Jiang jiangxgcaueducn

Received 18 September 2014 Accepted 15 December 2014

Academic Editor Chenfeng Li

Copyright copy 2015 Qiongqiong Cao et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

For beam dynamic finite element analysis according to differential equation of motion of beam with distributed mass generalanalytical solution of displacement equation for the beam vibration is obtained By applying displacement element constructionprinciple the general solution of displacement equation is conversed to themode expressed by beam end displacements And takingthe mode as displacement trial function element stiffness matrix and element mass matrix for beam flexural vibration and axialvibration are established respectively by applying principle of minimum potential energy After accurate integral explicit form ofelement matrix is obtained The comparison results show that the series of relative error between the solution of analytical trialfunction element and theoretical solution is about 1times10minus9 and the accuracy and efficiency are superior to that of interpolation trialfunction element The reason is that interpolation trial function cannot accurately simulate the displacement mode of vibratingbeamThe accuracy of dynamic stiffness matrix method is almost identical with that of analytical trial function But the applicationof dynamic stiffness matrix method in engineering is limited The beam dynamic element obtained in this paper is analytical andaccurate and can be applied in practice

1 Introduction

Dynamic structural analysis is essential in structure engi-neering design It is especially important for large-scalestructure in earthquake area such as high-rise buildingsdam hydropower station oil pipelines and gas pipelinesDynamic problem of beam structure is common in engi-neering Vibration happens at beam under earthquake gas-liquid flows and impact Resonance occurs when frequencyof external load is close to natural frequency of structureIt is a great threat for structure safety Therefore perfectingaccuracy and efficiency of dynamic structural analysis is toensure structure safety and reliability

A large number of theories and methods for dynamicstructural analysis have been suggested The analysis meth-ods include direct solving method energy method andnumerical method Clough and Penzien [1] have obtainedanalytical solution of displacement equation for flexuralvibrating beam and presented the first three-order vibration

mode and the corresponding frequency of cantilever beamand simply supported beam Jin [2] provided analyticalsolution of Timoshenko beam clamped two ends and sub-jected to uniformly distributed load according to differentjump condition Guo et al [3] established structure elementproperty matrix by energy principle considering flexuraland torsional deformation of T-beam and the effect ofbridge local member (such as diaphragm plate) Fang andWang [4] attained beam natural vibration frequency byanalyzing dynamic property of external prestressing beamusing energy method and the solution has a better matchwith numerical solution Lou and Hong [5] derived theapproximation analysis technique for dynamic characteristicsof the prestressed beam by applying the mode perturbationmethod Carrer et al [6] analyzed the dynamic behavior ofTimoshenko beam by using boundary element method Wu[7] analyzed the dynamic behavior of two-dimension framewith stiffening bar randomly distributed by using elastic-rigid composite beam element De Rosa et al [8] studied

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 582326 10 pageshttpdxdoiorg1011552015582326

2 Mathematical Problems in Engineering

the dynamic behavior of slender beam with concentratedmass at beam end and numerical solution of frequencyequation was obtained Among them numerical analysisrepresented by finite element is themain and efficientmethodfor dynamic analysis (such as natural vibration analysis andforced vibration analysis) Finite element method was firstlyproposed by Clough [9] in an article about plane elasticproblem and it is perfect in theory as a numerical method

At present themethods of constructing dynamic elementfor beam include dynamic stiffness matrix method Galerkinmethod Ritz method energy variation method Dynamicstiffness matrix can accurately solve differential equationof motion according to initial displacement field withoutany assumption and then accurate results can be obtainedirrespective of element number This method was proposedby Kolousek [10] To gain more accurate results stiffnessmatrix of tensile torsion bar and Euler beam about frequencythat is dynamic stiffness matrix was firstly derived fromanalytical solution for studying vibration characteristics ofplane truss A lot of work on the research and developmentof dynamic stiffness matrix method was also done by Longand Bao [11] Hashemi and Richard [12] Chen et al [13]Banerjee et al [14] and Banerjee et al [15] Shavezipur andHashemi [16] put forward an accurate finite element methodIn this method closed form solution of differential equationof beam not coupling flexural vibration torsional vibrationwas obtained by merging Galerkin weighted residual methodand dynamic stiffness matrix (DSM) Result of dynamicstiffness matrix is more accurate but analytical solutionof differential equation cannot be derived when structureload or displacement boundary condition is too complexThen dynamic stiffness matrix method is not suitable anymore According to principle of virtual displacement a largenumber of research achievements on dynamic analysis ofthin-walled open section beam elastic foundation beamand composite beam have been conducted by Chopra [17]Hu and Dai [18] Wang et al [19] Hashemi and Richard[20] and Pagani et al [21] Nabi and Ganesan [22] putforward a finite element method based on free vibrationanalysis theory of composite beam with the first order sheardeformation Zhao and Chen [23] developed the dynamicanalysis of a unified stochastic variational principle andthe corresponding stochastic finite element method via theinstantaneous minimum potential energy principle and thesmall parameter perturbation technique On the basis ofenergy variation principleWang et al [24] derived governingdifferential equation and natural boundary condition ofdynamic response for I-shaped beam and obtained closedsolution of the corresponding generalized dynamic displace-ment

Accuracy and efficiency of beam element depend onbeam displacement trial function by applying potentialenergy variation principle to constructing beam displace-ment element For current beam element cubic polynomialdisplacement mode is used to obtain a series of static anddynamic element widely applied to the software such asANSYS and NASTRAN For dynamic analysis vibratingbeam displacement mode has a big difference from polyno-mial mode Precision requirements cannot be met by taking

polynomial function as vibrating beam displacement trailfunction The basic analytical solution is used as the elementtrial function in analytical trial function method Discretefinite element method takes advantage of analytical solutionIt embodies the superiority of trial function using basicanalytical solution

This paper focuses on constructing element for beamdynamic analysis using analytical deflection trail functionbased on variational method of principle of minimum poten-tial energy and displacement element construction theoryThe fruits are useful to beam dynamic analysis

2 Displacement Trial Function forBeam Element

Selecting displacement trial function is one of the maincontents of constructing displacement element Appropri-ate displacement trial function should be in accordancewith element deformation behavior and should be easy forintegral of element energy functional Element accuracy isdetermined by the accuracy of displacement trial functionThe corresponding functional integral has a direct effect onelement calculation efficiency and accuracy

With regard to beam element displacement mode basedon interpolation function is used for element displacementtrial function of all kinds of problems Linear polynomialLagrange interpolation function meeting the continuity con-dition at 119862

0

is used for axial displacement Cubic polynomialHermit interpolation function meeting the continuity condi-tion at 119862

1

is used for flexural deformationTake flexural deformation for example for the static

problems of uniform cross section beam subjected to uniformdistributed load 119864119868(1198894119908(119909)1198891199094) = 119902 is stiffness equilibriumequation about deflection w(x) Here 119864119868 is beam sectionflexural stiffness and q is the uniform distributed load Theaccurate solution of this equation is a cubic polynomialCubic polynomial Hermit interpolation function meetingthe continuity condition of beam end displacement is actualdisplacement of element And the corresponding potentialenergy functional has analytic derivable integrable solutionFor this reason static beam element derived from Hermitinterpolation shape function is accurate element

However for dynamic and nonlinear straight beam andevery nonlinear beam deflection equation is not cubicpolynomial due to the change of equilibrium differentialequation mode Therefore element constructed by Hermitinterpolation trial function is an approximate element Thekey of constructing accurate element for all kinds of problemsof beam is to search analytical trial function having functionalintegrability

3 Analytical Trial Function of Displacementfor Vibrating Beam Element

Themethod to construct analytical trial function of displace-ment for vibrating beam element is as follows

(1) to deduce the general solution of displacement equa-tion for beam vibration containing undetermined

Mathematical Problems in Engineering 3

O x

y

V1

N1

M1

N2

M2

V21205792minus1205791

120601w(x)

Figure 1 Coordinate system and parameter positive direction

parameters according to differential equation of equi-librium for beam vibration

(2) to determine the undetermined parameters in dis-placement equation according to displacement con-dition of vibrating beam end

(3) towrite out displacement equation for beamvibrationexpressed by beam end displacement and then toobtain displacement trial function for vibrating beam

To construct beam dynamic element local coordinateas shown in Figure 1 is established It is in accordancewith general beam element The positive direction of lineardisplacement of beam end and force is in accordance withcoordinate direction and the positive direction of rotationangle and moment is in accordance with clockwise direction

31 Governing Equation for Beam Free Vibrationand Displacement General Solution

311 Beam Flexural Vibration Equation Dynamic equilib-rium governing equation for beam flexural vibration isexpressed as [1]

1198641198681205974V (119909 119905)1205971199094

+ 1198981205972V (119909 119905)1205971199052

= 0 (1)

In (1) V(119909 119905) is the displacement response for beamflexural vibration

By applying the method of separation of variables ana-lytical solution of deflection equation for beam flexuralvibration is obtained [1]

120601119908

(119909) = 1198621

sin120572119909 + 1198622

cos120572119909 + 1198623

sh120572119909 + 1198624

ch120572119909 (2)

In (2) flexural vibration parameter 120572 is connected withcircular frequency for beam flexural vibration 120596

119908

120596119908

=

1205722

radic119864119868119898 EI is beam section flexural stiffness and119898 is beammass per unit length

312 Beam Axial Vibration Equation Dynamic equilibriumgoverning equation for beam axial vibration is expressed as[1]

1198641198601198892

119906

1198891199092

minus 1198981205972

119906

1205971199052

= 0 (3)

In (3) 119906(119909 119905) is the displacement response for beam axialvibration

By applying themethod of separation of variables analyt-ical solution of deflection equation for beam axial vibration isobtained [1]

120601119906

(119909) = 1198631

cos120573119909 + 1198632

sin120573119909 (4)

In (4) axial vibration parameter 120573 is connected with cir-cular frequency for beam axial vibration 120596

119906

120596119906

= 120573radic119864119860119898and 119864119860 is beam section tensile (compressive) rigidity

32 Analytical Trial Function of Displacement for DynamicBeam Element

321 Analytical Trial Function for Beam Flexural Vibration

(1) Parameter Determination of Displacement Function Equa-tion (2) can be further expressed as

120601119908

(119909) =

4

sum

119894=1

120593119894

(119909) 119862119894

= 120593 119862 (5)

where

119862 = 1198621

1198622

1198623

1198624

119879

120593 = 1205931

(119909) 1205932

(119909) 1205933

(119909) 1205934

(119909)

= sin120572119909 cos120572119909 sh120572119909 ch120572119909

(6)

Beam end displacement is defined as 120575119908

119890

=

1205961

1205791

1205962

1205792

119879

According to (5) beam end displacement becomes

120575119908

119890

=

[[[[[

[

1205931

(0) 1205932

(0) 1205933

(0) 1205934

(0)

minus1205931015840

1

(0) minus1205931015840

2

(0) minus1205931015840

3

(0) minus1205931015840

4

(0)

1205931

(119897) 1205932

(119897) 1205933

(119897) 1205934

(119897)

minus1205931015840

1

(119897) minus1205931015840

2

(119897) minus1205931015840

3

(119897) minus1205931015840

4

(119897)

]]]]]

]

sdot

1198621

1198622

1198623

1198624

= [119860] 119862

(7)

If 120582 = 120572119897 with (6) [A] can be expressed as

[119860] =

[[[[[

[

0 1 0 1

minus120572 0 minus120572 0

sin 120582 cos 120582 sh120582 ch120582minus120572 cos 120582 120572 sin 120582 minus120572ch120582 minus120572sh120582

]]]]]

]

(8)

With (7) undetermined parameter of displacement func-tion is expressed as

119862 = [119860]minus1

120575119908

119890

(9)

4 Mathematical Problems in Engineering

To perform matrix inversion then

[119860]minus1

=

[[[[[

[

11988811

11988812

11988813

11988814

11988821

11988822

11988823

11988824

11988831

11988832

11988833

11988834

11988841

11988842

11988843

11988844

]]]]]

]

(10)

where

11988811

= minus11988831

=cos 120582 sinh 120582 + cosh 120582 sin 120582

119878

11988812

= minuscos 120582 cosh 120582 + sinh 120582 sin 120582 minus 1

120572119878

11988813

= minus11988833

= minussin 120582 + sinh 120582

119878

11988814

= minus11988834

= minuscosh 120582 minus cos 120582

120572119878

11988821

=1 minus sinh 120582 sin 120582 + cos 120582 cosh 120582

119878

11988822

= minus11988842

=cosh 120582 sin 120582 minus cos 120582 sinh 120582

119878

11988823

= minus11988843

=cosh 120582 minus cos 120582

119878

11988824

= minus11988844

=sinh 120582 minus sin 120582

120572119878

11988832

=1 + sinh 120582 sin 120582 minus cos 120582 cosh 120582

120572119878

11988841

=cos 120582 cosh 120582 + sin 120582 sinh 120582 minus 1

119878

119878 = 2 (cos 120582 cosh 120582 minus 1)

(11)

(2) Element Displacement Shape Function By substituting(9) into (5) displacement function for element flexural vibra-tion expressed by beam end displacement can be obtained

120601119908

(119909) = 120593 [119860]minus1

120575119908

119890

(12)

If

[119873119908

] = 120593 [119860]minus1

(13)

(12) becomes

120601119908

(119909) = [119873119908

] 120575119908

119890

(14)

[119873119908

] is flexural vibration displacement trial function fordynamic beam element With (13) it can be expressed as

[119873119908

] = 120593 [119860]minus1

= [1198731

1198732

1198733

1198734

] (15)

where 1198731

1198732

1198733

and 1198734

are deflection shape functionsAfter matrix operation they can be expressed as

1198731

=1

119878[sinh 120582 sin (120572119909 minus 120582) + cosh 120582 cos (120572119909 minus 120582)

minus cos (120572119909) + cos 120582 cosh (120572119909 minus 120582)

minus sin 120582 sinh (120572119909 minus 120582) minus cosh (120572119909)]

1198732

=1

120572119878[minus cosh 120582 sin (120572119909 minus 120582) minus sinh 120582 cos (120572119909 minus 120582)

+ sin (120572119909) + sinh (120572119909) minus sin 120582 cosh (120572119909 minus 120582)

minus cos 120582 sinh (120572119909 minus 120582)]

1198733

=1

119878[minus sin (120572119909) sinh 120582 minus cos (120572119909 minus 120582)

+ cos (120572119909) cosh 120582 minus cosh (120572119909 minus 120582)

+ sin 120582 sinh (120572119909) + cosh (120572119909) cos 120582]

1198734

=1

120572119878[minus sin (120572119909) cosh 120582 + sin (120572119909 minus 120582)

+ cos (120572119909) sinh 120582 + sinh (120572119909 minus 120582)

minus sinh (120572119909) cos 120582 + cosh (120572119909) sin 120582]

(16)

322 Analytical Trial Function for Beam Axial Vibration

(1) Parameter Determination of Displacement Function Equa-tion (4) can be further expressed as

120601119906

(119909) =

4

sum

119894=1

120595119894

(119909)119863119894

= 120595 119863 (17)

where

119863 = 1198631

1198632

119879

120595 = 1205951

(119909) 1205952

(119909)

= cos120573119909 sin120573119909

(18)

Beam end axial displacement is defined as 120575119906

119890

=

1199061

1199062

119879

According to (17) beam end axial displacement isobtained

120575119906

119890

= [

1205951

(0) 1205952

(0)

1205951

(119897) 1205952

(119897)

]

1198631

1198632

= [119861] 119863 (19)

If 120574 = 120573119897 with (18) [119861] can be expressed as

[119861] = [

1 0

cos 120574 sin 120574] (20)

With (19) undetermined parameter of axial displacementfunction is expressed as

119863 = [119861]minus1

120575119906

119890

(21)

Mathematical Problems in Engineering 5

To perform matrix inversion then

[119861]minus1

=[[

[

1 0

minuscos 120574sin 120574

1

sin 120574

]]

]

(22)

(2) ElementDisplacement Shape FunctionBy substituting (21)into (17) the following can be obtained

120601119906

(119909) = 120595 [119861]minus1

120575119906

119890

(23)

If

[119873119906

] = 120595 [119861]minus1

(24)

(23) becomes

120601119906

(119909) = [119873119906

] 120575119906

119890

(25)

[119873119906

] is axial vibration displacement trial function fordynamic beam element With (23) it can be expressed as

[119873119906

] = 120595 [119861]minus1

= [1198735

1198736

] (26)

where 1198735

1198736

are axial vibration displacement shape func-tions After matrix operation they can be expressed as

1198735

=cos120573119909 minus sin120573119909 cos 120574

sin 120574

1198736

=sin120573119909sin 120574

(27)

4 Element Potential EnergyFunctional and Variation

Finite element formulation of dynamic beam element isconstructed by principle of minimum potential energy andanalytical trial function

41 Potential Energy Functional In the light of potentialenergy functional of total potential energy is given by

prod

119901

= 119880 + 119881

=1

2int

119897

0

11986411986012060110158402

119906

(119909) 119889119909 +1

2int

119897

0

119864119868120601101584010158402

119908

(119909) 119889119909

minus1

2int

119897

0

1198981205962

119906

1206012

119906

(119909) 119889119909 minus1

2int

119897

0

1198981205962

119908

1206012

119908

(119909) 119889119909

(28)

where119880 is element strain energy and119881 is potential energy ofinertia force

With element displacement the following can beobtained

prod

119901

=1

2int

119897

0

119864119860120575119906

119890

119879

[1198731015840

119906

]119879

[1198731015840

119906

] 120575119906

119890

119889119909

+1

2int

119897

0

119864119868120575119908

119890

119879

[11987310158401015840

119908

]119879

[11987310158401015840

119908

] 120575119908

119890

119889119909

minus1

2int

119897

0

1198981205962

119906

120575119906

119890

119879

[119873119906

]119879

[119873119906

] 120575119906

119890

119889119909

minus1

2int

119897

0

1198981205962

119908

120575119908

119890

119879

[119873119908

]119879

[119873119908

] 120575119908

119890

119889119909

(29)

42 Functional Variation and Transformation In the light ofprinciple of minimum potential energy that is 120597prod

119901

120597120575119890

=

0 element equilibrium equation is obtained

[119870]119890

120575119890

minus 1205962

[119872]119890

120575119890

= 0 (30)

where [119870]119890 and [119872]119890 are dynamic stiffness matrix and massmatrix of beam element respectively

[119870]119890

= int

119897

0

119864119860 [1198731015840

119906

]119879

[1198731015840

119906

] 119889119909 + int

119897

0

119864119868 [11987310158401015840

119908

]119879

[11987310158401015840

119908

] 119889119909

[119872119908

]119890

= int

119897

0

119898[119873119906

]119879

[119873119906

] 119889119909 + int

119897

0

119898[119873119908

]119879

[119873119908

] 119889119909

(31)

5 Element Matrix

Flexural vibration and axial vibration are independent of oneanother irrespective of large deformation Element matrixesat flexural vibration and axial vibration are obtained byapplying variation on flexural vibration displacement andaxial vibration displacement respectively

51 Element Matrix for Beam Flexural Vibration

511 Stiffness Matrix After matrix operation elementdynamic stiffness matrix for beam flexural vibration isexpressed as

[119870119908

]119890

= 119867

[[[[[

[

11989611990811

11989611990812

11989611990813

11989611990814

11989611990821

11989611990822

11989611990823

11989611990824

11989611990831

11989611990832

11989611990833

11989611990834

11989611990841

11989611990842

11989611990843

11989611990844

]]]]]

]

(32)

where

119896119908119894119895

=1

119867int

119897

0

11987310158401015840

119908119894

11987310158401015840

119908119894

119889119909

119867 =1198641198681205723

119890minus2120582

119866

119866 = 16 (1 + cos2120582ch2120582 minus 2 cos 120582ch120582)

(33)

6 Mathematical Problems in Engineering

After integral each stiffness matrix coefficient can beobtained such as

11989611990811

= (cosh 120582 sin 120582 + cos 120582 sin 120582 minus sinh 120582 cosh 120582

minus2cosh2120582 sin 120582 cos 120582)

sdot (21198902120582

+ 1198904120582

+ 1)

+ 4 (119890120582

minus 1198903120582

) cosh 120582 (sinh 120582 sin 120582 + cosh 120582 cos 120582)

minus 2 (1 + 1198904120582

) cos 120582 sin 120582 cosh 120582 sinh 120582

+ 41198902120582cos2120582 (120582 minus sinh 120582 cosh 120582)

+ (61198902120582

+ 1198904120582

+ 1) cos 120582 sinh 120582

+ (1 minus 1198904120582

minus 81198902120582

120582) cos 120582 cosh 120582 + 4 (119890120582 + 1198903120582)

sdot (cos 120582 sinh 120582 cosh 120582 minus cos2120582 sinh120582 minus sin 120582

+cosh2120582 sin 120582 minus cos2120582 cosh 120582)

+ (41205821198902120582

+ 1198904120582

minus 1) cosh2120582 + (1 minus 1198904120582) sin 120582 sinh 120582(34)

512 Mass Matrix Mass matrix of element for beam flexuralvibration is given by

[119872119908

] = 119875

[[[[[

[

11989811990811

11989811990812

11989811990813

11989811990814

11989811990821

11989811990822

11989811990823

11989811990824

11989811990831

11989811990832

11989811990833

11989811990834

11989811990841

11989811990842

11989811990843

11989811990844

]]]]]

]

(35)

where

119898119908119894119895

=1

119875int

119897

0

119873119908119894

119873119908119895

119889119909 119875 =119898119890minus2120582

119866

119866 = 16 (1 + cos2120582ch2120582 minus 2 cos 120582ch120582) (36)

After integral each mass matrix coefficient can beobtained such as

11989811990811

= 120572minus1

[4 (119890120582

+ 1198903120582

)

sdot (cos2120582 sinh 120582 minus cos 120582 sinh120582 cosh 120582

minus sin 120582cosh2120582 + sin 120582)

+ (41205821198902120582

+ 1198904120582

minus 1) cosh2120582

+ (1 minus 1198904120582

) sin 120582

sdot (sinh 120582 minus 2 cos 120582 sinh 120582 cosh 120582)

+ (1198904120582

minus 101198902120582

+ 1) cos 120582 sinh120582

minus (81205821198902120582

+ 1198904120582

minus 1) cos 120582 cosh 120582

+ 4 (1198903120582

minus 119890120582

)

sdot cosh 120582 (sin 120582 sinh 120582 minus cos2120582 + 4 cos 120582 cosh 120582)

+ (21198902120582

+ 119890120582

+ 1) (sin 120582 cos 120582 minus sinh 120582 cosh 120582)

+ 31198902120582cos2120582 (4 sinh 120582 cosh 120582 + 120582)

+ 2 (61198902120582

minus 1 minus 1198904120582

) sin 120582 cos 120582cosh2120582

+ (1198904120582

minus 141198902120582

+ 1) sin 120582 cosh 120582] (37)

52 Element Matrix for Beam Axial Vibration After matrixoperation element dynamic stiffness matrix for beam axialvibration is expressed as

[119870119906

]119890

= 119864119860119888

[[[

[

minuscos 120574 sin 120574 + 1205742 (cos2120574 minus 1)

120574 cos 120574 + sin 1205742 (cos2120574 minus 1)

120574 cos 120574 + sin 1205742 (cos2120574 minus 1)

minuscos 120574 sin 120574 + 1205742 (cos2120574 minus 1)

]]]

]

(38)

Mass matrix of element for beam axial vibration is givenby

[119872119906

] = 119898

[[[

[

sin 120574 cos 120574 minus 1205742119888 (cos2120574 minus 1)

120574 cos 120574 minus sin 1205742119888 (cos2120574 minus 1)

120574 cos 120574 minus sin 1205742119888 (cos2120574 minus 1)

sin 120574 cos 120574 minus 1205742119888 (cos2120574 minus 1)

]]]

]

(39)

53 Dynamic Beam ElementMatrix Because of the indepen-dence of axial vibration and flexural vibration 6 by 6 matrixis obtained by adding two rows and two columns to the above4 by 4 matrix considering axial vibration without coupling ofaxial vibration and flexural vibration It is element stiffnessmatrix

Element stiffness matrix is given by

[119870]119890

=

[[[[[[[[[[[

[

11989611

0 0 11989614

0 0

11989622

11989623

0 11989625

11989626

11989633

0 11989635

11989636

11989644

0 0

symmetry 11989655

11989656

11989666

]]]]]]]]]]]

]

(40)

Mathematical Problems in Engineering 7

where

11989611

= minus119864119860119888 (cos 120574 sin 120574 + 120574)

2 (cos2120574 minus 1)

11989614

=119864119860119888 (120574 cos 120574 + sin 120574)

2 (cos2120574 minus 1)

11989622

= 11986711989611990811

11989623

= 11986711989611990812

11989625

= 11986711989611990813

11989626

= 11986711989611990814

11989633

= 11986711989611990822

11989635

= 11986711989611990823

11989636

= 11986711989611990824

11989644

= minus119864119860119888 (cos 120574 sin 120574 + 120574)

2 (cos2120574 minus 1)

11989655

= 11986711989611990833

11989656

= 11986711989611990834

11989666

= 11986711989611990844

(41)

Element mass matrix is given by

[119872]119890

=

[[[[[[[[[[[

[

11989811

0 0 11989814

0 0

11989822

11989823

0 11989825

11989826

11989833

0 11989835

11989836

11989844

0 0

symmetry 11989855

11989856

11989866

]]]]]]]]]]]

]

(42)

where

11989811

=119898 (sin 120574 cos 120574 minus 120574)2119888 (cos2120574 minus 1)

11989814

=119898 (120574 cos 120574 minus sin 120574)2119888 (cos2120574 minus 1)

11989822

= 11987511989811990811

11989823

= 11987511989811990812

11989825

= 11987511989811990813

11989826

= 11987511989811990814

11989833

= 11987511989811990822

11989835

= 11987511989811990823

11989836

= 11987511989811990824

11989844

=119898 (sin 120574 cos 120574 minus 120574)2119888 (cos2120574 minus 1)

11989855

= 11987511989811990833

11989856

= 11987511989811990834

11989866

= 11987511989811990844

(43)

6 Example and Comparison

To verify the dynamic beam element constructed by ana-lytical trial function method the comparisons between thecalculation results of this element general beam element andtheoretical solution are conducted

61 Calculation Model

(1) Four Kinds of Typical Beams Free vibration of four kindsof typical beams are analyzed including cantilever beamsimply supported beam one end clamped and another simplysupported beam and clamped-clamped beam

(2) Structure and Material Parameters The parameters ofcalculation model are as follows beam dimension is 6mtimes 02m times 03m material elastic modulus 119864 = 210GPaPoissonrsquos ratio 120592 = 03 and material density 120588 = 7800 kgm3

(3) Theoretical Solution According to theoretical equation offree vibration for distributed mass beam base frequencies offree vibration for all kinds of beam are obtained by usinganalytic method [1]

(4) Dynamic StiffnessMatrixMethodAccording to governingequation of free vibration for distributedmass beam stiffnesscoefficient [11] of beam vibration is provided and thenbase frequencies of free vibration for all kinds of beam areobtained

(5) Finite Element Model of General Beam Element Byusing the software ANSYS finite element model of beamvibration is established and modal analysis is conductedElement BEAM3 is used to simulate beam BEAM3 is generaltwo-dimension elastic beam element and uniaxial elementbearing tension pressure and bend Every node has threedegrees of freedom that is 119909-axis linear displacement 119910-axislinear displacement and 119911-axis angular displacement Cubicpolynomial interpolating function is used as its displacementtrial function Beam is divided into one beam element BlockLanczos method is used for modal extraction

(6) Model in This Paper Equilibrium equation of beamflexural free vibration is given by

119870 (120596) Δ (119905) + 119872 (120596) Δ (119905) = 0 (44)

where 119870(120596) is structure stiffness matrix with boundarydisplacement restraint and 119872(120596) is structure mass matrixwith boundary displacement restraint According to natural

8 Mathematical Problems in Engineering

Table 1 Base frequency of beam free vibration and its comparison (rads)

Beam type Theoreticalsolution

Dynamic stiffness matrixmethod

Interpolation trialfunction element

Analytical trial functionelement

Absolutevalue

Relative error()

Absolutevalue

Relative error()

Absolutevalue

Relative error()

Cantilever beam 438875389 438875390 289119864 minus 07 440740 0423 438875391 312119864 minus 07

Simply supportedbeam 1231941891 1231941888 minus859119864 minus 08 1365902 10874 1231941889 minus637119864 minus 08

One end clamped andanother simplysupported beam

1924528355 1924528348 minus150119864 minus 07 2554366 32727 1924528349 minus153119864 minus 07

Clamped-clampedbeam 2792673996 2792673991 minus132119864 minus 07 2834156 1485 2792673992 minus127119864 minus 07

Note calculation of eigenvalue and eigenvector cannot be conducted due to all DOFs of two ends of clamped-clamped beam being restrained If the beam isdivided into one element semistructure is used for vibration simulation of clamped-clamped beam

Table 2 Comparison of different element results (relative error )

Beam typeElement number for analytical

trial function Element number for interpolation trial function

1 1 2 10 20Cantilever beam 312119864 minus 07 04249 minus00003 minus00489 minus00489Simply supported beam minus637119864 minus 08 108739 02909 minus01018 minus01018One end clamped and anothersimply supported beam minus153119864 minus 07 327269 08037 minus01169 minus01202

Clamped-clamped beam minus127119864 minus 07 14854 14854 minus01255 minus01278

vibration governing equation of beam beam natural vibra-tion is harmonic vibration The following assumption can beobtained

Δ (119905) = 119884 sin120596119905 (45)

Substituting (45) into (44) yields

119872minus1

(120596)119870 (120596) 119884 = 1205962

119884 (46)

Assuming 120578 = 1205962 then 119864(120578) = 119872minus1(120578)119870(120578)The following equation can be obtained

119864 (120578) 119884 = 120578119884 (47)

Beamnatural vibration analysis is converted to eigenvalueand eigenvector analysis of matrix 119864(120578) = 119872minus1(120578)119870(120578)

Iteration method is used to calculate base frequency ofbeam vibration Through circular calculations the iterationstops when frequency relative error reaches 120581

119910

= (120596119894+1

minus

120596119894

)120596119894

lt 10minus8

62 Result Comparison and Analysis

621 Element Accuracy Vibration analysis is conducted bydividing beam into one element Table 1 shows base frequencyof beam free vibration and the relative error between calcu-lated solution and theoretical solution

As shown in Table 1 the following results can be obtained(1) By applying analytical trial function element to the

analysis a more accurate solution can be attained when the

relative error is less than 10minus8 The series of relative errorbetween this solution and theoretical solution is about 1 times10minus9 It can be supposed that this error is calculation error ofiteration efficiency

(2) By applying interpolation trial function element tothe analysis the maximum relative error between the solu-tion and theoretical solution reaches 30 It indicates thatdynamic beam element constructed by taking polynomialfunction as displacement trial function has bigger error

(3)The solution of dynamic stiffness matrix method [11]is close to theoretical solution The relative error is almostidentical with that calculated by analytical trial functionmethod The reason is that their displacement curves forvibrating beam are both derived from dynamic equilibriumgoverning equation Element load value and node balancecondition need to be analyzed in dynamic stiffness matrixmethod Dynamic stiffness matrix of beam element can bedirectly obtained in analytical trial function method Thelatter is more simple and intuitive and widely applied

622 Element Efficiency The beam is divided into 1 element2 elements 10 elements and 20 elements respectively Beamvibration numerical simulation is conducted by applyinginterpolation trial function method The relative error ofcalculated base frequency of free vibration for every typicalbeam and theoretical solution is obtained Table 2 presentsthe results

As shown in Table 2 the result of interpolation trialfunction is closer to theoretical solution with the increasing

Mathematical Problems in Engineering 9

of element number But it is obvious that base frequencybecomes smaller and its difference with theoretical solution islargerThe reason is that the more the elements are the lowerthe structure calculation stiffness isThe greater the differencebetween the actual stiffness and calculation stiffness Thenthe calculated base frequency further has large differencewith theoretical solution It indicates that actual deformationcurve for vibrating beam is not polynomial and polynomialfunction cannot be taken as displacement trial function forvibrating beam

7 Conclusions

Based on the results of this investigation the followingconclusions can be drawn

(1) The result of interpolation trial function elementfor simulating beam vibration is not in accordance withtheoretical solution and the relative error is larger Withelement number increasing base frequency becomes smallerand has large difference with theoretical solution It indicatesthat vibrating beam displacement mode is different frompolynomial mode and the precision requirement cannot bemet by taking polynomial function as displacement trialfunction for vibrating beam

(2)The solution of dynamic stiffness matrix method forsimulating beam vibration is close to theoretical solutionBut in this method element load value and node balancecondition need to be specially analyzed and it is not easy toderive nonstructural question Its application in engineeringis limited

(3)Dynamic stiffness matrix of beam element is obtainedby applying analytical trial function put forward in thispaper Base frequencies of four typical beams are attained byanalyzing eigenvalue and eigenvector and are compared withtheoretical solutionThe results show that the series of relativeerror is about 1 times 10minus9 and it is actually calculation error ofiteration efficiency The dynamic beam element in the lightof analytical trial function put forward in this paper is high-precision element

Conflict of Interests

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

Acknowledgments

This research is supported by National Natural ScienceFunds no 51279206 Beijing Natural Science Foundation no3144029 Chinese Universities Scientific Fund no 2011JS126and the Specialized Research Fund for the Doctoral Programof Higher Education of China no 20110008120017

References

[1] A K Clough and J Penzien Dynamic of Structures McGraw-Hill New York NY USA 1995

[2] Q Jin ldquoAn analytical solution of dynamic response for therigid perfectly plastic Timoshenko beamrdquo Chinese Journal ofTheoretical and Applied Mechanics vol 16 no 5 pp 504ndash5111984

[3] X Guo H Chen and Q Zeng ldquoDynamic characteristicsanalysis model of prestressed concrete T-type beamrdquo ChineseJournal of Computational Mechanics vol 17 no 2 pp 176ndash1832000

[4] D-P Fang and Q-F Wang ldquoDynamic behavior analysis of anexternally prestressed beam with energy methodrdquo Journal ofVibration and Shock vol 31 no 1 pp 177ndash181 2012

[5] M Lou and T Hong ldquoAnalytical approach for dynamic charac-teristics of prestressed beam with external tendonsrdquo Journal ofTongji University Natural Science vol 34 no 10 pp 1284ndash12882006

[6] J A Carrer S A Fleischfresser L F Garcia and W J MansurldquoDynamic analysis of Timoshenko beams by the boundary ele-ment methodrdquo Engineering Analysis with Boundary Elementsvol 37 no 12 pp 1602ndash1616 2013

[7] J-J Wu ldquoUse of the elastic-and-rigid-combined beam elementfor dynamic analysis of a two-dimensional frame with arbi-trarily distributed rigid beam segmentsrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 3 pp 1240ndash1251 2011

[8] M A De Rosa C Franciosi and M J Maurizi ldquoOn thedynamic behaviour of slender beams with elastic ends carryinga concentrated massrdquo Computers and Structures vol 58 no 6pp 1145ndash1159 1996

[9] R W Clough ldquoThe finite element method in plane stress anal-ysisrdquo in Proceedings of the 2nd ASCE Conference on ElectronicComputation Pittsburgh Pa USA 1960

[10] V Kolousek ldquoAnwendung des Gesetzes der virtuellen Verschiebungen and des in der Reziprozitatssatzesrdquo Stab weksDynamik Lngenieur Archiv vol 12 pp 363ndash370 1941

[11] Y-Q Long and S-H Bao Structural Mechanics II HigherEducation Press Beijing China 2nd edition 1996

[12] S M Hashemi and M J Richard ldquoFree vibrational analysis ofaxially loaded bending-torsion coupled beams a dynamic finiteelementrdquo Computers amp Structures vol 77 no 6 pp 711ndash7242000

[13] S Chen M Geradin and E Lamine ldquoAn improved dynamicstiffness method and modal analysis for beam-like structuresrdquoComputers amp Structures vol 60 no 5 pp 725ndash731 1996

[14] J R Banerjee H Su and C Jayatunga ldquoA dynamic stiffnesselement for free vibration analysis of composite beams and itsapplication to aircraft wingsrdquo Computers amp Structures vol 86no 6 pp 573ndash579 2008

[15] J R Banerjee S Guo and W P Howson ldquoExact dynamicstiffness matrix of a bending-torsion coupled beam includingwarpingrdquo Computers and Structures vol 59 no 4 pp 613ndash6211996

[16] M Shavezipur and S M Hashemi ldquoFree vibration of triplycoupled centrifugally stiffened nonuniform beams using arefined dynamic finite element methodrdquo Aerospace Science andTechnology vol 13 no 1 pp 59ndash70 2009

[17] A K ChopraDynamics of Structures PrenticeHall EnglewoodCliffs NJ USA 2000

[18] X-L Hu and Z-M Dai ldquoOn the dynamic stress concentrationsin orthotropic plates with an arbitrary cutoutrdquo Chinese Journalof Applied Mechanics vol 15 no 1 pp 12ndash17 1998

10 Mathematical Problems in Engineering

[19] X Wang L Yang and W Gao ldquoDynamic FEM analysis forthe integration ballast structure based on variation principlerdquoJournal of Vibration and Shock vol 24 no 4 pp 99ndash102 2005

[20] S M Hashemi and M J Richard ldquoA dynamic finite element(DFE) method for free vibrations of bending-torsion coupledbeamsrdquo Aerospace Science and Technology vol 4 no 1 pp 41ndash55 2000

[21] A Pagani E Carrera M Boscolo and J R Banerjee ldquoRefineddynamic stiffness elements applied to free vibration analysis ofgenerally laminated composite beams with arbitrary boundaryconditionsrdquo Composite Structures vol 110 no 1 pp 305ndash3162014

[22] S M Nabi and N Ganesan ldquoA generalized element for thefree vibration analysis of composite beamsrdquo Computers andStructures vol 51 no 5 pp 607ndash610 1994

[23] L Zhao and Q Chen ldquoDynamic analysis of the stochasticvariational principle and stochastic finite element methodfor structures with random parametersrdquo Chinese Journal ofComputational Mechanics vol 15 no 3 pp 263ndash274 1998

[24] G-H Wang Y-N Gan and Z-B Wang ldquoEnergy-variationalmethod for the dynamic response of thin-walled I-beams withwide flangerdquo Engineering Mechanics vol 27 no 8 pp 15ndash202010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

the dynamic behavior of slender beam with concentratedmass at beam end and numerical solution of frequencyequation was obtained Among them numerical analysisrepresented by finite element is themain and efficientmethodfor dynamic analysis (such as natural vibration analysis andforced vibration analysis) Finite element method was firstlyproposed by Clough [9] in an article about plane elasticproblem and it is perfect in theory as a numerical method

At present themethods of constructing dynamic elementfor beam include dynamic stiffness matrix method Galerkinmethod Ritz method energy variation method Dynamicstiffness matrix can accurately solve differential equationof motion according to initial displacement field withoutany assumption and then accurate results can be obtainedirrespective of element number This method was proposedby Kolousek [10] To gain more accurate results stiffnessmatrix of tensile torsion bar and Euler beam about frequencythat is dynamic stiffness matrix was firstly derived fromanalytical solution for studying vibration characteristics ofplane truss A lot of work on the research and developmentof dynamic stiffness matrix method was also done by Longand Bao [11] Hashemi and Richard [12] Chen et al [13]Banerjee et al [14] and Banerjee et al [15] Shavezipur andHashemi [16] put forward an accurate finite element methodIn this method closed form solution of differential equationof beam not coupling flexural vibration torsional vibrationwas obtained by merging Galerkin weighted residual methodand dynamic stiffness matrix (DSM) Result of dynamicstiffness matrix is more accurate but analytical solutionof differential equation cannot be derived when structureload or displacement boundary condition is too complexThen dynamic stiffness matrix method is not suitable anymore According to principle of virtual displacement a largenumber of research achievements on dynamic analysis ofthin-walled open section beam elastic foundation beamand composite beam have been conducted by Chopra [17]Hu and Dai [18] Wang et al [19] Hashemi and Richard[20] and Pagani et al [21] Nabi and Ganesan [22] putforward a finite element method based on free vibrationanalysis theory of composite beam with the first order sheardeformation Zhao and Chen [23] developed the dynamicanalysis of a unified stochastic variational principle andthe corresponding stochastic finite element method via theinstantaneous minimum potential energy principle and thesmall parameter perturbation technique On the basis ofenergy variation principleWang et al [24] derived governingdifferential equation and natural boundary condition ofdynamic response for I-shaped beam and obtained closedsolution of the corresponding generalized dynamic displace-ment

Accuracy and efficiency of beam element depend onbeam displacement trial function by applying potentialenergy variation principle to constructing beam displace-ment element For current beam element cubic polynomialdisplacement mode is used to obtain a series of static anddynamic element widely applied to the software such asANSYS and NASTRAN For dynamic analysis vibratingbeam displacement mode has a big difference from polyno-mial mode Precision requirements cannot be met by taking

polynomial function as vibrating beam displacement trailfunction The basic analytical solution is used as the elementtrial function in analytical trial function method Discretefinite element method takes advantage of analytical solutionIt embodies the superiority of trial function using basicanalytical solution

This paper focuses on constructing element for beamdynamic analysis using analytical deflection trail functionbased on variational method of principle of minimum poten-tial energy and displacement element construction theoryThe fruits are useful to beam dynamic analysis

2 Displacement Trial Function forBeam Element

Selecting displacement trial function is one of the maincontents of constructing displacement element Appropri-ate displacement trial function should be in accordancewith element deformation behavior and should be easy forintegral of element energy functional Element accuracy isdetermined by the accuracy of displacement trial functionThe corresponding functional integral has a direct effect onelement calculation efficiency and accuracy

With regard to beam element displacement mode basedon interpolation function is used for element displacementtrial function of all kinds of problems Linear polynomialLagrange interpolation function meeting the continuity con-dition at 119862

0

is used for axial displacement Cubic polynomialHermit interpolation function meeting the continuity condi-tion at 119862

1

is used for flexural deformationTake flexural deformation for example for the static

problems of uniform cross section beam subjected to uniformdistributed load 119864119868(1198894119908(119909)1198891199094) = 119902 is stiffness equilibriumequation about deflection w(x) Here 119864119868 is beam sectionflexural stiffness and q is the uniform distributed load Theaccurate solution of this equation is a cubic polynomialCubic polynomial Hermit interpolation function meetingthe continuity condition of beam end displacement is actualdisplacement of element And the corresponding potentialenergy functional has analytic derivable integrable solutionFor this reason static beam element derived from Hermitinterpolation shape function is accurate element

However for dynamic and nonlinear straight beam andevery nonlinear beam deflection equation is not cubicpolynomial due to the change of equilibrium differentialequation mode Therefore element constructed by Hermitinterpolation trial function is an approximate element Thekey of constructing accurate element for all kinds of problemsof beam is to search analytical trial function having functionalintegrability

3 Analytical Trial Function of Displacementfor Vibrating Beam Element

Themethod to construct analytical trial function of displace-ment for vibrating beam element is as follows

(1) to deduce the general solution of displacement equa-tion for beam vibration containing undetermined

Mathematical Problems in Engineering 3

O x

y

V1

N1

M1

N2

M2

V21205792minus1205791

120601w(x)

Figure 1 Coordinate system and parameter positive direction

parameters according to differential equation of equi-librium for beam vibration

(2) to determine the undetermined parameters in dis-placement equation according to displacement con-dition of vibrating beam end

(3) towrite out displacement equation for beamvibrationexpressed by beam end displacement and then toobtain displacement trial function for vibrating beam

To construct beam dynamic element local coordinateas shown in Figure 1 is established It is in accordancewith general beam element The positive direction of lineardisplacement of beam end and force is in accordance withcoordinate direction and the positive direction of rotationangle and moment is in accordance with clockwise direction

31 Governing Equation for Beam Free Vibrationand Displacement General Solution

311 Beam Flexural Vibration Equation Dynamic equilib-rium governing equation for beam flexural vibration isexpressed as [1]

1198641198681205974V (119909 119905)1205971199094

+ 1198981205972V (119909 119905)1205971199052

= 0 (1)

In (1) V(119909 119905) is the displacement response for beamflexural vibration

By applying the method of separation of variables ana-lytical solution of deflection equation for beam flexuralvibration is obtained [1]

120601119908

(119909) = 1198621

sin120572119909 + 1198622

cos120572119909 + 1198623

sh120572119909 + 1198624

ch120572119909 (2)

In (2) flexural vibration parameter 120572 is connected withcircular frequency for beam flexural vibration 120596

119908

120596119908

=

1205722

radic119864119868119898 EI is beam section flexural stiffness and119898 is beammass per unit length

312 Beam Axial Vibration Equation Dynamic equilibriumgoverning equation for beam axial vibration is expressed as[1]

1198641198601198892

119906

1198891199092

minus 1198981205972

119906

1205971199052

= 0 (3)

In (3) 119906(119909 119905) is the displacement response for beam axialvibration

By applying themethod of separation of variables analyt-ical solution of deflection equation for beam axial vibration isobtained [1]

120601119906

(119909) = 1198631

cos120573119909 + 1198632

sin120573119909 (4)

In (4) axial vibration parameter 120573 is connected with cir-cular frequency for beam axial vibration 120596

119906

120596119906

= 120573radic119864119860119898and 119864119860 is beam section tensile (compressive) rigidity

32 Analytical Trial Function of Displacement for DynamicBeam Element

321 Analytical Trial Function for Beam Flexural Vibration

(1) Parameter Determination of Displacement Function Equa-tion (2) can be further expressed as

120601119908

(119909) =

4

sum

119894=1

120593119894

(119909) 119862119894

= 120593 119862 (5)

where

119862 = 1198621

1198622

1198623

1198624

119879

120593 = 1205931

(119909) 1205932

(119909) 1205933

(119909) 1205934

(119909)

= sin120572119909 cos120572119909 sh120572119909 ch120572119909

(6)

Beam end displacement is defined as 120575119908

119890

=

1205961

1205791

1205962

1205792

119879

According to (5) beam end displacement becomes

120575119908

119890

=

[[[[[

[

1205931

(0) 1205932

(0) 1205933

(0) 1205934

(0)

minus1205931015840

1

(0) minus1205931015840

2

(0) minus1205931015840

3

(0) minus1205931015840

4

(0)

1205931

(119897) 1205932

(119897) 1205933

(119897) 1205934

(119897)

minus1205931015840

1

(119897) minus1205931015840

2

(119897) minus1205931015840

3

(119897) minus1205931015840

4

(119897)

]]]]]

]

sdot

1198621

1198622

1198623

1198624

= [119860] 119862

(7)

If 120582 = 120572119897 with (6) [A] can be expressed as

[119860] =

[[[[[

[

0 1 0 1

minus120572 0 minus120572 0

sin 120582 cos 120582 sh120582 ch120582minus120572 cos 120582 120572 sin 120582 minus120572ch120582 minus120572sh120582

]]]]]

]

(8)

With (7) undetermined parameter of displacement func-tion is expressed as

119862 = [119860]minus1

120575119908

119890

(9)

4 Mathematical Problems in Engineering

To perform matrix inversion then

[119860]minus1

=

[[[[[

[

11988811

11988812

11988813

11988814

11988821

11988822

11988823

11988824

11988831

11988832

11988833

11988834

11988841

11988842

11988843

11988844

]]]]]

]

(10)

where

11988811

= minus11988831

=cos 120582 sinh 120582 + cosh 120582 sin 120582

119878

11988812

= minuscos 120582 cosh 120582 + sinh 120582 sin 120582 minus 1

120572119878

11988813

= minus11988833

= minussin 120582 + sinh 120582

119878

11988814

= minus11988834

= minuscosh 120582 minus cos 120582

120572119878

11988821

=1 minus sinh 120582 sin 120582 + cos 120582 cosh 120582

119878

11988822

= minus11988842

=cosh 120582 sin 120582 minus cos 120582 sinh 120582

119878

11988823

= minus11988843

=cosh 120582 minus cos 120582

119878

11988824

= minus11988844

=sinh 120582 minus sin 120582

120572119878

11988832

=1 + sinh 120582 sin 120582 minus cos 120582 cosh 120582

120572119878

11988841

=cos 120582 cosh 120582 + sin 120582 sinh 120582 minus 1

119878

119878 = 2 (cos 120582 cosh 120582 minus 1)

(11)

(2) Element Displacement Shape Function By substituting(9) into (5) displacement function for element flexural vibra-tion expressed by beam end displacement can be obtained

120601119908

(119909) = 120593 [119860]minus1

120575119908

119890

(12)

If

[119873119908

] = 120593 [119860]minus1

(13)

(12) becomes

120601119908

(119909) = [119873119908

] 120575119908

119890

(14)

[119873119908

] is flexural vibration displacement trial function fordynamic beam element With (13) it can be expressed as

[119873119908

] = 120593 [119860]minus1

= [1198731

1198732

1198733

1198734

] (15)

where 1198731

1198732

1198733

and 1198734

are deflection shape functionsAfter matrix operation they can be expressed as

1198731

=1

119878[sinh 120582 sin (120572119909 minus 120582) + cosh 120582 cos (120572119909 minus 120582)

minus cos (120572119909) + cos 120582 cosh (120572119909 minus 120582)

minus sin 120582 sinh (120572119909 minus 120582) minus cosh (120572119909)]

1198732

=1

120572119878[minus cosh 120582 sin (120572119909 minus 120582) minus sinh 120582 cos (120572119909 minus 120582)

+ sin (120572119909) + sinh (120572119909) minus sin 120582 cosh (120572119909 minus 120582)

minus cos 120582 sinh (120572119909 minus 120582)]

1198733

=1

119878[minus sin (120572119909) sinh 120582 minus cos (120572119909 minus 120582)

+ cos (120572119909) cosh 120582 minus cosh (120572119909 minus 120582)

+ sin 120582 sinh (120572119909) + cosh (120572119909) cos 120582]

1198734

=1

120572119878[minus sin (120572119909) cosh 120582 + sin (120572119909 minus 120582)

+ cos (120572119909) sinh 120582 + sinh (120572119909 minus 120582)

minus sinh (120572119909) cos 120582 + cosh (120572119909) sin 120582]

(16)

322 Analytical Trial Function for Beam Axial Vibration

(1) Parameter Determination of Displacement Function Equa-tion (4) can be further expressed as

120601119906

(119909) =

4

sum

119894=1

120595119894

(119909)119863119894

= 120595 119863 (17)

where

119863 = 1198631

1198632

119879

120595 = 1205951

(119909) 1205952

(119909)

= cos120573119909 sin120573119909

(18)

Beam end axial displacement is defined as 120575119906

119890

=

1199061

1199062

119879

According to (17) beam end axial displacement isobtained

120575119906

119890

= [

1205951

(0) 1205952

(0)

1205951

(119897) 1205952

(119897)

]

1198631

1198632

= [119861] 119863 (19)

If 120574 = 120573119897 with (18) [119861] can be expressed as

[119861] = [

1 0

cos 120574 sin 120574] (20)

With (19) undetermined parameter of axial displacementfunction is expressed as

119863 = [119861]minus1

120575119906

119890

(21)

Mathematical Problems in Engineering 5

To perform matrix inversion then

[119861]minus1

=[[

[

1 0

minuscos 120574sin 120574

1

sin 120574

]]

]

(22)

(2) ElementDisplacement Shape FunctionBy substituting (21)into (17) the following can be obtained

120601119906

(119909) = 120595 [119861]minus1

120575119906

119890

(23)

If

[119873119906

] = 120595 [119861]minus1

(24)

(23) becomes

120601119906

(119909) = [119873119906

] 120575119906

119890

(25)

[119873119906

] is axial vibration displacement trial function fordynamic beam element With (23) it can be expressed as

[119873119906

] = 120595 [119861]minus1

= [1198735

1198736

] (26)

where 1198735

1198736

are axial vibration displacement shape func-tions After matrix operation they can be expressed as

1198735

=cos120573119909 minus sin120573119909 cos 120574

sin 120574

1198736

=sin120573119909sin 120574

(27)

4 Element Potential EnergyFunctional and Variation

Finite element formulation of dynamic beam element isconstructed by principle of minimum potential energy andanalytical trial function

41 Potential Energy Functional In the light of potentialenergy functional of total potential energy is given by

prod

119901

= 119880 + 119881

=1

2int

119897

0

11986411986012060110158402

119906

(119909) 119889119909 +1

2int

119897

0

119864119868120601101584010158402

119908

(119909) 119889119909

minus1

2int

119897

0

1198981205962

119906

1206012

119906

(119909) 119889119909 minus1

2int

119897

0

1198981205962

119908

1206012

119908

(119909) 119889119909

(28)

where119880 is element strain energy and119881 is potential energy ofinertia force

With element displacement the following can beobtained

prod

119901

=1

2int

119897

0

119864119860120575119906

119890

119879

[1198731015840

119906

]119879

[1198731015840

119906

] 120575119906

119890

119889119909

+1

2int

119897

0

119864119868120575119908

119890

119879

[11987310158401015840

119908

]119879

[11987310158401015840

119908

] 120575119908

119890

119889119909

minus1

2int

119897

0

1198981205962

119906

120575119906

119890

119879

[119873119906

]119879

[119873119906

] 120575119906

119890

119889119909

minus1

2int

119897

0

1198981205962

119908

120575119908

119890

119879

[119873119908

]119879

[119873119908

] 120575119908

119890

119889119909

(29)

42 Functional Variation and Transformation In the light ofprinciple of minimum potential energy that is 120597prod

119901

120597120575119890

=

0 element equilibrium equation is obtained

[119870]119890

120575119890

minus 1205962

[119872]119890

120575119890

= 0 (30)

where [119870]119890 and [119872]119890 are dynamic stiffness matrix and massmatrix of beam element respectively

[119870]119890

= int

119897

0

119864119860 [1198731015840

119906

]119879

[1198731015840

119906

] 119889119909 + int

119897

0

119864119868 [11987310158401015840

119908

]119879

[11987310158401015840

119908

] 119889119909

[119872119908

]119890

= int

119897

0

119898[119873119906

]119879

[119873119906

] 119889119909 + int

119897

0

119898[119873119908

]119879

[119873119908

] 119889119909

(31)

5 Element Matrix

Flexural vibration and axial vibration are independent of oneanother irrespective of large deformation Element matrixesat flexural vibration and axial vibration are obtained byapplying variation on flexural vibration displacement andaxial vibration displacement respectively

51 Element Matrix for Beam Flexural Vibration

511 Stiffness Matrix After matrix operation elementdynamic stiffness matrix for beam flexural vibration isexpressed as

[119870119908

]119890

= 119867

[[[[[

[

11989611990811

11989611990812

11989611990813

11989611990814

11989611990821

11989611990822

11989611990823

11989611990824

11989611990831

11989611990832

11989611990833

11989611990834

11989611990841

11989611990842

11989611990843

11989611990844

]]]]]

]

(32)

where

119896119908119894119895

=1

119867int

119897

0

11987310158401015840

119908119894

11987310158401015840

119908119894

119889119909

119867 =1198641198681205723

119890minus2120582

119866

119866 = 16 (1 + cos2120582ch2120582 minus 2 cos 120582ch120582)

(33)

6 Mathematical Problems in Engineering

After integral each stiffness matrix coefficient can beobtained such as

11989611990811

= (cosh 120582 sin 120582 + cos 120582 sin 120582 minus sinh 120582 cosh 120582

minus2cosh2120582 sin 120582 cos 120582)

sdot (21198902120582

+ 1198904120582

+ 1)

+ 4 (119890120582

minus 1198903120582

) cosh 120582 (sinh 120582 sin 120582 + cosh 120582 cos 120582)

minus 2 (1 + 1198904120582

) cos 120582 sin 120582 cosh 120582 sinh 120582

+ 41198902120582cos2120582 (120582 minus sinh 120582 cosh 120582)

+ (61198902120582

+ 1198904120582

+ 1) cos 120582 sinh 120582

+ (1 minus 1198904120582

minus 81198902120582

120582) cos 120582 cosh 120582 + 4 (119890120582 + 1198903120582)

sdot (cos 120582 sinh 120582 cosh 120582 minus cos2120582 sinh120582 minus sin 120582

+cosh2120582 sin 120582 minus cos2120582 cosh 120582)

+ (41205821198902120582

+ 1198904120582

minus 1) cosh2120582 + (1 minus 1198904120582) sin 120582 sinh 120582(34)

512 Mass Matrix Mass matrix of element for beam flexuralvibration is given by

[119872119908

] = 119875

[[[[[

[

11989811990811

11989811990812

11989811990813

11989811990814

11989811990821

11989811990822

11989811990823

11989811990824

11989811990831

11989811990832

11989811990833

11989811990834

11989811990841

11989811990842

11989811990843

11989811990844

]]]]]

]

(35)

where

119898119908119894119895

=1

119875int

119897

0

119873119908119894

119873119908119895

119889119909 119875 =119898119890minus2120582

119866

119866 = 16 (1 + cos2120582ch2120582 minus 2 cos 120582ch120582) (36)

After integral each mass matrix coefficient can beobtained such as

11989811990811

= 120572minus1

[4 (119890120582

+ 1198903120582

)

sdot (cos2120582 sinh 120582 minus cos 120582 sinh120582 cosh 120582

minus sin 120582cosh2120582 + sin 120582)

+ (41205821198902120582

+ 1198904120582

minus 1) cosh2120582

+ (1 minus 1198904120582

) sin 120582

sdot (sinh 120582 minus 2 cos 120582 sinh 120582 cosh 120582)

+ (1198904120582

minus 101198902120582

+ 1) cos 120582 sinh120582

minus (81205821198902120582

+ 1198904120582

minus 1) cos 120582 cosh 120582

+ 4 (1198903120582

minus 119890120582

)

sdot cosh 120582 (sin 120582 sinh 120582 minus cos2120582 + 4 cos 120582 cosh 120582)

+ (21198902120582

+ 119890120582

+ 1) (sin 120582 cos 120582 minus sinh 120582 cosh 120582)

+ 31198902120582cos2120582 (4 sinh 120582 cosh 120582 + 120582)

+ 2 (61198902120582

minus 1 minus 1198904120582

) sin 120582 cos 120582cosh2120582

+ (1198904120582

minus 141198902120582

+ 1) sin 120582 cosh 120582] (37)

52 Element Matrix for Beam Axial Vibration After matrixoperation element dynamic stiffness matrix for beam axialvibration is expressed as

[119870119906

]119890

= 119864119860119888

[[[

[

minuscos 120574 sin 120574 + 1205742 (cos2120574 minus 1)

120574 cos 120574 + sin 1205742 (cos2120574 minus 1)

120574 cos 120574 + sin 1205742 (cos2120574 minus 1)

minuscos 120574 sin 120574 + 1205742 (cos2120574 minus 1)

]]]

]

(38)

Mass matrix of element for beam axial vibration is givenby

[119872119906

] = 119898

[[[

[

sin 120574 cos 120574 minus 1205742119888 (cos2120574 minus 1)

120574 cos 120574 minus sin 1205742119888 (cos2120574 minus 1)

120574 cos 120574 minus sin 1205742119888 (cos2120574 minus 1)

sin 120574 cos 120574 minus 1205742119888 (cos2120574 minus 1)

]]]

]

(39)

53 Dynamic Beam ElementMatrix Because of the indepen-dence of axial vibration and flexural vibration 6 by 6 matrixis obtained by adding two rows and two columns to the above4 by 4 matrix considering axial vibration without coupling ofaxial vibration and flexural vibration It is element stiffnessmatrix

Element stiffness matrix is given by

[119870]119890

=

[[[[[[[[[[[

[

11989611

0 0 11989614

0 0

11989622

11989623

0 11989625

11989626

11989633

0 11989635

11989636

11989644

0 0

symmetry 11989655

11989656

11989666

]]]]]]]]]]]

]

(40)

Mathematical Problems in Engineering 7

where

11989611

= minus119864119860119888 (cos 120574 sin 120574 + 120574)

2 (cos2120574 minus 1)

11989614

=119864119860119888 (120574 cos 120574 + sin 120574)

2 (cos2120574 minus 1)

11989622

= 11986711989611990811

11989623

= 11986711989611990812

11989625

= 11986711989611990813

11989626

= 11986711989611990814

11989633

= 11986711989611990822

11989635

= 11986711989611990823

11989636

= 11986711989611990824

11989644

= minus119864119860119888 (cos 120574 sin 120574 + 120574)

2 (cos2120574 minus 1)

11989655

= 11986711989611990833

11989656

= 11986711989611990834

11989666

= 11986711989611990844

(41)

Element mass matrix is given by

[119872]119890

=

[[[[[[[[[[[

[

11989811

0 0 11989814

0 0

11989822

11989823

0 11989825

11989826

11989833

0 11989835

11989836

11989844

0 0

symmetry 11989855

11989856

11989866

]]]]]]]]]]]

]

(42)

where

11989811

=119898 (sin 120574 cos 120574 minus 120574)2119888 (cos2120574 minus 1)

11989814

=119898 (120574 cos 120574 minus sin 120574)2119888 (cos2120574 minus 1)

11989822

= 11987511989811990811

11989823

= 11987511989811990812

11989825

= 11987511989811990813

11989826

= 11987511989811990814

11989833

= 11987511989811990822

11989835

= 11987511989811990823

11989836

= 11987511989811990824

11989844

=119898 (sin 120574 cos 120574 minus 120574)2119888 (cos2120574 minus 1)

11989855

= 11987511989811990833

11989856

= 11987511989811990834

11989866

= 11987511989811990844

(43)

6 Example and Comparison

To verify the dynamic beam element constructed by ana-lytical trial function method the comparisons between thecalculation results of this element general beam element andtheoretical solution are conducted

61 Calculation Model

(1) Four Kinds of Typical Beams Free vibration of four kindsof typical beams are analyzed including cantilever beamsimply supported beam one end clamped and another simplysupported beam and clamped-clamped beam

(2) Structure and Material Parameters The parameters ofcalculation model are as follows beam dimension is 6mtimes 02m times 03m material elastic modulus 119864 = 210GPaPoissonrsquos ratio 120592 = 03 and material density 120588 = 7800 kgm3

(3) Theoretical Solution According to theoretical equation offree vibration for distributed mass beam base frequencies offree vibration for all kinds of beam are obtained by usinganalytic method [1]

(4) Dynamic StiffnessMatrixMethodAccording to governingequation of free vibration for distributedmass beam stiffnesscoefficient [11] of beam vibration is provided and thenbase frequencies of free vibration for all kinds of beam areobtained

(5) Finite Element Model of General Beam Element Byusing the software ANSYS finite element model of beamvibration is established and modal analysis is conductedElement BEAM3 is used to simulate beam BEAM3 is generaltwo-dimension elastic beam element and uniaxial elementbearing tension pressure and bend Every node has threedegrees of freedom that is 119909-axis linear displacement 119910-axislinear displacement and 119911-axis angular displacement Cubicpolynomial interpolating function is used as its displacementtrial function Beam is divided into one beam element BlockLanczos method is used for modal extraction

(6) Model in This Paper Equilibrium equation of beamflexural free vibration is given by

119870 (120596) Δ (119905) + 119872 (120596) Δ (119905) = 0 (44)

where 119870(120596) is structure stiffness matrix with boundarydisplacement restraint and 119872(120596) is structure mass matrixwith boundary displacement restraint According to natural

8 Mathematical Problems in Engineering

Table 1 Base frequency of beam free vibration and its comparison (rads)

Beam type Theoreticalsolution

Dynamic stiffness matrixmethod

Interpolation trialfunction element

Analytical trial functionelement

Absolutevalue

Relative error()

Absolutevalue

Relative error()

Absolutevalue

Relative error()

Cantilever beam 438875389 438875390 289119864 minus 07 440740 0423 438875391 312119864 minus 07

Simply supportedbeam 1231941891 1231941888 minus859119864 minus 08 1365902 10874 1231941889 minus637119864 minus 08

One end clamped andanother simplysupported beam

1924528355 1924528348 minus150119864 minus 07 2554366 32727 1924528349 minus153119864 minus 07

Clamped-clampedbeam 2792673996 2792673991 minus132119864 minus 07 2834156 1485 2792673992 minus127119864 minus 07

Note calculation of eigenvalue and eigenvector cannot be conducted due to all DOFs of two ends of clamped-clamped beam being restrained If the beam isdivided into one element semistructure is used for vibration simulation of clamped-clamped beam

Table 2 Comparison of different element results (relative error )

Beam typeElement number for analytical

trial function Element number for interpolation trial function

1 1 2 10 20Cantilever beam 312119864 minus 07 04249 minus00003 minus00489 minus00489Simply supported beam minus637119864 minus 08 108739 02909 minus01018 minus01018One end clamped and anothersimply supported beam minus153119864 minus 07 327269 08037 minus01169 minus01202

Clamped-clamped beam minus127119864 minus 07 14854 14854 minus01255 minus01278

vibration governing equation of beam beam natural vibra-tion is harmonic vibration The following assumption can beobtained

Δ (119905) = 119884 sin120596119905 (45)

Substituting (45) into (44) yields

119872minus1

(120596)119870 (120596) 119884 = 1205962

119884 (46)

Assuming 120578 = 1205962 then 119864(120578) = 119872minus1(120578)119870(120578)The following equation can be obtained

119864 (120578) 119884 = 120578119884 (47)

Beamnatural vibration analysis is converted to eigenvalueand eigenvector analysis of matrix 119864(120578) = 119872minus1(120578)119870(120578)

Iteration method is used to calculate base frequency ofbeam vibration Through circular calculations the iterationstops when frequency relative error reaches 120581

119910

= (120596119894+1

minus

120596119894

)120596119894

lt 10minus8

62 Result Comparison and Analysis

621 Element Accuracy Vibration analysis is conducted bydividing beam into one element Table 1 shows base frequencyof beam free vibration and the relative error between calcu-lated solution and theoretical solution

As shown in Table 1 the following results can be obtained(1) By applying analytical trial function element to the

analysis a more accurate solution can be attained when the

relative error is less than 10minus8 The series of relative errorbetween this solution and theoretical solution is about 1 times10minus9 It can be supposed that this error is calculation error ofiteration efficiency

(2) By applying interpolation trial function element tothe analysis the maximum relative error between the solu-tion and theoretical solution reaches 30 It indicates thatdynamic beam element constructed by taking polynomialfunction as displacement trial function has bigger error

(3)The solution of dynamic stiffness matrix method [11]is close to theoretical solution The relative error is almostidentical with that calculated by analytical trial functionmethod The reason is that their displacement curves forvibrating beam are both derived from dynamic equilibriumgoverning equation Element load value and node balancecondition need to be analyzed in dynamic stiffness matrixmethod Dynamic stiffness matrix of beam element can bedirectly obtained in analytical trial function method Thelatter is more simple and intuitive and widely applied

622 Element Efficiency The beam is divided into 1 element2 elements 10 elements and 20 elements respectively Beamvibration numerical simulation is conducted by applyinginterpolation trial function method The relative error ofcalculated base frequency of free vibration for every typicalbeam and theoretical solution is obtained Table 2 presentsthe results

As shown in Table 2 the result of interpolation trialfunction is closer to theoretical solution with the increasing

Mathematical Problems in Engineering 9

of element number But it is obvious that base frequencybecomes smaller and its difference with theoretical solution islargerThe reason is that the more the elements are the lowerthe structure calculation stiffness isThe greater the differencebetween the actual stiffness and calculation stiffness Thenthe calculated base frequency further has large differencewith theoretical solution It indicates that actual deformationcurve for vibrating beam is not polynomial and polynomialfunction cannot be taken as displacement trial function forvibrating beam

7 Conclusions

Based on the results of this investigation the followingconclusions can be drawn

(1) The result of interpolation trial function elementfor simulating beam vibration is not in accordance withtheoretical solution and the relative error is larger Withelement number increasing base frequency becomes smallerand has large difference with theoretical solution It indicatesthat vibrating beam displacement mode is different frompolynomial mode and the precision requirement cannot bemet by taking polynomial function as displacement trialfunction for vibrating beam

(2)The solution of dynamic stiffness matrix method forsimulating beam vibration is close to theoretical solutionBut in this method element load value and node balancecondition need to be specially analyzed and it is not easy toderive nonstructural question Its application in engineeringis limited

(3)Dynamic stiffness matrix of beam element is obtainedby applying analytical trial function put forward in thispaper Base frequencies of four typical beams are attained byanalyzing eigenvalue and eigenvector and are compared withtheoretical solutionThe results show that the series of relativeerror is about 1 times 10minus9 and it is actually calculation error ofiteration efficiency The dynamic beam element in the lightof analytical trial function put forward in this paper is high-precision element

Conflict of Interests

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

Acknowledgments

This research is supported by National Natural ScienceFunds no 51279206 Beijing Natural Science Foundation no3144029 Chinese Universities Scientific Fund no 2011JS126and the Specialized Research Fund for the Doctoral Programof Higher Education of China no 20110008120017

References

[1] A K Clough and J Penzien Dynamic of Structures McGraw-Hill New York NY USA 1995

[2] Q Jin ldquoAn analytical solution of dynamic response for therigid perfectly plastic Timoshenko beamrdquo Chinese Journal ofTheoretical and Applied Mechanics vol 16 no 5 pp 504ndash5111984

[3] X Guo H Chen and Q Zeng ldquoDynamic characteristicsanalysis model of prestressed concrete T-type beamrdquo ChineseJournal of Computational Mechanics vol 17 no 2 pp 176ndash1832000

[4] D-P Fang and Q-F Wang ldquoDynamic behavior analysis of anexternally prestressed beam with energy methodrdquo Journal ofVibration and Shock vol 31 no 1 pp 177ndash181 2012

[5] M Lou and T Hong ldquoAnalytical approach for dynamic charac-teristics of prestressed beam with external tendonsrdquo Journal ofTongji University Natural Science vol 34 no 10 pp 1284ndash12882006

[6] J A Carrer S A Fleischfresser L F Garcia and W J MansurldquoDynamic analysis of Timoshenko beams by the boundary ele-ment methodrdquo Engineering Analysis with Boundary Elementsvol 37 no 12 pp 1602ndash1616 2013

[7] J-J Wu ldquoUse of the elastic-and-rigid-combined beam elementfor dynamic analysis of a two-dimensional frame with arbi-trarily distributed rigid beam segmentsrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 3 pp 1240ndash1251 2011

[8] M A De Rosa C Franciosi and M J Maurizi ldquoOn thedynamic behaviour of slender beams with elastic ends carryinga concentrated massrdquo Computers and Structures vol 58 no 6pp 1145ndash1159 1996

[9] R W Clough ldquoThe finite element method in plane stress anal-ysisrdquo in Proceedings of the 2nd ASCE Conference on ElectronicComputation Pittsburgh Pa USA 1960

[10] V Kolousek ldquoAnwendung des Gesetzes der virtuellen Verschiebungen and des in der Reziprozitatssatzesrdquo Stab weksDynamik Lngenieur Archiv vol 12 pp 363ndash370 1941

[11] Y-Q Long and S-H Bao Structural Mechanics II HigherEducation Press Beijing China 2nd edition 1996

[12] S M Hashemi and M J Richard ldquoFree vibrational analysis ofaxially loaded bending-torsion coupled beams a dynamic finiteelementrdquo Computers amp Structures vol 77 no 6 pp 711ndash7242000

[13] S Chen M Geradin and E Lamine ldquoAn improved dynamicstiffness method and modal analysis for beam-like structuresrdquoComputers amp Structures vol 60 no 5 pp 725ndash731 1996

[14] J R Banerjee H Su and C Jayatunga ldquoA dynamic stiffnesselement for free vibration analysis of composite beams and itsapplication to aircraft wingsrdquo Computers amp Structures vol 86no 6 pp 573ndash579 2008

[15] J R Banerjee S Guo and W P Howson ldquoExact dynamicstiffness matrix of a bending-torsion coupled beam includingwarpingrdquo Computers and Structures vol 59 no 4 pp 613ndash6211996

[16] M Shavezipur and S M Hashemi ldquoFree vibration of triplycoupled centrifugally stiffened nonuniform beams using arefined dynamic finite element methodrdquo Aerospace Science andTechnology vol 13 no 1 pp 59ndash70 2009

[17] A K ChopraDynamics of Structures PrenticeHall EnglewoodCliffs NJ USA 2000

[18] X-L Hu and Z-M Dai ldquoOn the dynamic stress concentrationsin orthotropic plates with an arbitrary cutoutrdquo Chinese Journalof Applied Mechanics vol 15 no 1 pp 12ndash17 1998

10 Mathematical Problems in Engineering

[19] X Wang L Yang and W Gao ldquoDynamic FEM analysis forthe integration ballast structure based on variation principlerdquoJournal of Vibration and Shock vol 24 no 4 pp 99ndash102 2005

[20] S M Hashemi and M J Richard ldquoA dynamic finite element(DFE) method for free vibrations of bending-torsion coupledbeamsrdquo Aerospace Science and Technology vol 4 no 1 pp 41ndash55 2000

[21] A Pagani E Carrera M Boscolo and J R Banerjee ldquoRefineddynamic stiffness elements applied to free vibration analysis ofgenerally laminated composite beams with arbitrary boundaryconditionsrdquo Composite Structures vol 110 no 1 pp 305ndash3162014

[22] S M Nabi and N Ganesan ldquoA generalized element for thefree vibration analysis of composite beamsrdquo Computers andStructures vol 51 no 5 pp 607ndash610 1994

[23] L Zhao and Q Chen ldquoDynamic analysis of the stochasticvariational principle and stochastic finite element methodfor structures with random parametersrdquo Chinese Journal ofComputational Mechanics vol 15 no 3 pp 263ndash274 1998

[24] G-H Wang Y-N Gan and Z-B Wang ldquoEnergy-variationalmethod for the dynamic response of thin-walled I-beams withwide flangerdquo Engineering Mechanics vol 27 no 8 pp 15ndash202010

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

O x

y

V1

N1

M1

N2

M2

V21205792minus1205791

120601w(x)

Figure 1 Coordinate system and parameter positive direction

parameters according to differential equation of equi-librium for beam vibration

(2) to determine the undetermined parameters in dis-placement equation according to displacement con-dition of vibrating beam end

(3) towrite out displacement equation for beamvibrationexpressed by beam end displacement and then toobtain displacement trial function for vibrating beam

To construct beam dynamic element local coordinateas shown in Figure 1 is established It is in accordancewith general beam element The positive direction of lineardisplacement of beam end and force is in accordance withcoordinate direction and the positive direction of rotationangle and moment is in accordance with clockwise direction

31 Governing Equation for Beam Free Vibrationand Displacement General Solution

311 Beam Flexural Vibration Equation Dynamic equilib-rium governing equation for beam flexural vibration isexpressed as [1]

1198641198681205974V (119909 119905)1205971199094

+ 1198981205972V (119909 119905)1205971199052

= 0 (1)

In (1) V(119909 119905) is the displacement response for beamflexural vibration

By applying the method of separation of variables ana-lytical solution of deflection equation for beam flexuralvibration is obtained [1]

120601119908

(119909) = 1198621

sin120572119909 + 1198622

cos120572119909 + 1198623

sh120572119909 + 1198624

ch120572119909 (2)

In (2) flexural vibration parameter 120572 is connected withcircular frequency for beam flexural vibration 120596

119908

120596119908

=

1205722

radic119864119868119898 EI is beam section flexural stiffness and119898 is beammass per unit length

312 Beam Axial Vibration Equation Dynamic equilibriumgoverning equation for beam axial vibration is expressed as[1]

1198641198601198892

119906

1198891199092

minus 1198981205972

119906

1205971199052

= 0 (3)

In (3) 119906(119909 119905) is the displacement response for beam axialvibration

By applying themethod of separation of variables analyt-ical solution of deflection equation for beam axial vibration isobtained [1]

120601119906

(119909) = 1198631

cos120573119909 + 1198632

sin120573119909 (4)

In (4) axial vibration parameter 120573 is connected with cir-cular frequency for beam axial vibration 120596

119906

120596119906

= 120573radic119864119860119898and 119864119860 is beam section tensile (compressive) rigidity

32 Analytical Trial Function of Displacement for DynamicBeam Element

321 Analytical Trial Function for Beam Flexural Vibration

(1) Parameter Determination of Displacement Function Equa-tion (2) can be further expressed as

120601119908

(119909) =

4

sum

119894=1

120593119894

(119909) 119862119894

= 120593 119862 (5)

where

119862 = 1198621

1198622

1198623

1198624

119879

120593 = 1205931

(119909) 1205932

(119909) 1205933

(119909) 1205934

(119909)

= sin120572119909 cos120572119909 sh120572119909 ch120572119909

(6)

Beam end displacement is defined as 120575119908

119890

=

1205961

1205791

1205962

1205792

119879

According to (5) beam end displacement becomes

120575119908

119890

=

[[[[[

[

1205931

(0) 1205932

(0) 1205933

(0) 1205934

(0)

minus1205931015840

1

(0) minus1205931015840

2

(0) minus1205931015840

3

(0) minus1205931015840

4

(0)

1205931

(119897) 1205932

(119897) 1205933

(119897) 1205934

(119897)

minus1205931015840

1

(119897) minus1205931015840

2

(119897) minus1205931015840

3

(119897) minus1205931015840

4

(119897)

]]]]]

]

sdot

1198621

1198622

1198623

1198624

= [119860] 119862

(7)

If 120582 = 120572119897 with (6) [A] can be expressed as

[119860] =

[[[[[

[

0 1 0 1

minus120572 0 minus120572 0

sin 120582 cos 120582 sh120582 ch120582minus120572 cos 120582 120572 sin 120582 minus120572ch120582 minus120572sh120582

]]]]]

]

(8)

With (7) undetermined parameter of displacement func-tion is expressed as

119862 = [119860]minus1

120575119908

119890

(9)

4 Mathematical Problems in Engineering

To perform matrix inversion then

[119860]minus1

=

[[[[[

[

11988811

11988812

11988813

11988814

11988821

11988822

11988823

11988824

11988831

11988832

11988833

11988834

11988841

11988842

11988843

11988844

]]]]]

]

(10)

where

11988811

= minus11988831

=cos 120582 sinh 120582 + cosh 120582 sin 120582

119878

11988812

= minuscos 120582 cosh 120582 + sinh 120582 sin 120582 minus 1

120572119878

11988813

= minus11988833

= minussin 120582 + sinh 120582

119878

11988814

= minus11988834

= minuscosh 120582 minus cos 120582

120572119878

11988821

=1 minus sinh 120582 sin 120582 + cos 120582 cosh 120582

119878

11988822

= minus11988842

=cosh 120582 sin 120582 minus cos 120582 sinh 120582

119878

11988823

= minus11988843

=cosh 120582 minus cos 120582

119878

11988824

= minus11988844

=sinh 120582 minus sin 120582

120572119878

11988832

=1 + sinh 120582 sin 120582 minus cos 120582 cosh 120582

120572119878

11988841

=cos 120582 cosh 120582 + sin 120582 sinh 120582 minus 1

119878

119878 = 2 (cos 120582 cosh 120582 minus 1)

(11)

(2) Element Displacement Shape Function By substituting(9) into (5) displacement function for element flexural vibra-tion expressed by beam end displacement can be obtained

120601119908

(119909) = 120593 [119860]minus1

120575119908

119890

(12)

If

[119873119908

] = 120593 [119860]minus1

(13)

(12) becomes

120601119908

(119909) = [119873119908

] 120575119908

119890

(14)

[119873119908

] is flexural vibration displacement trial function fordynamic beam element With (13) it can be expressed as

[119873119908

] = 120593 [119860]minus1

= [1198731

1198732

1198733

1198734

] (15)

where 1198731

1198732

1198733

and 1198734

are deflection shape functionsAfter matrix operation they can be expressed as

1198731

=1

119878[sinh 120582 sin (120572119909 minus 120582) + cosh 120582 cos (120572119909 minus 120582)

minus cos (120572119909) + cos 120582 cosh (120572119909 minus 120582)

minus sin 120582 sinh (120572119909 minus 120582) minus cosh (120572119909)]

1198732

=1

120572119878[minus cosh 120582 sin (120572119909 minus 120582) minus sinh 120582 cos (120572119909 minus 120582)

+ sin (120572119909) + sinh (120572119909) minus sin 120582 cosh (120572119909 minus 120582)

minus cos 120582 sinh (120572119909 minus 120582)]

1198733

=1

119878[minus sin (120572119909) sinh 120582 minus cos (120572119909 minus 120582)

+ cos (120572119909) cosh 120582 minus cosh (120572119909 minus 120582)

+ sin 120582 sinh (120572119909) + cosh (120572119909) cos 120582]

1198734

=1

120572119878[minus sin (120572119909) cosh 120582 + sin (120572119909 minus 120582)

+ cos (120572119909) sinh 120582 + sinh (120572119909 minus 120582)

minus sinh (120572119909) cos 120582 + cosh (120572119909) sin 120582]

(16)

322 Analytical Trial Function for Beam Axial Vibration

(1) Parameter Determination of Displacement Function Equa-tion (4) can be further expressed as

120601119906

(119909) =

4

sum

119894=1

120595119894

(119909)119863119894

= 120595 119863 (17)

where

119863 = 1198631

1198632

119879

120595 = 1205951

(119909) 1205952

(119909)

= cos120573119909 sin120573119909

(18)

Beam end axial displacement is defined as 120575119906

119890

=

1199061

1199062

119879

According to (17) beam end axial displacement isobtained

120575119906

119890

= [

1205951

(0) 1205952

(0)

1205951

(119897) 1205952

(119897)

]

1198631

1198632

= [119861] 119863 (19)

If 120574 = 120573119897 with (18) [119861] can be expressed as

[119861] = [

1 0

cos 120574 sin 120574] (20)

With (19) undetermined parameter of axial displacementfunction is expressed as

119863 = [119861]minus1

120575119906

119890

(21)

Mathematical Problems in Engineering 5

To perform matrix inversion then

[119861]minus1

=[[

[

1 0

minuscos 120574sin 120574

1

sin 120574

]]

]

(22)

(2) ElementDisplacement Shape FunctionBy substituting (21)into (17) the following can be obtained

120601119906

(119909) = 120595 [119861]minus1

120575119906

119890

(23)

If

[119873119906

] = 120595 [119861]minus1

(24)

(23) becomes

120601119906

(119909) = [119873119906

] 120575119906

119890

(25)

[119873119906

] is axial vibration displacement trial function fordynamic beam element With (23) it can be expressed as

[119873119906

] = 120595 [119861]minus1

= [1198735

1198736

] (26)

where 1198735

1198736

are axial vibration displacement shape func-tions After matrix operation they can be expressed as

1198735

=cos120573119909 minus sin120573119909 cos 120574

sin 120574

1198736

=sin120573119909sin 120574

(27)

4 Element Potential EnergyFunctional and Variation

Finite element formulation of dynamic beam element isconstructed by principle of minimum potential energy andanalytical trial function

41 Potential Energy Functional In the light of potentialenergy functional of total potential energy is given by

prod

119901

= 119880 + 119881

=1

2int

119897

0

11986411986012060110158402

119906

(119909) 119889119909 +1

2int

119897

0

119864119868120601101584010158402

119908

(119909) 119889119909

minus1

2int

119897

0

1198981205962

119906

1206012

119906

(119909) 119889119909 minus1

2int

119897

0

1198981205962

119908

1206012

119908

(119909) 119889119909

(28)

where119880 is element strain energy and119881 is potential energy ofinertia force

With element displacement the following can beobtained

prod

119901

=1

2int

119897

0

119864119860120575119906

119890

119879

[1198731015840

119906

]119879

[1198731015840

119906

] 120575119906

119890

119889119909

+1

2int

119897

0

119864119868120575119908

119890

119879

[11987310158401015840

119908

]119879

[11987310158401015840

119908

] 120575119908

119890

119889119909

minus1

2int

119897

0

1198981205962

119906

120575119906

119890

119879

[119873119906

]119879

[119873119906

] 120575119906

119890

119889119909

minus1

2int

119897

0

1198981205962

119908

120575119908

119890

119879

[119873119908

]119879

[119873119908

] 120575119908

119890

119889119909

(29)

42 Functional Variation and Transformation In the light ofprinciple of minimum potential energy that is 120597prod

119901

120597120575119890

=

0 element equilibrium equation is obtained

[119870]119890

120575119890

minus 1205962

[119872]119890

120575119890

= 0 (30)

where [119870]119890 and [119872]119890 are dynamic stiffness matrix and massmatrix of beam element respectively

[119870]119890

= int

119897

0

119864119860 [1198731015840

119906

]119879

[1198731015840

119906

] 119889119909 + int

119897

0

119864119868 [11987310158401015840

119908

]119879

[11987310158401015840

119908

] 119889119909

[119872119908

]119890

= int

119897

0

119898[119873119906

]119879

[119873119906

] 119889119909 + int

119897

0

119898[119873119908

]119879

[119873119908

] 119889119909

(31)

5 Element Matrix

Flexural vibration and axial vibration are independent of oneanother irrespective of large deformation Element matrixesat flexural vibration and axial vibration are obtained byapplying variation on flexural vibration displacement andaxial vibration displacement respectively

51 Element Matrix for Beam Flexural Vibration

511 Stiffness Matrix After matrix operation elementdynamic stiffness matrix for beam flexural vibration isexpressed as

[119870119908

]119890

= 119867

[[[[[

[

11989611990811

11989611990812

11989611990813

11989611990814

11989611990821

11989611990822

11989611990823

11989611990824

11989611990831

11989611990832

11989611990833

11989611990834

11989611990841

11989611990842

11989611990843

11989611990844

]]]]]

]

(32)

where

119896119908119894119895

=1

119867int

119897

0

11987310158401015840

119908119894

11987310158401015840

119908119894

119889119909

119867 =1198641198681205723

119890minus2120582

119866

119866 = 16 (1 + cos2120582ch2120582 minus 2 cos 120582ch120582)

(33)

6 Mathematical Problems in Engineering

After integral each stiffness matrix coefficient can beobtained such as

11989611990811

= (cosh 120582 sin 120582 + cos 120582 sin 120582 minus sinh 120582 cosh 120582

minus2cosh2120582 sin 120582 cos 120582)

sdot (21198902120582

+ 1198904120582

+ 1)

+ 4 (119890120582

minus 1198903120582

) cosh 120582 (sinh 120582 sin 120582 + cosh 120582 cos 120582)

minus 2 (1 + 1198904120582

) cos 120582 sin 120582 cosh 120582 sinh 120582

+ 41198902120582cos2120582 (120582 minus sinh 120582 cosh 120582)

+ (61198902120582

+ 1198904120582

+ 1) cos 120582 sinh 120582

+ (1 minus 1198904120582

minus 81198902120582

120582) cos 120582 cosh 120582 + 4 (119890120582 + 1198903120582)

sdot (cos 120582 sinh 120582 cosh 120582 minus cos2120582 sinh120582 minus sin 120582

+cosh2120582 sin 120582 minus cos2120582 cosh 120582)

+ (41205821198902120582

+ 1198904120582

minus 1) cosh2120582 + (1 minus 1198904120582) sin 120582 sinh 120582(34)

512 Mass Matrix Mass matrix of element for beam flexuralvibration is given by

[119872119908

] = 119875

[[[[[

[

11989811990811

11989811990812

11989811990813

11989811990814

11989811990821

11989811990822

11989811990823

11989811990824

11989811990831

11989811990832

11989811990833

11989811990834

11989811990841

11989811990842

11989811990843

11989811990844

]]]]]

]

(35)

where

119898119908119894119895

=1

119875int

119897

0

119873119908119894

119873119908119895

119889119909 119875 =119898119890minus2120582

119866

119866 = 16 (1 + cos2120582ch2120582 minus 2 cos 120582ch120582) (36)

After integral each mass matrix coefficient can beobtained such as

11989811990811

= 120572minus1

[4 (119890120582

+ 1198903120582

)

sdot (cos2120582 sinh 120582 minus cos 120582 sinh120582 cosh 120582

minus sin 120582cosh2120582 + sin 120582)

+ (41205821198902120582

+ 1198904120582

minus 1) cosh2120582

+ (1 minus 1198904120582

) sin 120582

sdot (sinh 120582 minus 2 cos 120582 sinh 120582 cosh 120582)

+ (1198904120582

minus 101198902120582

+ 1) cos 120582 sinh120582

minus (81205821198902120582

+ 1198904120582

minus 1) cos 120582 cosh 120582

+ 4 (1198903120582

minus 119890120582

)

sdot cosh 120582 (sin 120582 sinh 120582 minus cos2120582 + 4 cos 120582 cosh 120582)

+ (21198902120582

+ 119890120582

+ 1) (sin 120582 cos 120582 minus sinh 120582 cosh 120582)

+ 31198902120582cos2120582 (4 sinh 120582 cosh 120582 + 120582)

+ 2 (61198902120582

minus 1 minus 1198904120582

) sin 120582 cos 120582cosh2120582

+ (1198904120582

minus 141198902120582

+ 1) sin 120582 cosh 120582] (37)

52 Element Matrix for Beam Axial Vibration After matrixoperation element dynamic stiffness matrix for beam axialvibration is expressed as

[119870119906

]119890

= 119864119860119888

[[[

[

minuscos 120574 sin 120574 + 1205742 (cos2120574 minus 1)

120574 cos 120574 + sin 1205742 (cos2120574 minus 1)

120574 cos 120574 + sin 1205742 (cos2120574 minus 1)

minuscos 120574 sin 120574 + 1205742 (cos2120574 minus 1)

]]]

]

(38)

Mass matrix of element for beam axial vibration is givenby

[119872119906

] = 119898

[[[

[

sin 120574 cos 120574 minus 1205742119888 (cos2120574 minus 1)

120574 cos 120574 minus sin 1205742119888 (cos2120574 minus 1)

120574 cos 120574 minus sin 1205742119888 (cos2120574 minus 1)

sin 120574 cos 120574 minus 1205742119888 (cos2120574 minus 1)

]]]

]

(39)

53 Dynamic Beam ElementMatrix Because of the indepen-dence of axial vibration and flexural vibration 6 by 6 matrixis obtained by adding two rows and two columns to the above4 by 4 matrix considering axial vibration without coupling ofaxial vibration and flexural vibration It is element stiffnessmatrix

Element stiffness matrix is given by

[119870]119890

=

[[[[[[[[[[[

[

11989611

0 0 11989614

0 0

11989622

11989623

0 11989625

11989626

11989633

0 11989635

11989636

11989644

0 0

symmetry 11989655

11989656

11989666

]]]]]]]]]]]

]

(40)

Mathematical Problems in Engineering 7

where

11989611

= minus119864119860119888 (cos 120574 sin 120574 + 120574)

2 (cos2120574 minus 1)

11989614

=119864119860119888 (120574 cos 120574 + sin 120574)

2 (cos2120574 minus 1)

11989622

= 11986711989611990811

11989623

= 11986711989611990812

11989625

= 11986711989611990813

11989626

= 11986711989611990814

11989633

= 11986711989611990822

11989635

= 11986711989611990823

11989636

= 11986711989611990824

11989644

= minus119864119860119888 (cos 120574 sin 120574 + 120574)

2 (cos2120574 minus 1)

11989655

= 11986711989611990833

11989656

= 11986711989611990834

11989666

= 11986711989611990844

(41)

Element mass matrix is given by

[119872]119890

=

[[[[[[[[[[[

[

11989811

0 0 11989814

0 0

11989822

11989823

0 11989825

11989826

11989833

0 11989835

11989836

11989844

0 0

symmetry 11989855

11989856

11989866

]]]]]]]]]]]

]

(42)

where

11989811

=119898 (sin 120574 cos 120574 minus 120574)2119888 (cos2120574 minus 1)

11989814

=119898 (120574 cos 120574 minus sin 120574)2119888 (cos2120574 minus 1)

11989822

= 11987511989811990811

11989823

= 11987511989811990812

11989825

= 11987511989811990813

11989826

= 11987511989811990814

11989833

= 11987511989811990822

11989835

= 11987511989811990823

11989836

= 11987511989811990824

11989844

=119898 (sin 120574 cos 120574 minus 120574)2119888 (cos2120574 minus 1)

11989855

= 11987511989811990833

11989856

= 11987511989811990834

11989866

= 11987511989811990844

(43)

6 Example and Comparison

To verify the dynamic beam element constructed by ana-lytical trial function method the comparisons between thecalculation results of this element general beam element andtheoretical solution are conducted

61 Calculation Model

(1) Four Kinds of Typical Beams Free vibration of four kindsof typical beams are analyzed including cantilever beamsimply supported beam one end clamped and another simplysupported beam and clamped-clamped beam

(2) Structure and Material Parameters The parameters ofcalculation model are as follows beam dimension is 6mtimes 02m times 03m material elastic modulus 119864 = 210GPaPoissonrsquos ratio 120592 = 03 and material density 120588 = 7800 kgm3

(3) Theoretical Solution According to theoretical equation offree vibration for distributed mass beam base frequencies offree vibration for all kinds of beam are obtained by usinganalytic method [1]

(4) Dynamic StiffnessMatrixMethodAccording to governingequation of free vibration for distributedmass beam stiffnesscoefficient [11] of beam vibration is provided and thenbase frequencies of free vibration for all kinds of beam areobtained

(5) Finite Element Model of General Beam Element Byusing the software ANSYS finite element model of beamvibration is established and modal analysis is conductedElement BEAM3 is used to simulate beam BEAM3 is generaltwo-dimension elastic beam element and uniaxial elementbearing tension pressure and bend Every node has threedegrees of freedom that is 119909-axis linear displacement 119910-axislinear displacement and 119911-axis angular displacement Cubicpolynomial interpolating function is used as its displacementtrial function Beam is divided into one beam element BlockLanczos method is used for modal extraction

(6) Model in This Paper Equilibrium equation of beamflexural free vibration is given by

119870 (120596) Δ (119905) + 119872 (120596) Δ (119905) = 0 (44)

where 119870(120596) is structure stiffness matrix with boundarydisplacement restraint and 119872(120596) is structure mass matrixwith boundary displacement restraint According to natural

8 Mathematical Problems in Engineering

Table 1 Base frequency of beam free vibration and its comparison (rads)

Beam type Theoreticalsolution

Dynamic stiffness matrixmethod

Interpolation trialfunction element

Analytical trial functionelement

Absolutevalue

Relative error()

Absolutevalue

Relative error()

Absolutevalue

Relative error()

Cantilever beam 438875389 438875390 289119864 minus 07 440740 0423 438875391 312119864 minus 07

Simply supportedbeam 1231941891 1231941888 minus859119864 minus 08 1365902 10874 1231941889 minus637119864 minus 08

One end clamped andanother simplysupported beam

1924528355 1924528348 minus150119864 minus 07 2554366 32727 1924528349 minus153119864 minus 07

Clamped-clampedbeam 2792673996 2792673991 minus132119864 minus 07 2834156 1485 2792673992 minus127119864 minus 07

Note calculation of eigenvalue and eigenvector cannot be conducted due to all DOFs of two ends of clamped-clamped beam being restrained If the beam isdivided into one element semistructure is used for vibration simulation of clamped-clamped beam

Table 2 Comparison of different element results (relative error )

Beam typeElement number for analytical

trial function Element number for interpolation trial function

1 1 2 10 20Cantilever beam 312119864 minus 07 04249 minus00003 minus00489 minus00489Simply supported beam minus637119864 minus 08 108739 02909 minus01018 minus01018One end clamped and anothersimply supported beam minus153119864 minus 07 327269 08037 minus01169 minus01202

Clamped-clamped beam minus127119864 minus 07 14854 14854 minus01255 minus01278

vibration governing equation of beam beam natural vibra-tion is harmonic vibration The following assumption can beobtained

Δ (119905) = 119884 sin120596119905 (45)

Substituting (45) into (44) yields

119872minus1

(120596)119870 (120596) 119884 = 1205962

119884 (46)

Assuming 120578 = 1205962 then 119864(120578) = 119872minus1(120578)119870(120578)The following equation can be obtained

119864 (120578) 119884 = 120578119884 (47)

Beamnatural vibration analysis is converted to eigenvalueand eigenvector analysis of matrix 119864(120578) = 119872minus1(120578)119870(120578)

Iteration method is used to calculate base frequency ofbeam vibration Through circular calculations the iterationstops when frequency relative error reaches 120581

119910

= (120596119894+1

minus

120596119894

)120596119894

lt 10minus8

62 Result Comparison and Analysis

621 Element Accuracy Vibration analysis is conducted bydividing beam into one element Table 1 shows base frequencyof beam free vibration and the relative error between calcu-lated solution and theoretical solution

As shown in Table 1 the following results can be obtained(1) By applying analytical trial function element to the

analysis a more accurate solution can be attained when the

relative error is less than 10minus8 The series of relative errorbetween this solution and theoretical solution is about 1 times10minus9 It can be supposed that this error is calculation error ofiteration efficiency

(2) By applying interpolation trial function element tothe analysis the maximum relative error between the solu-tion and theoretical solution reaches 30 It indicates thatdynamic beam element constructed by taking polynomialfunction as displacement trial function has bigger error

(3)The solution of dynamic stiffness matrix method [11]is close to theoretical solution The relative error is almostidentical with that calculated by analytical trial functionmethod The reason is that their displacement curves forvibrating beam are both derived from dynamic equilibriumgoverning equation Element load value and node balancecondition need to be analyzed in dynamic stiffness matrixmethod Dynamic stiffness matrix of beam element can bedirectly obtained in analytical trial function method Thelatter is more simple and intuitive and widely applied

622 Element Efficiency The beam is divided into 1 element2 elements 10 elements and 20 elements respectively Beamvibration numerical simulation is conducted by applyinginterpolation trial function method The relative error ofcalculated base frequency of free vibration for every typicalbeam and theoretical solution is obtained Table 2 presentsthe results

As shown in Table 2 the result of interpolation trialfunction is closer to theoretical solution with the increasing

Mathematical Problems in Engineering 9

of element number But it is obvious that base frequencybecomes smaller and its difference with theoretical solution islargerThe reason is that the more the elements are the lowerthe structure calculation stiffness isThe greater the differencebetween the actual stiffness and calculation stiffness Thenthe calculated base frequency further has large differencewith theoretical solution It indicates that actual deformationcurve for vibrating beam is not polynomial and polynomialfunction cannot be taken as displacement trial function forvibrating beam

7 Conclusions

Based on the results of this investigation the followingconclusions can be drawn

(1) The result of interpolation trial function elementfor simulating beam vibration is not in accordance withtheoretical solution and the relative error is larger Withelement number increasing base frequency becomes smallerand has large difference with theoretical solution It indicatesthat vibrating beam displacement mode is different frompolynomial mode and the precision requirement cannot bemet by taking polynomial function as displacement trialfunction for vibrating beam

(2)The solution of dynamic stiffness matrix method forsimulating beam vibration is close to theoretical solutionBut in this method element load value and node balancecondition need to be specially analyzed and it is not easy toderive nonstructural question Its application in engineeringis limited

(3)Dynamic stiffness matrix of beam element is obtainedby applying analytical trial function put forward in thispaper Base frequencies of four typical beams are attained byanalyzing eigenvalue and eigenvector and are compared withtheoretical solutionThe results show that the series of relativeerror is about 1 times 10minus9 and it is actually calculation error ofiteration efficiency The dynamic beam element in the lightof analytical trial function put forward in this paper is high-precision element

Conflict of Interests

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

Acknowledgments

This research is supported by National Natural ScienceFunds no 51279206 Beijing Natural Science Foundation no3144029 Chinese Universities Scientific Fund no 2011JS126and the Specialized Research Fund for the Doctoral Programof Higher Education of China no 20110008120017

References

[1] A K Clough and J Penzien Dynamic of Structures McGraw-Hill New York NY USA 1995

[2] Q Jin ldquoAn analytical solution of dynamic response for therigid perfectly plastic Timoshenko beamrdquo Chinese Journal ofTheoretical and Applied Mechanics vol 16 no 5 pp 504ndash5111984

[3] X Guo H Chen and Q Zeng ldquoDynamic characteristicsanalysis model of prestressed concrete T-type beamrdquo ChineseJournal of Computational Mechanics vol 17 no 2 pp 176ndash1832000

[4] D-P Fang and Q-F Wang ldquoDynamic behavior analysis of anexternally prestressed beam with energy methodrdquo Journal ofVibration and Shock vol 31 no 1 pp 177ndash181 2012

[5] M Lou and T Hong ldquoAnalytical approach for dynamic charac-teristics of prestressed beam with external tendonsrdquo Journal ofTongji University Natural Science vol 34 no 10 pp 1284ndash12882006

[6] J A Carrer S A Fleischfresser L F Garcia and W J MansurldquoDynamic analysis of Timoshenko beams by the boundary ele-ment methodrdquo Engineering Analysis with Boundary Elementsvol 37 no 12 pp 1602ndash1616 2013

[7] J-J Wu ldquoUse of the elastic-and-rigid-combined beam elementfor dynamic analysis of a two-dimensional frame with arbi-trarily distributed rigid beam segmentsrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 3 pp 1240ndash1251 2011

[8] M A De Rosa C Franciosi and M J Maurizi ldquoOn thedynamic behaviour of slender beams with elastic ends carryinga concentrated massrdquo Computers and Structures vol 58 no 6pp 1145ndash1159 1996

[9] R W Clough ldquoThe finite element method in plane stress anal-ysisrdquo in Proceedings of the 2nd ASCE Conference on ElectronicComputation Pittsburgh Pa USA 1960

[10] V Kolousek ldquoAnwendung des Gesetzes der virtuellen Verschiebungen and des in der Reziprozitatssatzesrdquo Stab weksDynamik Lngenieur Archiv vol 12 pp 363ndash370 1941

[11] Y-Q Long and S-H Bao Structural Mechanics II HigherEducation Press Beijing China 2nd edition 1996

[12] S M Hashemi and M J Richard ldquoFree vibrational analysis ofaxially loaded bending-torsion coupled beams a dynamic finiteelementrdquo Computers amp Structures vol 77 no 6 pp 711ndash7242000

[13] S Chen M Geradin and E Lamine ldquoAn improved dynamicstiffness method and modal analysis for beam-like structuresrdquoComputers amp Structures vol 60 no 5 pp 725ndash731 1996

[14] J R Banerjee H Su and C Jayatunga ldquoA dynamic stiffnesselement for free vibration analysis of composite beams and itsapplication to aircraft wingsrdquo Computers amp Structures vol 86no 6 pp 573ndash579 2008

[15] J R Banerjee S Guo and W P Howson ldquoExact dynamicstiffness matrix of a bending-torsion coupled beam includingwarpingrdquo Computers and Structures vol 59 no 4 pp 613ndash6211996

[16] M Shavezipur and S M Hashemi ldquoFree vibration of triplycoupled centrifugally stiffened nonuniform beams using arefined dynamic finite element methodrdquo Aerospace Science andTechnology vol 13 no 1 pp 59ndash70 2009

[17] A K ChopraDynamics of Structures PrenticeHall EnglewoodCliffs NJ USA 2000

[18] X-L Hu and Z-M Dai ldquoOn the dynamic stress concentrationsin orthotropic plates with an arbitrary cutoutrdquo Chinese Journalof Applied Mechanics vol 15 no 1 pp 12ndash17 1998

10 Mathematical Problems in Engineering

[19] X Wang L Yang and W Gao ldquoDynamic FEM analysis forthe integration ballast structure based on variation principlerdquoJournal of Vibration and Shock vol 24 no 4 pp 99ndash102 2005

[20] S M Hashemi and M J Richard ldquoA dynamic finite element(DFE) method for free vibrations of bending-torsion coupledbeamsrdquo Aerospace Science and Technology vol 4 no 1 pp 41ndash55 2000

[21] A Pagani E Carrera M Boscolo and J R Banerjee ldquoRefineddynamic stiffness elements applied to free vibration analysis ofgenerally laminated composite beams with arbitrary boundaryconditionsrdquo Composite Structures vol 110 no 1 pp 305ndash3162014

[22] S M Nabi and N Ganesan ldquoA generalized element for thefree vibration analysis of composite beamsrdquo Computers andStructures vol 51 no 5 pp 607ndash610 1994

[23] L Zhao and Q Chen ldquoDynamic analysis of the stochasticvariational principle and stochastic finite element methodfor structures with random parametersrdquo Chinese Journal ofComputational Mechanics vol 15 no 3 pp 263ndash274 1998

[24] G-H Wang Y-N Gan and Z-B Wang ldquoEnergy-variationalmethod for the dynamic response of thin-walled I-beams withwide flangerdquo Engineering Mechanics vol 27 no 8 pp 15ndash202010

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 Mathematical Problems in Engineering

To perform matrix inversion then

[119860]minus1

=

[[[[[

[

11988811

11988812

11988813

11988814

11988821

11988822

11988823

11988824

11988831

11988832

11988833

11988834

11988841

11988842

11988843

11988844

]]]]]

]

(10)

where

11988811

= minus11988831

=cos 120582 sinh 120582 + cosh 120582 sin 120582

119878

11988812

= minuscos 120582 cosh 120582 + sinh 120582 sin 120582 minus 1

120572119878

11988813

= minus11988833

= minussin 120582 + sinh 120582

119878

11988814

= minus11988834

= minuscosh 120582 minus cos 120582

120572119878

11988821

=1 minus sinh 120582 sin 120582 + cos 120582 cosh 120582

119878

11988822

= minus11988842

=cosh 120582 sin 120582 minus cos 120582 sinh 120582

119878

11988823

= minus11988843

=cosh 120582 minus cos 120582

119878

11988824

= minus11988844

=sinh 120582 minus sin 120582

120572119878

11988832

=1 + sinh 120582 sin 120582 minus cos 120582 cosh 120582

120572119878

11988841

=cos 120582 cosh 120582 + sin 120582 sinh 120582 minus 1

119878

119878 = 2 (cos 120582 cosh 120582 minus 1)

(11)

(2) Element Displacement Shape Function By substituting(9) into (5) displacement function for element flexural vibra-tion expressed by beam end displacement can be obtained

120601119908

(119909) = 120593 [119860]minus1

120575119908

119890

(12)

If

[119873119908

] = 120593 [119860]minus1

(13)

(12) becomes

120601119908

(119909) = [119873119908

] 120575119908

119890

(14)

[119873119908

] is flexural vibration displacement trial function fordynamic beam element With (13) it can be expressed as

[119873119908

] = 120593 [119860]minus1

= [1198731

1198732

1198733

1198734

] (15)

where 1198731

1198732

1198733

and 1198734

are deflection shape functionsAfter matrix operation they can be expressed as

1198731

=1

119878[sinh 120582 sin (120572119909 minus 120582) + cosh 120582 cos (120572119909 minus 120582)

minus cos (120572119909) + cos 120582 cosh (120572119909 minus 120582)

minus sin 120582 sinh (120572119909 minus 120582) minus cosh (120572119909)]

1198732

=1

120572119878[minus cosh 120582 sin (120572119909 minus 120582) minus sinh 120582 cos (120572119909 minus 120582)

+ sin (120572119909) + sinh (120572119909) minus sin 120582 cosh (120572119909 minus 120582)

minus cos 120582 sinh (120572119909 minus 120582)]

1198733

=1

119878[minus sin (120572119909) sinh 120582 minus cos (120572119909 minus 120582)

+ cos (120572119909) cosh 120582 minus cosh (120572119909 minus 120582)

+ sin 120582 sinh (120572119909) + cosh (120572119909) cos 120582]

1198734

=1

120572119878[minus sin (120572119909) cosh 120582 + sin (120572119909 minus 120582)

+ cos (120572119909) sinh 120582 + sinh (120572119909 minus 120582)

minus sinh (120572119909) cos 120582 + cosh (120572119909) sin 120582]

(16)

322 Analytical Trial Function for Beam Axial Vibration

(1) Parameter Determination of Displacement Function Equa-tion (4) can be further expressed as

120601119906

(119909) =

4

sum

119894=1

120595119894

(119909)119863119894

= 120595 119863 (17)

where

119863 = 1198631

1198632

119879

120595 = 1205951

(119909) 1205952

(119909)

= cos120573119909 sin120573119909

(18)

Beam end axial displacement is defined as 120575119906

119890

=

1199061

1199062

119879

According to (17) beam end axial displacement isobtained

120575119906

119890

= [

1205951

(0) 1205952

(0)

1205951

(119897) 1205952

(119897)

]

1198631

1198632

= [119861] 119863 (19)

If 120574 = 120573119897 with (18) [119861] can be expressed as

[119861] = [

1 0

cos 120574 sin 120574] (20)

With (19) undetermined parameter of axial displacementfunction is expressed as

119863 = [119861]minus1

120575119906

119890

(21)

Mathematical Problems in Engineering 5

To perform matrix inversion then

[119861]minus1

=[[

[

1 0

minuscos 120574sin 120574

1

sin 120574

]]

]

(22)

(2) ElementDisplacement Shape FunctionBy substituting (21)into (17) the following can be obtained

120601119906

(119909) = 120595 [119861]minus1

120575119906

119890

(23)

If

[119873119906

] = 120595 [119861]minus1

(24)

(23) becomes

120601119906

(119909) = [119873119906

] 120575119906

119890

(25)

[119873119906

] is axial vibration displacement trial function fordynamic beam element With (23) it can be expressed as

[119873119906

] = 120595 [119861]minus1

= [1198735

1198736

] (26)

where 1198735

1198736

are axial vibration displacement shape func-tions After matrix operation they can be expressed as

1198735

=cos120573119909 minus sin120573119909 cos 120574

sin 120574

1198736

=sin120573119909sin 120574

(27)

4 Element Potential EnergyFunctional and Variation

Finite element formulation of dynamic beam element isconstructed by principle of minimum potential energy andanalytical trial function

41 Potential Energy Functional In the light of potentialenergy functional of total potential energy is given by

prod

119901

= 119880 + 119881

=1

2int

119897

0

11986411986012060110158402

119906

(119909) 119889119909 +1

2int

119897

0

119864119868120601101584010158402

119908

(119909) 119889119909

minus1

2int

119897

0

1198981205962

119906

1206012

119906

(119909) 119889119909 minus1

2int

119897

0

1198981205962

119908

1206012

119908

(119909) 119889119909

(28)

where119880 is element strain energy and119881 is potential energy ofinertia force

With element displacement the following can beobtained

prod

119901

=1

2int

119897

0

119864119860120575119906

119890

119879

[1198731015840

119906

]119879

[1198731015840

119906

] 120575119906

119890

119889119909

+1

2int

119897

0

119864119868120575119908

119890

119879

[11987310158401015840

119908

]119879

[11987310158401015840

119908

] 120575119908

119890

119889119909

minus1

2int

119897

0

1198981205962

119906

120575119906

119890

119879

[119873119906

]119879

[119873119906

] 120575119906

119890

119889119909

minus1

2int

119897

0

1198981205962

119908

120575119908

119890

119879

[119873119908

]119879

[119873119908

] 120575119908

119890

119889119909

(29)

42 Functional Variation and Transformation In the light ofprinciple of minimum potential energy that is 120597prod

119901

120597120575119890

=

0 element equilibrium equation is obtained

[119870]119890

120575119890

minus 1205962

[119872]119890

120575119890

= 0 (30)

where [119870]119890 and [119872]119890 are dynamic stiffness matrix and massmatrix of beam element respectively

[119870]119890

= int

119897

0

119864119860 [1198731015840

119906

]119879

[1198731015840

119906

] 119889119909 + int

119897

0

119864119868 [11987310158401015840

119908

]119879

[11987310158401015840

119908

] 119889119909

[119872119908

]119890

= int

119897

0

119898[119873119906

]119879

[119873119906

] 119889119909 + int

119897

0

119898[119873119908

]119879

[119873119908

] 119889119909

(31)

5 Element Matrix

Flexural vibration and axial vibration are independent of oneanother irrespective of large deformation Element matrixesat flexural vibration and axial vibration are obtained byapplying variation on flexural vibration displacement andaxial vibration displacement respectively

51 Element Matrix for Beam Flexural Vibration

511 Stiffness Matrix After matrix operation elementdynamic stiffness matrix for beam flexural vibration isexpressed as

[119870119908

]119890

= 119867

[[[[[

[

11989611990811

11989611990812

11989611990813

11989611990814

11989611990821

11989611990822

11989611990823

11989611990824

11989611990831

11989611990832

11989611990833

11989611990834

11989611990841

11989611990842

11989611990843

11989611990844

]]]]]

]

(32)

where

119896119908119894119895

=1

119867int

119897

0

11987310158401015840

119908119894

11987310158401015840

119908119894

119889119909

119867 =1198641198681205723

119890minus2120582

119866

119866 = 16 (1 + cos2120582ch2120582 minus 2 cos 120582ch120582)

(33)

6 Mathematical Problems in Engineering

After integral each stiffness matrix coefficient can beobtained such as

11989611990811

= (cosh 120582 sin 120582 + cos 120582 sin 120582 minus sinh 120582 cosh 120582

minus2cosh2120582 sin 120582 cos 120582)

sdot (21198902120582

+ 1198904120582

+ 1)

+ 4 (119890120582

minus 1198903120582

) cosh 120582 (sinh 120582 sin 120582 + cosh 120582 cos 120582)

minus 2 (1 + 1198904120582

) cos 120582 sin 120582 cosh 120582 sinh 120582

+ 41198902120582cos2120582 (120582 minus sinh 120582 cosh 120582)

+ (61198902120582

+ 1198904120582

+ 1) cos 120582 sinh 120582

+ (1 minus 1198904120582

minus 81198902120582

120582) cos 120582 cosh 120582 + 4 (119890120582 + 1198903120582)

sdot (cos 120582 sinh 120582 cosh 120582 minus cos2120582 sinh120582 minus sin 120582

+cosh2120582 sin 120582 minus cos2120582 cosh 120582)

+ (41205821198902120582

+ 1198904120582

minus 1) cosh2120582 + (1 minus 1198904120582) sin 120582 sinh 120582(34)

512 Mass Matrix Mass matrix of element for beam flexuralvibration is given by

[119872119908

] = 119875

[[[[[

[

11989811990811

11989811990812

11989811990813

11989811990814

11989811990821

11989811990822

11989811990823

11989811990824

11989811990831

11989811990832

11989811990833

11989811990834

11989811990841

11989811990842

11989811990843

11989811990844

]]]]]

]

(35)

where

119898119908119894119895

=1

119875int

119897

0

119873119908119894

119873119908119895

119889119909 119875 =119898119890minus2120582

119866

119866 = 16 (1 + cos2120582ch2120582 minus 2 cos 120582ch120582) (36)

After integral each mass matrix coefficient can beobtained such as

11989811990811

= 120572minus1

[4 (119890120582

+ 1198903120582

)

sdot (cos2120582 sinh 120582 minus cos 120582 sinh120582 cosh 120582

minus sin 120582cosh2120582 + sin 120582)

+ (41205821198902120582

+ 1198904120582

minus 1) cosh2120582

+ (1 minus 1198904120582

) sin 120582

sdot (sinh 120582 minus 2 cos 120582 sinh 120582 cosh 120582)

+ (1198904120582

minus 101198902120582

+ 1) cos 120582 sinh120582

minus (81205821198902120582

+ 1198904120582

minus 1) cos 120582 cosh 120582

+ 4 (1198903120582

minus 119890120582

)

sdot cosh 120582 (sin 120582 sinh 120582 minus cos2120582 + 4 cos 120582 cosh 120582)

+ (21198902120582

+ 119890120582

+ 1) (sin 120582 cos 120582 minus sinh 120582 cosh 120582)

+ 31198902120582cos2120582 (4 sinh 120582 cosh 120582 + 120582)

+ 2 (61198902120582

minus 1 minus 1198904120582

) sin 120582 cos 120582cosh2120582

+ (1198904120582

minus 141198902120582

+ 1) sin 120582 cosh 120582] (37)

52 Element Matrix for Beam Axial Vibration After matrixoperation element dynamic stiffness matrix for beam axialvibration is expressed as

[119870119906

]119890

= 119864119860119888

[[[

[

minuscos 120574 sin 120574 + 1205742 (cos2120574 minus 1)

120574 cos 120574 + sin 1205742 (cos2120574 minus 1)

120574 cos 120574 + sin 1205742 (cos2120574 minus 1)

minuscos 120574 sin 120574 + 1205742 (cos2120574 minus 1)

]]]

]

(38)

Mass matrix of element for beam axial vibration is givenby

[119872119906

] = 119898

[[[

[

sin 120574 cos 120574 minus 1205742119888 (cos2120574 minus 1)

120574 cos 120574 minus sin 1205742119888 (cos2120574 minus 1)

120574 cos 120574 minus sin 1205742119888 (cos2120574 minus 1)

sin 120574 cos 120574 minus 1205742119888 (cos2120574 minus 1)

]]]

]

(39)

53 Dynamic Beam ElementMatrix Because of the indepen-dence of axial vibration and flexural vibration 6 by 6 matrixis obtained by adding two rows and two columns to the above4 by 4 matrix considering axial vibration without coupling ofaxial vibration and flexural vibration It is element stiffnessmatrix

Element stiffness matrix is given by

[119870]119890

=

[[[[[[[[[[[

[

11989611

0 0 11989614

0 0

11989622

11989623

0 11989625

11989626

11989633

0 11989635

11989636

11989644

0 0

symmetry 11989655

11989656

11989666

]]]]]]]]]]]

]

(40)

Mathematical Problems in Engineering 7

where

11989611

= minus119864119860119888 (cos 120574 sin 120574 + 120574)

2 (cos2120574 minus 1)

11989614

=119864119860119888 (120574 cos 120574 + sin 120574)

2 (cos2120574 minus 1)

11989622

= 11986711989611990811

11989623

= 11986711989611990812

11989625

= 11986711989611990813

11989626

= 11986711989611990814

11989633

= 11986711989611990822

11989635

= 11986711989611990823

11989636

= 11986711989611990824

11989644

= minus119864119860119888 (cos 120574 sin 120574 + 120574)

2 (cos2120574 minus 1)

11989655

= 11986711989611990833

11989656

= 11986711989611990834

11989666

= 11986711989611990844

(41)

Element mass matrix is given by

[119872]119890

=

[[[[[[[[[[[

[

11989811

0 0 11989814

0 0

11989822

11989823

0 11989825

11989826

11989833

0 11989835

11989836

11989844

0 0

symmetry 11989855

11989856

11989866

]]]]]]]]]]]

]

(42)

where

11989811

=119898 (sin 120574 cos 120574 minus 120574)2119888 (cos2120574 minus 1)

11989814

=119898 (120574 cos 120574 minus sin 120574)2119888 (cos2120574 minus 1)

11989822

= 11987511989811990811

11989823

= 11987511989811990812

11989825

= 11987511989811990813

11989826

= 11987511989811990814

11989833

= 11987511989811990822

11989835

= 11987511989811990823

11989836

= 11987511989811990824

11989844

=119898 (sin 120574 cos 120574 minus 120574)2119888 (cos2120574 minus 1)

11989855

= 11987511989811990833

11989856

= 11987511989811990834

11989866

= 11987511989811990844

(43)

6 Example and Comparison

To verify the dynamic beam element constructed by ana-lytical trial function method the comparisons between thecalculation results of this element general beam element andtheoretical solution are conducted

61 Calculation Model

(1) Four Kinds of Typical Beams Free vibration of four kindsof typical beams are analyzed including cantilever beamsimply supported beam one end clamped and another simplysupported beam and clamped-clamped beam

(2) Structure and Material Parameters The parameters ofcalculation model are as follows beam dimension is 6mtimes 02m times 03m material elastic modulus 119864 = 210GPaPoissonrsquos ratio 120592 = 03 and material density 120588 = 7800 kgm3

(3) Theoretical Solution According to theoretical equation offree vibration for distributed mass beam base frequencies offree vibration for all kinds of beam are obtained by usinganalytic method [1]

(4) Dynamic StiffnessMatrixMethodAccording to governingequation of free vibration for distributedmass beam stiffnesscoefficient [11] of beam vibration is provided and thenbase frequencies of free vibration for all kinds of beam areobtained

(5) Finite Element Model of General Beam Element Byusing the software ANSYS finite element model of beamvibration is established and modal analysis is conductedElement BEAM3 is used to simulate beam BEAM3 is generaltwo-dimension elastic beam element and uniaxial elementbearing tension pressure and bend Every node has threedegrees of freedom that is 119909-axis linear displacement 119910-axislinear displacement and 119911-axis angular displacement Cubicpolynomial interpolating function is used as its displacementtrial function Beam is divided into one beam element BlockLanczos method is used for modal extraction

(6) Model in This Paper Equilibrium equation of beamflexural free vibration is given by

119870 (120596) Δ (119905) + 119872 (120596) Δ (119905) = 0 (44)

where 119870(120596) is structure stiffness matrix with boundarydisplacement restraint and 119872(120596) is structure mass matrixwith boundary displacement restraint According to natural

8 Mathematical Problems in Engineering

Table 1 Base frequency of beam free vibration and its comparison (rads)

Beam type Theoreticalsolution

Dynamic stiffness matrixmethod

Interpolation trialfunction element

Analytical trial functionelement

Absolutevalue

Relative error()

Absolutevalue

Relative error()

Absolutevalue

Relative error()

Cantilever beam 438875389 438875390 289119864 minus 07 440740 0423 438875391 312119864 minus 07

Simply supportedbeam 1231941891 1231941888 minus859119864 minus 08 1365902 10874 1231941889 minus637119864 minus 08

One end clamped andanother simplysupported beam

1924528355 1924528348 minus150119864 minus 07 2554366 32727 1924528349 minus153119864 minus 07

Clamped-clampedbeam 2792673996 2792673991 minus132119864 minus 07 2834156 1485 2792673992 minus127119864 minus 07

Note calculation of eigenvalue and eigenvector cannot be conducted due to all DOFs of two ends of clamped-clamped beam being restrained If the beam isdivided into one element semistructure is used for vibration simulation of clamped-clamped beam

Table 2 Comparison of different element results (relative error )

Beam typeElement number for analytical

trial function Element number for interpolation trial function

1 1 2 10 20Cantilever beam 312119864 minus 07 04249 minus00003 minus00489 minus00489Simply supported beam minus637119864 minus 08 108739 02909 minus01018 minus01018One end clamped and anothersimply supported beam minus153119864 minus 07 327269 08037 minus01169 minus01202

Clamped-clamped beam minus127119864 minus 07 14854 14854 minus01255 minus01278

vibration governing equation of beam beam natural vibra-tion is harmonic vibration The following assumption can beobtained

Δ (119905) = 119884 sin120596119905 (45)

Substituting (45) into (44) yields

119872minus1

(120596)119870 (120596) 119884 = 1205962

119884 (46)

Assuming 120578 = 1205962 then 119864(120578) = 119872minus1(120578)119870(120578)The following equation can be obtained

119864 (120578) 119884 = 120578119884 (47)

Beamnatural vibration analysis is converted to eigenvalueand eigenvector analysis of matrix 119864(120578) = 119872minus1(120578)119870(120578)

Iteration method is used to calculate base frequency ofbeam vibration Through circular calculations the iterationstops when frequency relative error reaches 120581

119910

= (120596119894+1

minus

120596119894

)120596119894

lt 10minus8

62 Result Comparison and Analysis

621 Element Accuracy Vibration analysis is conducted bydividing beam into one element Table 1 shows base frequencyof beam free vibration and the relative error between calcu-lated solution and theoretical solution

As shown in Table 1 the following results can be obtained(1) By applying analytical trial function element to the

analysis a more accurate solution can be attained when the

relative error is less than 10minus8 The series of relative errorbetween this solution and theoretical solution is about 1 times10minus9 It can be supposed that this error is calculation error ofiteration efficiency

(2) By applying interpolation trial function element tothe analysis the maximum relative error between the solu-tion and theoretical solution reaches 30 It indicates thatdynamic beam element constructed by taking polynomialfunction as displacement trial function has bigger error

(3)The solution of dynamic stiffness matrix method [11]is close to theoretical solution The relative error is almostidentical with that calculated by analytical trial functionmethod The reason is that their displacement curves forvibrating beam are both derived from dynamic equilibriumgoverning equation Element load value and node balancecondition need to be analyzed in dynamic stiffness matrixmethod Dynamic stiffness matrix of beam element can bedirectly obtained in analytical trial function method Thelatter is more simple and intuitive and widely applied

622 Element Efficiency The beam is divided into 1 element2 elements 10 elements and 20 elements respectively Beamvibration numerical simulation is conducted by applyinginterpolation trial function method The relative error ofcalculated base frequency of free vibration for every typicalbeam and theoretical solution is obtained Table 2 presentsthe results

As shown in Table 2 the result of interpolation trialfunction is closer to theoretical solution with the increasing

Mathematical Problems in Engineering 9

of element number But it is obvious that base frequencybecomes smaller and its difference with theoretical solution islargerThe reason is that the more the elements are the lowerthe structure calculation stiffness isThe greater the differencebetween the actual stiffness and calculation stiffness Thenthe calculated base frequency further has large differencewith theoretical solution It indicates that actual deformationcurve for vibrating beam is not polynomial and polynomialfunction cannot be taken as displacement trial function forvibrating beam

7 Conclusions

Based on the results of this investigation the followingconclusions can be drawn

(1) The result of interpolation trial function elementfor simulating beam vibration is not in accordance withtheoretical solution and the relative error is larger Withelement number increasing base frequency becomes smallerand has large difference with theoretical solution It indicatesthat vibrating beam displacement mode is different frompolynomial mode and the precision requirement cannot bemet by taking polynomial function as displacement trialfunction for vibrating beam

(2)The solution of dynamic stiffness matrix method forsimulating beam vibration is close to theoretical solutionBut in this method element load value and node balancecondition need to be specially analyzed and it is not easy toderive nonstructural question Its application in engineeringis limited

(3)Dynamic stiffness matrix of beam element is obtainedby applying analytical trial function put forward in thispaper Base frequencies of four typical beams are attained byanalyzing eigenvalue and eigenvector and are compared withtheoretical solutionThe results show that the series of relativeerror is about 1 times 10minus9 and it is actually calculation error ofiteration efficiency The dynamic beam element in the lightof analytical trial function put forward in this paper is high-precision element

Conflict of Interests

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

Acknowledgments

This research is supported by National Natural ScienceFunds no 51279206 Beijing Natural Science Foundation no3144029 Chinese Universities Scientific Fund no 2011JS126and the Specialized Research Fund for the Doctoral Programof Higher Education of China no 20110008120017

References

[1] A K Clough and J Penzien Dynamic of Structures McGraw-Hill New York NY USA 1995

[2] Q Jin ldquoAn analytical solution of dynamic response for therigid perfectly plastic Timoshenko beamrdquo Chinese Journal ofTheoretical and Applied Mechanics vol 16 no 5 pp 504ndash5111984

[3] X Guo H Chen and Q Zeng ldquoDynamic characteristicsanalysis model of prestressed concrete T-type beamrdquo ChineseJournal of Computational Mechanics vol 17 no 2 pp 176ndash1832000

[4] D-P Fang and Q-F Wang ldquoDynamic behavior analysis of anexternally prestressed beam with energy methodrdquo Journal ofVibration and Shock vol 31 no 1 pp 177ndash181 2012

[5] M Lou and T Hong ldquoAnalytical approach for dynamic charac-teristics of prestressed beam with external tendonsrdquo Journal ofTongji University Natural Science vol 34 no 10 pp 1284ndash12882006

[6] J A Carrer S A Fleischfresser L F Garcia and W J MansurldquoDynamic analysis of Timoshenko beams by the boundary ele-ment methodrdquo Engineering Analysis with Boundary Elementsvol 37 no 12 pp 1602ndash1616 2013

[7] J-J Wu ldquoUse of the elastic-and-rigid-combined beam elementfor dynamic analysis of a two-dimensional frame with arbi-trarily distributed rigid beam segmentsrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 3 pp 1240ndash1251 2011

[8] M A De Rosa C Franciosi and M J Maurizi ldquoOn thedynamic behaviour of slender beams with elastic ends carryinga concentrated massrdquo Computers and Structures vol 58 no 6pp 1145ndash1159 1996

[9] R W Clough ldquoThe finite element method in plane stress anal-ysisrdquo in Proceedings of the 2nd ASCE Conference on ElectronicComputation Pittsburgh Pa USA 1960

[10] V Kolousek ldquoAnwendung des Gesetzes der virtuellen Verschiebungen and des in der Reziprozitatssatzesrdquo Stab weksDynamik Lngenieur Archiv vol 12 pp 363ndash370 1941

[11] Y-Q Long and S-H Bao Structural Mechanics II HigherEducation Press Beijing China 2nd edition 1996

[12] S M Hashemi and M J Richard ldquoFree vibrational analysis ofaxially loaded bending-torsion coupled beams a dynamic finiteelementrdquo Computers amp Structures vol 77 no 6 pp 711ndash7242000

[13] S Chen M Geradin and E Lamine ldquoAn improved dynamicstiffness method and modal analysis for beam-like structuresrdquoComputers amp Structures vol 60 no 5 pp 725ndash731 1996

[14] J R Banerjee H Su and C Jayatunga ldquoA dynamic stiffnesselement for free vibration analysis of composite beams and itsapplication to aircraft wingsrdquo Computers amp Structures vol 86no 6 pp 573ndash579 2008

[15] J R Banerjee S Guo and W P Howson ldquoExact dynamicstiffness matrix of a bending-torsion coupled beam includingwarpingrdquo Computers and Structures vol 59 no 4 pp 613ndash6211996

[16] M Shavezipur and S M Hashemi ldquoFree vibration of triplycoupled centrifugally stiffened nonuniform beams using arefined dynamic finite element methodrdquo Aerospace Science andTechnology vol 13 no 1 pp 59ndash70 2009

[17] A K ChopraDynamics of Structures PrenticeHall EnglewoodCliffs NJ USA 2000

[18] X-L Hu and Z-M Dai ldquoOn the dynamic stress concentrationsin orthotropic plates with an arbitrary cutoutrdquo Chinese Journalof Applied Mechanics vol 15 no 1 pp 12ndash17 1998

10 Mathematical Problems in Engineering

[19] X Wang L Yang and W Gao ldquoDynamic FEM analysis forthe integration ballast structure based on variation principlerdquoJournal of Vibration and Shock vol 24 no 4 pp 99ndash102 2005

[20] S M Hashemi and M J Richard ldquoA dynamic finite element(DFE) method for free vibrations of bending-torsion coupledbeamsrdquo Aerospace Science and Technology vol 4 no 1 pp 41ndash55 2000

[21] A Pagani E Carrera M Boscolo and J R Banerjee ldquoRefineddynamic stiffness elements applied to free vibration analysis ofgenerally laminated composite beams with arbitrary boundaryconditionsrdquo Composite Structures vol 110 no 1 pp 305ndash3162014

[22] S M Nabi and N Ganesan ldquoA generalized element for thefree vibration analysis of composite beamsrdquo Computers andStructures vol 51 no 5 pp 607ndash610 1994

[23] L Zhao and Q Chen ldquoDynamic analysis of the stochasticvariational principle and stochastic finite element methodfor structures with random parametersrdquo Chinese Journal ofComputational Mechanics vol 15 no 3 pp 263ndash274 1998

[24] G-H Wang Y-N Gan and Z-B Wang ldquoEnergy-variationalmethod for the dynamic response of thin-walled I-beams withwide flangerdquo Engineering Mechanics vol 27 no 8 pp 15ndash202010

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

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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

Mathematical Problems in Engineering 5

To perform matrix inversion then

[119861]minus1

=[[

[

1 0

minuscos 120574sin 120574

1

sin 120574

]]

]

(22)

(2) ElementDisplacement Shape FunctionBy substituting (21)into (17) the following can be obtained

120601119906

(119909) = 120595 [119861]minus1

120575119906

119890

(23)

If

[119873119906

] = 120595 [119861]minus1

(24)

(23) becomes

120601119906

(119909) = [119873119906

] 120575119906

119890

(25)

[119873119906

] is axial vibration displacement trial function fordynamic beam element With (23) it can be expressed as

[119873119906

] = 120595 [119861]minus1

= [1198735

1198736

] (26)

where 1198735

1198736

are axial vibration displacement shape func-tions After matrix operation they can be expressed as

1198735

=cos120573119909 minus sin120573119909 cos 120574

sin 120574

1198736

=sin120573119909sin 120574

(27)

4 Element Potential EnergyFunctional and Variation

Finite element formulation of dynamic beam element isconstructed by principle of minimum potential energy andanalytical trial function

41 Potential Energy Functional In the light of potentialenergy functional of total potential energy is given by

prod

119901

= 119880 + 119881

=1

2int

119897

0

11986411986012060110158402

119906

(119909) 119889119909 +1

2int

119897

0

119864119868120601101584010158402

119908

(119909) 119889119909

minus1

2int

119897

0

1198981205962

119906

1206012

119906

(119909) 119889119909 minus1

2int

119897

0

1198981205962

119908

1206012

119908

(119909) 119889119909

(28)

where119880 is element strain energy and119881 is potential energy ofinertia force

With element displacement the following can beobtained

prod

119901

=1

2int

119897

0

119864119860120575119906

119890

119879

[1198731015840

119906

]119879

[1198731015840

119906

] 120575119906

119890

119889119909

+1

2int

119897

0

119864119868120575119908

119890

119879

[11987310158401015840

119908

]119879

[11987310158401015840

119908

] 120575119908

119890

119889119909

minus1

2int

119897

0

1198981205962

119906

120575119906

119890

119879

[119873119906

]119879

[119873119906

] 120575119906

119890

119889119909

minus1

2int

119897

0

1198981205962

119908

120575119908

119890

119879

[119873119908

]119879

[119873119908

] 120575119908

119890

119889119909

(29)

42 Functional Variation and Transformation In the light ofprinciple of minimum potential energy that is 120597prod

119901

120597120575119890

=

0 element equilibrium equation is obtained

[119870]119890

120575119890

minus 1205962

[119872]119890

120575119890

= 0 (30)

where [119870]119890 and [119872]119890 are dynamic stiffness matrix and massmatrix of beam element respectively

[119870]119890

= int

119897

0

119864119860 [1198731015840

119906

]119879

[1198731015840

119906

] 119889119909 + int

119897

0

119864119868 [11987310158401015840

119908

]119879

[11987310158401015840

119908

] 119889119909

[119872119908

]119890

= int

119897

0

119898[119873119906

]119879

[119873119906

] 119889119909 + int

119897

0

119898[119873119908

]119879

[119873119908

] 119889119909

(31)

5 Element Matrix

Flexural vibration and axial vibration are independent of oneanother irrespective of large deformation Element matrixesat flexural vibration and axial vibration are obtained byapplying variation on flexural vibration displacement andaxial vibration displacement respectively

51 Element Matrix for Beam Flexural Vibration

511 Stiffness Matrix After matrix operation elementdynamic stiffness matrix for beam flexural vibration isexpressed as

[119870119908

]119890

= 119867

[[[[[

[

11989611990811

11989611990812

11989611990813

11989611990814

11989611990821

11989611990822

11989611990823

11989611990824

11989611990831

11989611990832

11989611990833

11989611990834

11989611990841

11989611990842

11989611990843

11989611990844

]]]]]

]

(32)

where

119896119908119894119895

=1

119867int

119897

0

11987310158401015840

119908119894

11987310158401015840

119908119894

119889119909

119867 =1198641198681205723

119890minus2120582

119866

119866 = 16 (1 + cos2120582ch2120582 minus 2 cos 120582ch120582)

(33)

6 Mathematical Problems in Engineering

After integral each stiffness matrix coefficient can beobtained such as

11989611990811

= (cosh 120582 sin 120582 + cos 120582 sin 120582 minus sinh 120582 cosh 120582

minus2cosh2120582 sin 120582 cos 120582)

sdot (21198902120582

+ 1198904120582

+ 1)

+ 4 (119890120582

minus 1198903120582

) cosh 120582 (sinh 120582 sin 120582 + cosh 120582 cos 120582)

minus 2 (1 + 1198904120582

) cos 120582 sin 120582 cosh 120582 sinh 120582

+ 41198902120582cos2120582 (120582 minus sinh 120582 cosh 120582)

+ (61198902120582

+ 1198904120582

+ 1) cos 120582 sinh 120582

+ (1 minus 1198904120582

minus 81198902120582

120582) cos 120582 cosh 120582 + 4 (119890120582 + 1198903120582)

sdot (cos 120582 sinh 120582 cosh 120582 minus cos2120582 sinh120582 minus sin 120582

+cosh2120582 sin 120582 minus cos2120582 cosh 120582)

+ (41205821198902120582

+ 1198904120582

minus 1) cosh2120582 + (1 minus 1198904120582) sin 120582 sinh 120582(34)

512 Mass Matrix Mass matrix of element for beam flexuralvibration is given by

[119872119908

] = 119875

[[[[[

[

11989811990811

11989811990812

11989811990813

11989811990814

11989811990821

11989811990822

11989811990823

11989811990824

11989811990831

11989811990832

11989811990833

11989811990834

11989811990841

11989811990842

11989811990843

11989811990844

]]]]]

]

(35)

where

119898119908119894119895

=1

119875int

119897

0

119873119908119894

119873119908119895

119889119909 119875 =119898119890minus2120582

119866

119866 = 16 (1 + cos2120582ch2120582 minus 2 cos 120582ch120582) (36)

After integral each mass matrix coefficient can beobtained such as

11989811990811

= 120572minus1

[4 (119890120582

+ 1198903120582

)

sdot (cos2120582 sinh 120582 minus cos 120582 sinh120582 cosh 120582

minus sin 120582cosh2120582 + sin 120582)

+ (41205821198902120582

+ 1198904120582

minus 1) cosh2120582

+ (1 minus 1198904120582

) sin 120582

sdot (sinh 120582 minus 2 cos 120582 sinh 120582 cosh 120582)

+ (1198904120582

minus 101198902120582

+ 1) cos 120582 sinh120582

minus (81205821198902120582

+ 1198904120582

minus 1) cos 120582 cosh 120582

+ 4 (1198903120582

minus 119890120582

)

sdot cosh 120582 (sin 120582 sinh 120582 minus cos2120582 + 4 cos 120582 cosh 120582)

+ (21198902120582

+ 119890120582

+ 1) (sin 120582 cos 120582 minus sinh 120582 cosh 120582)

+ 31198902120582cos2120582 (4 sinh 120582 cosh 120582 + 120582)

+ 2 (61198902120582

minus 1 minus 1198904120582

) sin 120582 cos 120582cosh2120582

+ (1198904120582

minus 141198902120582

+ 1) sin 120582 cosh 120582] (37)

52 Element Matrix for Beam Axial Vibration After matrixoperation element dynamic stiffness matrix for beam axialvibration is expressed as

[119870119906

]119890

= 119864119860119888

[[[

[

minuscos 120574 sin 120574 + 1205742 (cos2120574 minus 1)

120574 cos 120574 + sin 1205742 (cos2120574 minus 1)

120574 cos 120574 + sin 1205742 (cos2120574 minus 1)

minuscos 120574 sin 120574 + 1205742 (cos2120574 minus 1)

]]]

]

(38)

Mass matrix of element for beam axial vibration is givenby

[119872119906

] = 119898

[[[

[

sin 120574 cos 120574 minus 1205742119888 (cos2120574 minus 1)

120574 cos 120574 minus sin 1205742119888 (cos2120574 minus 1)

120574 cos 120574 minus sin 1205742119888 (cos2120574 minus 1)

sin 120574 cos 120574 minus 1205742119888 (cos2120574 minus 1)

]]]

]

(39)

53 Dynamic Beam ElementMatrix Because of the indepen-dence of axial vibration and flexural vibration 6 by 6 matrixis obtained by adding two rows and two columns to the above4 by 4 matrix considering axial vibration without coupling ofaxial vibration and flexural vibration It is element stiffnessmatrix

Element stiffness matrix is given by

[119870]119890

=

[[[[[[[[[[[

[

11989611

0 0 11989614

0 0

11989622

11989623

0 11989625

11989626

11989633

0 11989635

11989636

11989644

0 0

symmetry 11989655

11989656

11989666

]]]]]]]]]]]

]

(40)

Mathematical Problems in Engineering 7

where

11989611

= minus119864119860119888 (cos 120574 sin 120574 + 120574)

2 (cos2120574 minus 1)

11989614

=119864119860119888 (120574 cos 120574 + sin 120574)

2 (cos2120574 minus 1)

11989622

= 11986711989611990811

11989623

= 11986711989611990812

11989625

= 11986711989611990813

11989626

= 11986711989611990814

11989633

= 11986711989611990822

11989635

= 11986711989611990823

11989636

= 11986711989611990824

11989644

= minus119864119860119888 (cos 120574 sin 120574 + 120574)

2 (cos2120574 minus 1)

11989655

= 11986711989611990833

11989656

= 11986711989611990834

11989666

= 11986711989611990844

(41)

Element mass matrix is given by

[119872]119890

=

[[[[[[[[[[[

[

11989811

0 0 11989814

0 0

11989822

11989823

0 11989825

11989826

11989833

0 11989835

11989836

11989844

0 0

symmetry 11989855

11989856

11989866

]]]]]]]]]]]

]

(42)

where

11989811

=119898 (sin 120574 cos 120574 minus 120574)2119888 (cos2120574 minus 1)

11989814

=119898 (120574 cos 120574 minus sin 120574)2119888 (cos2120574 minus 1)

11989822

= 11987511989811990811

11989823

= 11987511989811990812

11989825

= 11987511989811990813

11989826

= 11987511989811990814

11989833

= 11987511989811990822

11989835

= 11987511989811990823

11989836

= 11987511989811990824

11989844

=119898 (sin 120574 cos 120574 minus 120574)2119888 (cos2120574 minus 1)

11989855

= 11987511989811990833

11989856

= 11987511989811990834

11989866

= 11987511989811990844

(43)

6 Example and Comparison

To verify the dynamic beam element constructed by ana-lytical trial function method the comparisons between thecalculation results of this element general beam element andtheoretical solution are conducted

61 Calculation Model

(1) Four Kinds of Typical Beams Free vibration of four kindsof typical beams are analyzed including cantilever beamsimply supported beam one end clamped and another simplysupported beam and clamped-clamped beam

(2) Structure and Material Parameters The parameters ofcalculation model are as follows beam dimension is 6mtimes 02m times 03m material elastic modulus 119864 = 210GPaPoissonrsquos ratio 120592 = 03 and material density 120588 = 7800 kgm3

(3) Theoretical Solution According to theoretical equation offree vibration for distributed mass beam base frequencies offree vibration for all kinds of beam are obtained by usinganalytic method [1]

(4) Dynamic StiffnessMatrixMethodAccording to governingequation of free vibration for distributedmass beam stiffnesscoefficient [11] of beam vibration is provided and thenbase frequencies of free vibration for all kinds of beam areobtained

(5) Finite Element Model of General Beam Element Byusing the software ANSYS finite element model of beamvibration is established and modal analysis is conductedElement BEAM3 is used to simulate beam BEAM3 is generaltwo-dimension elastic beam element and uniaxial elementbearing tension pressure and bend Every node has threedegrees of freedom that is 119909-axis linear displacement 119910-axislinear displacement and 119911-axis angular displacement Cubicpolynomial interpolating function is used as its displacementtrial function Beam is divided into one beam element BlockLanczos method is used for modal extraction

(6) Model in This Paper Equilibrium equation of beamflexural free vibration is given by

119870 (120596) Δ (119905) + 119872 (120596) Δ (119905) = 0 (44)

where 119870(120596) is structure stiffness matrix with boundarydisplacement restraint and 119872(120596) is structure mass matrixwith boundary displacement restraint According to natural

8 Mathematical Problems in Engineering

Table 1 Base frequency of beam free vibration and its comparison (rads)

Beam type Theoreticalsolution

Dynamic stiffness matrixmethod

Interpolation trialfunction element

Analytical trial functionelement

Absolutevalue

Relative error()

Absolutevalue

Relative error()

Absolutevalue

Relative error()

Cantilever beam 438875389 438875390 289119864 minus 07 440740 0423 438875391 312119864 minus 07

Simply supportedbeam 1231941891 1231941888 minus859119864 minus 08 1365902 10874 1231941889 minus637119864 minus 08

One end clamped andanother simplysupported beam

1924528355 1924528348 minus150119864 minus 07 2554366 32727 1924528349 minus153119864 minus 07

Clamped-clampedbeam 2792673996 2792673991 minus132119864 minus 07 2834156 1485 2792673992 minus127119864 minus 07

Note calculation of eigenvalue and eigenvector cannot be conducted due to all DOFs of two ends of clamped-clamped beam being restrained If the beam isdivided into one element semistructure is used for vibration simulation of clamped-clamped beam

Table 2 Comparison of different element results (relative error )

Beam typeElement number for analytical

trial function Element number for interpolation trial function

1 1 2 10 20Cantilever beam 312119864 minus 07 04249 minus00003 minus00489 minus00489Simply supported beam minus637119864 minus 08 108739 02909 minus01018 minus01018One end clamped and anothersimply supported beam minus153119864 minus 07 327269 08037 minus01169 minus01202

Clamped-clamped beam minus127119864 minus 07 14854 14854 minus01255 minus01278

vibration governing equation of beam beam natural vibra-tion is harmonic vibration The following assumption can beobtained

Δ (119905) = 119884 sin120596119905 (45)

Substituting (45) into (44) yields

119872minus1

(120596)119870 (120596) 119884 = 1205962

119884 (46)

Assuming 120578 = 1205962 then 119864(120578) = 119872minus1(120578)119870(120578)The following equation can be obtained

119864 (120578) 119884 = 120578119884 (47)

Beamnatural vibration analysis is converted to eigenvalueand eigenvector analysis of matrix 119864(120578) = 119872minus1(120578)119870(120578)

Iteration method is used to calculate base frequency ofbeam vibration Through circular calculations the iterationstops when frequency relative error reaches 120581

119910

= (120596119894+1

minus

120596119894

)120596119894

lt 10minus8

62 Result Comparison and Analysis

621 Element Accuracy Vibration analysis is conducted bydividing beam into one element Table 1 shows base frequencyof beam free vibration and the relative error between calcu-lated solution and theoretical solution

As shown in Table 1 the following results can be obtained(1) By applying analytical trial function element to the

analysis a more accurate solution can be attained when the

relative error is less than 10minus8 The series of relative errorbetween this solution and theoretical solution is about 1 times10minus9 It can be supposed that this error is calculation error ofiteration efficiency

(2) By applying interpolation trial function element tothe analysis the maximum relative error between the solu-tion and theoretical solution reaches 30 It indicates thatdynamic beam element constructed by taking polynomialfunction as displacement trial function has bigger error

(3)The solution of dynamic stiffness matrix method [11]is close to theoretical solution The relative error is almostidentical with that calculated by analytical trial functionmethod The reason is that their displacement curves forvibrating beam are both derived from dynamic equilibriumgoverning equation Element load value and node balancecondition need to be analyzed in dynamic stiffness matrixmethod Dynamic stiffness matrix of beam element can bedirectly obtained in analytical trial function method Thelatter is more simple and intuitive and widely applied

622 Element Efficiency The beam is divided into 1 element2 elements 10 elements and 20 elements respectively Beamvibration numerical simulation is conducted by applyinginterpolation trial function method The relative error ofcalculated base frequency of free vibration for every typicalbeam and theoretical solution is obtained Table 2 presentsthe results

As shown in Table 2 the result of interpolation trialfunction is closer to theoretical solution with the increasing

Mathematical Problems in Engineering 9

of element number But it is obvious that base frequencybecomes smaller and its difference with theoretical solution islargerThe reason is that the more the elements are the lowerthe structure calculation stiffness isThe greater the differencebetween the actual stiffness and calculation stiffness Thenthe calculated base frequency further has large differencewith theoretical solution It indicates that actual deformationcurve for vibrating beam is not polynomial and polynomialfunction cannot be taken as displacement trial function forvibrating beam

7 Conclusions

Based on the results of this investigation the followingconclusions can be drawn

(1) The result of interpolation trial function elementfor simulating beam vibration is not in accordance withtheoretical solution and the relative error is larger Withelement number increasing base frequency becomes smallerand has large difference with theoretical solution It indicatesthat vibrating beam displacement mode is different frompolynomial mode and the precision requirement cannot bemet by taking polynomial function as displacement trialfunction for vibrating beam

(2)The solution of dynamic stiffness matrix method forsimulating beam vibration is close to theoretical solutionBut in this method element load value and node balancecondition need to be specially analyzed and it is not easy toderive nonstructural question Its application in engineeringis limited

(3)Dynamic stiffness matrix of beam element is obtainedby applying analytical trial function put forward in thispaper Base frequencies of four typical beams are attained byanalyzing eigenvalue and eigenvector and are compared withtheoretical solutionThe results show that the series of relativeerror is about 1 times 10minus9 and it is actually calculation error ofiteration efficiency The dynamic beam element in the lightof analytical trial function put forward in this paper is high-precision element

Conflict of Interests

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

Acknowledgments

This research is supported by National Natural ScienceFunds no 51279206 Beijing Natural Science Foundation no3144029 Chinese Universities Scientific Fund no 2011JS126and the Specialized Research Fund for the Doctoral Programof Higher Education of China no 20110008120017

References

[1] A K Clough and J Penzien Dynamic of Structures McGraw-Hill New York NY USA 1995

[2] Q Jin ldquoAn analytical solution of dynamic response for therigid perfectly plastic Timoshenko beamrdquo Chinese Journal ofTheoretical and Applied Mechanics vol 16 no 5 pp 504ndash5111984

[3] X Guo H Chen and Q Zeng ldquoDynamic characteristicsanalysis model of prestressed concrete T-type beamrdquo ChineseJournal of Computational Mechanics vol 17 no 2 pp 176ndash1832000

[4] D-P Fang and Q-F Wang ldquoDynamic behavior analysis of anexternally prestressed beam with energy methodrdquo Journal ofVibration and Shock vol 31 no 1 pp 177ndash181 2012

[5] M Lou and T Hong ldquoAnalytical approach for dynamic charac-teristics of prestressed beam with external tendonsrdquo Journal ofTongji University Natural Science vol 34 no 10 pp 1284ndash12882006

[6] J A Carrer S A Fleischfresser L F Garcia and W J MansurldquoDynamic analysis of Timoshenko beams by the boundary ele-ment methodrdquo Engineering Analysis with Boundary Elementsvol 37 no 12 pp 1602ndash1616 2013

[7] J-J Wu ldquoUse of the elastic-and-rigid-combined beam elementfor dynamic analysis of a two-dimensional frame with arbi-trarily distributed rigid beam segmentsrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 3 pp 1240ndash1251 2011

[8] M A De Rosa C Franciosi and M J Maurizi ldquoOn thedynamic behaviour of slender beams with elastic ends carryinga concentrated massrdquo Computers and Structures vol 58 no 6pp 1145ndash1159 1996

[9] R W Clough ldquoThe finite element method in plane stress anal-ysisrdquo in Proceedings of the 2nd ASCE Conference on ElectronicComputation Pittsburgh Pa USA 1960

[10] V Kolousek ldquoAnwendung des Gesetzes der virtuellen Verschiebungen and des in der Reziprozitatssatzesrdquo Stab weksDynamik Lngenieur Archiv vol 12 pp 363ndash370 1941

[11] Y-Q Long and S-H Bao Structural Mechanics II HigherEducation Press Beijing China 2nd edition 1996

[12] S M Hashemi and M J Richard ldquoFree vibrational analysis ofaxially loaded bending-torsion coupled beams a dynamic finiteelementrdquo Computers amp Structures vol 77 no 6 pp 711ndash7242000

[13] S Chen M Geradin and E Lamine ldquoAn improved dynamicstiffness method and modal analysis for beam-like structuresrdquoComputers amp Structures vol 60 no 5 pp 725ndash731 1996

[14] J R Banerjee H Su and C Jayatunga ldquoA dynamic stiffnesselement for free vibration analysis of composite beams and itsapplication to aircraft wingsrdquo Computers amp Structures vol 86no 6 pp 573ndash579 2008

[15] J R Banerjee S Guo and W P Howson ldquoExact dynamicstiffness matrix of a bending-torsion coupled beam includingwarpingrdquo Computers and Structures vol 59 no 4 pp 613ndash6211996

[16] M Shavezipur and S M Hashemi ldquoFree vibration of triplycoupled centrifugally stiffened nonuniform beams using arefined dynamic finite element methodrdquo Aerospace Science andTechnology vol 13 no 1 pp 59ndash70 2009

[17] A K ChopraDynamics of Structures PrenticeHall EnglewoodCliffs NJ USA 2000

[18] X-L Hu and Z-M Dai ldquoOn the dynamic stress concentrationsin orthotropic plates with an arbitrary cutoutrdquo Chinese Journalof Applied Mechanics vol 15 no 1 pp 12ndash17 1998

10 Mathematical Problems in Engineering

[19] X Wang L Yang and W Gao ldquoDynamic FEM analysis forthe integration ballast structure based on variation principlerdquoJournal of Vibration and Shock vol 24 no 4 pp 99ndash102 2005

[20] S M Hashemi and M J Richard ldquoA dynamic finite element(DFE) method for free vibrations of bending-torsion coupledbeamsrdquo Aerospace Science and Technology vol 4 no 1 pp 41ndash55 2000

[21] A Pagani E Carrera M Boscolo and J R Banerjee ldquoRefineddynamic stiffness elements applied to free vibration analysis ofgenerally laminated composite beams with arbitrary boundaryconditionsrdquo Composite Structures vol 110 no 1 pp 305ndash3162014

[22] S M Nabi and N Ganesan ldquoA generalized element for thefree vibration analysis of composite beamsrdquo Computers andStructures vol 51 no 5 pp 607ndash610 1994

[23] L Zhao and Q Chen ldquoDynamic analysis of the stochasticvariational principle and stochastic finite element methodfor structures with random parametersrdquo Chinese Journal ofComputational Mechanics vol 15 no 3 pp 263ndash274 1998

[24] G-H Wang Y-N Gan and Z-B Wang ldquoEnergy-variationalmethod for the dynamic response of thin-walled I-beams withwide flangerdquo Engineering Mechanics vol 27 no 8 pp 15ndash202010

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 Mathematical Problems in Engineering

After integral each stiffness matrix coefficient can beobtained such as

11989611990811

= (cosh 120582 sin 120582 + cos 120582 sin 120582 minus sinh 120582 cosh 120582

minus2cosh2120582 sin 120582 cos 120582)

sdot (21198902120582

+ 1198904120582

+ 1)

+ 4 (119890120582

minus 1198903120582

) cosh 120582 (sinh 120582 sin 120582 + cosh 120582 cos 120582)

minus 2 (1 + 1198904120582

) cos 120582 sin 120582 cosh 120582 sinh 120582

+ 41198902120582cos2120582 (120582 minus sinh 120582 cosh 120582)

+ (61198902120582

+ 1198904120582

+ 1) cos 120582 sinh 120582

+ (1 minus 1198904120582

minus 81198902120582

120582) cos 120582 cosh 120582 + 4 (119890120582 + 1198903120582)

sdot (cos 120582 sinh 120582 cosh 120582 minus cos2120582 sinh120582 minus sin 120582

+cosh2120582 sin 120582 minus cos2120582 cosh 120582)

+ (41205821198902120582

+ 1198904120582

minus 1) cosh2120582 + (1 minus 1198904120582) sin 120582 sinh 120582(34)

512 Mass Matrix Mass matrix of element for beam flexuralvibration is given by

[119872119908

] = 119875

[[[[[

[

11989811990811

11989811990812

11989811990813

11989811990814

11989811990821

11989811990822

11989811990823

11989811990824

11989811990831

11989811990832

11989811990833

11989811990834

11989811990841

11989811990842

11989811990843

11989811990844

]]]]]

]

(35)

where

119898119908119894119895

=1

119875int

119897

0

119873119908119894

119873119908119895

119889119909 119875 =119898119890minus2120582

119866

119866 = 16 (1 + cos2120582ch2120582 minus 2 cos 120582ch120582) (36)

After integral each mass matrix coefficient can beobtained such as

11989811990811

= 120572minus1

[4 (119890120582

+ 1198903120582

)

sdot (cos2120582 sinh 120582 minus cos 120582 sinh120582 cosh 120582

minus sin 120582cosh2120582 + sin 120582)

+ (41205821198902120582

+ 1198904120582

minus 1) cosh2120582

+ (1 minus 1198904120582

) sin 120582

sdot (sinh 120582 minus 2 cos 120582 sinh 120582 cosh 120582)

+ (1198904120582

minus 101198902120582

+ 1) cos 120582 sinh120582

minus (81205821198902120582

+ 1198904120582

minus 1) cos 120582 cosh 120582

+ 4 (1198903120582

minus 119890120582

)

sdot cosh 120582 (sin 120582 sinh 120582 minus cos2120582 + 4 cos 120582 cosh 120582)

+ (21198902120582

+ 119890120582

+ 1) (sin 120582 cos 120582 minus sinh 120582 cosh 120582)

+ 31198902120582cos2120582 (4 sinh 120582 cosh 120582 + 120582)

+ 2 (61198902120582

minus 1 minus 1198904120582

) sin 120582 cos 120582cosh2120582

+ (1198904120582

minus 141198902120582

+ 1) sin 120582 cosh 120582] (37)

52 Element Matrix for Beam Axial Vibration After matrixoperation element dynamic stiffness matrix for beam axialvibration is expressed as

[119870119906

]119890

= 119864119860119888

[[[

[

minuscos 120574 sin 120574 + 1205742 (cos2120574 minus 1)

120574 cos 120574 + sin 1205742 (cos2120574 minus 1)

120574 cos 120574 + sin 1205742 (cos2120574 minus 1)

minuscos 120574 sin 120574 + 1205742 (cos2120574 minus 1)

]]]

]

(38)

Mass matrix of element for beam axial vibration is givenby

[119872119906

] = 119898

[[[

[

sin 120574 cos 120574 minus 1205742119888 (cos2120574 minus 1)

120574 cos 120574 minus sin 1205742119888 (cos2120574 minus 1)

120574 cos 120574 minus sin 1205742119888 (cos2120574 minus 1)

sin 120574 cos 120574 minus 1205742119888 (cos2120574 minus 1)

]]]

]

(39)

53 Dynamic Beam ElementMatrix Because of the indepen-dence of axial vibration and flexural vibration 6 by 6 matrixis obtained by adding two rows and two columns to the above4 by 4 matrix considering axial vibration without coupling ofaxial vibration and flexural vibration It is element stiffnessmatrix

Element stiffness matrix is given by

[119870]119890

=

[[[[[[[[[[[

[

11989611

0 0 11989614

0 0

11989622

11989623

0 11989625

11989626

11989633

0 11989635

11989636

11989644

0 0

symmetry 11989655

11989656

11989666

]]]]]]]]]]]

]

(40)

Mathematical Problems in Engineering 7

where

11989611

= minus119864119860119888 (cos 120574 sin 120574 + 120574)

2 (cos2120574 minus 1)

11989614

=119864119860119888 (120574 cos 120574 + sin 120574)

2 (cos2120574 minus 1)

11989622

= 11986711989611990811

11989623

= 11986711989611990812

11989625

= 11986711989611990813

11989626

= 11986711989611990814

11989633

= 11986711989611990822

11989635

= 11986711989611990823

11989636

= 11986711989611990824

11989644

= minus119864119860119888 (cos 120574 sin 120574 + 120574)

2 (cos2120574 minus 1)

11989655

= 11986711989611990833

11989656

= 11986711989611990834

11989666

= 11986711989611990844

(41)

Element mass matrix is given by

[119872]119890

=

[[[[[[[[[[[

[

11989811

0 0 11989814

0 0

11989822

11989823

0 11989825

11989826

11989833

0 11989835

11989836

11989844

0 0

symmetry 11989855

11989856

11989866

]]]]]]]]]]]

]

(42)

where

11989811

=119898 (sin 120574 cos 120574 minus 120574)2119888 (cos2120574 minus 1)

11989814

=119898 (120574 cos 120574 minus sin 120574)2119888 (cos2120574 minus 1)

11989822

= 11987511989811990811

11989823

= 11987511989811990812

11989825

= 11987511989811990813

11989826

= 11987511989811990814

11989833

= 11987511989811990822

11989835

= 11987511989811990823

11989836

= 11987511989811990824

11989844

=119898 (sin 120574 cos 120574 minus 120574)2119888 (cos2120574 minus 1)

11989855

= 11987511989811990833

11989856

= 11987511989811990834

11989866

= 11987511989811990844

(43)

6 Example and Comparison

To verify the dynamic beam element constructed by ana-lytical trial function method the comparisons between thecalculation results of this element general beam element andtheoretical solution are conducted

61 Calculation Model

(1) Four Kinds of Typical Beams Free vibration of four kindsof typical beams are analyzed including cantilever beamsimply supported beam one end clamped and another simplysupported beam and clamped-clamped beam

(2) Structure and Material Parameters The parameters ofcalculation model are as follows beam dimension is 6mtimes 02m times 03m material elastic modulus 119864 = 210GPaPoissonrsquos ratio 120592 = 03 and material density 120588 = 7800 kgm3

(3) Theoretical Solution According to theoretical equation offree vibration for distributed mass beam base frequencies offree vibration for all kinds of beam are obtained by usinganalytic method [1]

(4) Dynamic StiffnessMatrixMethodAccording to governingequation of free vibration for distributedmass beam stiffnesscoefficient [11] of beam vibration is provided and thenbase frequencies of free vibration for all kinds of beam areobtained

(5) Finite Element Model of General Beam Element Byusing the software ANSYS finite element model of beamvibration is established and modal analysis is conductedElement BEAM3 is used to simulate beam BEAM3 is generaltwo-dimension elastic beam element and uniaxial elementbearing tension pressure and bend Every node has threedegrees of freedom that is 119909-axis linear displacement 119910-axislinear displacement and 119911-axis angular displacement Cubicpolynomial interpolating function is used as its displacementtrial function Beam is divided into one beam element BlockLanczos method is used for modal extraction

(6) Model in This Paper Equilibrium equation of beamflexural free vibration is given by

119870 (120596) Δ (119905) + 119872 (120596) Δ (119905) = 0 (44)

where 119870(120596) is structure stiffness matrix with boundarydisplacement restraint and 119872(120596) is structure mass matrixwith boundary displacement restraint According to natural

8 Mathematical Problems in Engineering

Table 1 Base frequency of beam free vibration and its comparison (rads)

Beam type Theoreticalsolution

Dynamic stiffness matrixmethod

Interpolation trialfunction element

Analytical trial functionelement

Absolutevalue

Relative error()

Absolutevalue

Relative error()

Absolutevalue

Relative error()

Cantilever beam 438875389 438875390 289119864 minus 07 440740 0423 438875391 312119864 minus 07

Simply supportedbeam 1231941891 1231941888 minus859119864 minus 08 1365902 10874 1231941889 minus637119864 minus 08

One end clamped andanother simplysupported beam

1924528355 1924528348 minus150119864 minus 07 2554366 32727 1924528349 minus153119864 minus 07

Clamped-clampedbeam 2792673996 2792673991 minus132119864 minus 07 2834156 1485 2792673992 minus127119864 minus 07

Note calculation of eigenvalue and eigenvector cannot be conducted due to all DOFs of two ends of clamped-clamped beam being restrained If the beam isdivided into one element semistructure is used for vibration simulation of clamped-clamped beam

Table 2 Comparison of different element results (relative error )

Beam typeElement number for analytical

trial function Element number for interpolation trial function

1 1 2 10 20Cantilever beam 312119864 minus 07 04249 minus00003 minus00489 minus00489Simply supported beam minus637119864 minus 08 108739 02909 minus01018 minus01018One end clamped and anothersimply supported beam minus153119864 minus 07 327269 08037 minus01169 minus01202

Clamped-clamped beam minus127119864 minus 07 14854 14854 minus01255 minus01278

vibration governing equation of beam beam natural vibra-tion is harmonic vibration The following assumption can beobtained

Δ (119905) = 119884 sin120596119905 (45)

Substituting (45) into (44) yields

119872minus1

(120596)119870 (120596) 119884 = 1205962

119884 (46)

Assuming 120578 = 1205962 then 119864(120578) = 119872minus1(120578)119870(120578)The following equation can be obtained

119864 (120578) 119884 = 120578119884 (47)

Beamnatural vibration analysis is converted to eigenvalueand eigenvector analysis of matrix 119864(120578) = 119872minus1(120578)119870(120578)

Iteration method is used to calculate base frequency ofbeam vibration Through circular calculations the iterationstops when frequency relative error reaches 120581

119910

= (120596119894+1

minus

120596119894

)120596119894

lt 10minus8

62 Result Comparison and Analysis

621 Element Accuracy Vibration analysis is conducted bydividing beam into one element Table 1 shows base frequencyof beam free vibration and the relative error between calcu-lated solution and theoretical solution

As shown in Table 1 the following results can be obtained(1) By applying analytical trial function element to the

analysis a more accurate solution can be attained when the

relative error is less than 10minus8 The series of relative errorbetween this solution and theoretical solution is about 1 times10minus9 It can be supposed that this error is calculation error ofiteration efficiency

(2) By applying interpolation trial function element tothe analysis the maximum relative error between the solu-tion and theoretical solution reaches 30 It indicates thatdynamic beam element constructed by taking polynomialfunction as displacement trial function has bigger error

(3)The solution of dynamic stiffness matrix method [11]is close to theoretical solution The relative error is almostidentical with that calculated by analytical trial functionmethod The reason is that their displacement curves forvibrating beam are both derived from dynamic equilibriumgoverning equation Element load value and node balancecondition need to be analyzed in dynamic stiffness matrixmethod Dynamic stiffness matrix of beam element can bedirectly obtained in analytical trial function method Thelatter is more simple and intuitive and widely applied

622 Element Efficiency The beam is divided into 1 element2 elements 10 elements and 20 elements respectively Beamvibration numerical simulation is conducted by applyinginterpolation trial function method The relative error ofcalculated base frequency of free vibration for every typicalbeam and theoretical solution is obtained Table 2 presentsthe results

As shown in Table 2 the result of interpolation trialfunction is closer to theoretical solution with the increasing

Mathematical Problems in Engineering 9

of element number But it is obvious that base frequencybecomes smaller and its difference with theoretical solution islargerThe reason is that the more the elements are the lowerthe structure calculation stiffness isThe greater the differencebetween the actual stiffness and calculation stiffness Thenthe calculated base frequency further has large differencewith theoretical solution It indicates that actual deformationcurve for vibrating beam is not polynomial and polynomialfunction cannot be taken as displacement trial function forvibrating beam

7 Conclusions

Based on the results of this investigation the followingconclusions can be drawn

(1) The result of interpolation trial function elementfor simulating beam vibration is not in accordance withtheoretical solution and the relative error is larger Withelement number increasing base frequency becomes smallerand has large difference with theoretical solution It indicatesthat vibrating beam displacement mode is different frompolynomial mode and the precision requirement cannot bemet by taking polynomial function as displacement trialfunction for vibrating beam

(2)The solution of dynamic stiffness matrix method forsimulating beam vibration is close to theoretical solutionBut in this method element load value and node balancecondition need to be specially analyzed and it is not easy toderive nonstructural question Its application in engineeringis limited

(3)Dynamic stiffness matrix of beam element is obtainedby applying analytical trial function put forward in thispaper Base frequencies of four typical beams are attained byanalyzing eigenvalue and eigenvector and are compared withtheoretical solutionThe results show that the series of relativeerror is about 1 times 10minus9 and it is actually calculation error ofiteration efficiency The dynamic beam element in the lightof analytical trial function put forward in this paper is high-precision element

Conflict of Interests

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

Acknowledgments

This research is supported by National Natural ScienceFunds no 51279206 Beijing Natural Science Foundation no3144029 Chinese Universities Scientific Fund no 2011JS126and the Specialized Research Fund for the Doctoral Programof Higher Education of China no 20110008120017

References

[1] A K Clough and J Penzien Dynamic of Structures McGraw-Hill New York NY USA 1995

[2] Q Jin ldquoAn analytical solution of dynamic response for therigid perfectly plastic Timoshenko beamrdquo Chinese Journal ofTheoretical and Applied Mechanics vol 16 no 5 pp 504ndash5111984

[3] X Guo H Chen and Q Zeng ldquoDynamic characteristicsanalysis model of prestressed concrete T-type beamrdquo ChineseJournal of Computational Mechanics vol 17 no 2 pp 176ndash1832000

[4] D-P Fang and Q-F Wang ldquoDynamic behavior analysis of anexternally prestressed beam with energy methodrdquo Journal ofVibration and Shock vol 31 no 1 pp 177ndash181 2012

[5] M Lou and T Hong ldquoAnalytical approach for dynamic charac-teristics of prestressed beam with external tendonsrdquo Journal ofTongji University Natural Science vol 34 no 10 pp 1284ndash12882006

[6] J A Carrer S A Fleischfresser L F Garcia and W J MansurldquoDynamic analysis of Timoshenko beams by the boundary ele-ment methodrdquo Engineering Analysis with Boundary Elementsvol 37 no 12 pp 1602ndash1616 2013

[7] J-J Wu ldquoUse of the elastic-and-rigid-combined beam elementfor dynamic analysis of a two-dimensional frame with arbi-trarily distributed rigid beam segmentsrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 3 pp 1240ndash1251 2011

[8] M A De Rosa C Franciosi and M J Maurizi ldquoOn thedynamic behaviour of slender beams with elastic ends carryinga concentrated massrdquo Computers and Structures vol 58 no 6pp 1145ndash1159 1996

[9] R W Clough ldquoThe finite element method in plane stress anal-ysisrdquo in Proceedings of the 2nd ASCE Conference on ElectronicComputation Pittsburgh Pa USA 1960

[10] V Kolousek ldquoAnwendung des Gesetzes der virtuellen Verschiebungen and des in der Reziprozitatssatzesrdquo Stab weksDynamik Lngenieur Archiv vol 12 pp 363ndash370 1941

[11] Y-Q Long and S-H Bao Structural Mechanics II HigherEducation Press Beijing China 2nd edition 1996

[12] S M Hashemi and M J Richard ldquoFree vibrational analysis ofaxially loaded bending-torsion coupled beams a dynamic finiteelementrdquo Computers amp Structures vol 77 no 6 pp 711ndash7242000

[13] S Chen M Geradin and E Lamine ldquoAn improved dynamicstiffness method and modal analysis for beam-like structuresrdquoComputers amp Structures vol 60 no 5 pp 725ndash731 1996

[14] J R Banerjee H Su and C Jayatunga ldquoA dynamic stiffnesselement for free vibration analysis of composite beams and itsapplication to aircraft wingsrdquo Computers amp Structures vol 86no 6 pp 573ndash579 2008

[15] J R Banerjee S Guo and W P Howson ldquoExact dynamicstiffness matrix of a bending-torsion coupled beam includingwarpingrdquo Computers and Structures vol 59 no 4 pp 613ndash6211996

[16] M Shavezipur and S M Hashemi ldquoFree vibration of triplycoupled centrifugally stiffened nonuniform beams using arefined dynamic finite element methodrdquo Aerospace Science andTechnology vol 13 no 1 pp 59ndash70 2009

[17] A K ChopraDynamics of Structures PrenticeHall EnglewoodCliffs NJ USA 2000

[18] X-L Hu and Z-M Dai ldquoOn the dynamic stress concentrationsin orthotropic plates with an arbitrary cutoutrdquo Chinese Journalof Applied Mechanics vol 15 no 1 pp 12ndash17 1998

10 Mathematical Problems in Engineering

[19] X Wang L Yang and W Gao ldquoDynamic FEM analysis forthe integration ballast structure based on variation principlerdquoJournal of Vibration and Shock vol 24 no 4 pp 99ndash102 2005

[20] S M Hashemi and M J Richard ldquoA dynamic finite element(DFE) method for free vibrations of bending-torsion coupledbeamsrdquo Aerospace Science and Technology vol 4 no 1 pp 41ndash55 2000

[21] A Pagani E Carrera M Boscolo and J R Banerjee ldquoRefineddynamic stiffness elements applied to free vibration analysis ofgenerally laminated composite beams with arbitrary boundaryconditionsrdquo Composite Structures vol 110 no 1 pp 305ndash3162014

[22] S M Nabi and N Ganesan ldquoA generalized element for thefree vibration analysis of composite beamsrdquo Computers andStructures vol 51 no 5 pp 607ndash610 1994

[23] L Zhao and Q Chen ldquoDynamic analysis of the stochasticvariational principle and stochastic finite element methodfor structures with random parametersrdquo Chinese Journal ofComputational Mechanics vol 15 no 3 pp 263ndash274 1998

[24] G-H Wang Y-N Gan and Z-B Wang ldquoEnergy-variationalmethod for the dynamic response of thin-walled I-beams withwide flangerdquo Engineering Mechanics vol 27 no 8 pp 15ndash202010

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

Mathematical Problems in Engineering 7

where

11989611

= minus119864119860119888 (cos 120574 sin 120574 + 120574)

2 (cos2120574 minus 1)

11989614

=119864119860119888 (120574 cos 120574 + sin 120574)

2 (cos2120574 minus 1)

11989622

= 11986711989611990811

11989623

= 11986711989611990812

11989625

= 11986711989611990813

11989626

= 11986711989611990814

11989633

= 11986711989611990822

11989635

= 11986711989611990823

11989636

= 11986711989611990824

11989644

= minus119864119860119888 (cos 120574 sin 120574 + 120574)

2 (cos2120574 minus 1)

11989655

= 11986711989611990833

11989656

= 11986711989611990834

11989666

= 11986711989611990844

(41)

Element mass matrix is given by

[119872]119890

=

[[[[[[[[[[[

[

11989811

0 0 11989814

0 0

11989822

11989823

0 11989825

11989826

11989833

0 11989835

11989836

11989844

0 0

symmetry 11989855

11989856

11989866

]]]]]]]]]]]

]

(42)

where

11989811

=119898 (sin 120574 cos 120574 minus 120574)2119888 (cos2120574 minus 1)

11989814

=119898 (120574 cos 120574 minus sin 120574)2119888 (cos2120574 minus 1)

11989822

= 11987511989811990811

11989823

= 11987511989811990812

11989825

= 11987511989811990813

11989826

= 11987511989811990814

11989833

= 11987511989811990822

11989835

= 11987511989811990823

11989836

= 11987511989811990824

11989844

=119898 (sin 120574 cos 120574 minus 120574)2119888 (cos2120574 minus 1)

11989855

= 11987511989811990833

11989856

= 11987511989811990834

11989866

= 11987511989811990844

(43)

6 Example and Comparison

To verify the dynamic beam element constructed by ana-lytical trial function method the comparisons between thecalculation results of this element general beam element andtheoretical solution are conducted

61 Calculation Model

(1) Four Kinds of Typical Beams Free vibration of four kindsof typical beams are analyzed including cantilever beamsimply supported beam one end clamped and another simplysupported beam and clamped-clamped beam

(2) Structure and Material Parameters The parameters ofcalculation model are as follows beam dimension is 6mtimes 02m times 03m material elastic modulus 119864 = 210GPaPoissonrsquos ratio 120592 = 03 and material density 120588 = 7800 kgm3

(3) Theoretical Solution According to theoretical equation offree vibration for distributed mass beam base frequencies offree vibration for all kinds of beam are obtained by usinganalytic method [1]

(4) Dynamic StiffnessMatrixMethodAccording to governingequation of free vibration for distributedmass beam stiffnesscoefficient [11] of beam vibration is provided and thenbase frequencies of free vibration for all kinds of beam areobtained

(5) Finite Element Model of General Beam Element Byusing the software ANSYS finite element model of beamvibration is established and modal analysis is conductedElement BEAM3 is used to simulate beam BEAM3 is generaltwo-dimension elastic beam element and uniaxial elementbearing tension pressure and bend Every node has threedegrees of freedom that is 119909-axis linear displacement 119910-axislinear displacement and 119911-axis angular displacement Cubicpolynomial interpolating function is used as its displacementtrial function Beam is divided into one beam element BlockLanczos method is used for modal extraction

(6) Model in This Paper Equilibrium equation of beamflexural free vibration is given by

119870 (120596) Δ (119905) + 119872 (120596) Δ (119905) = 0 (44)

where 119870(120596) is structure stiffness matrix with boundarydisplacement restraint and 119872(120596) is structure mass matrixwith boundary displacement restraint According to natural

8 Mathematical Problems in Engineering

Table 1 Base frequency of beam free vibration and its comparison (rads)

Beam type Theoreticalsolution

Dynamic stiffness matrixmethod

Interpolation trialfunction element

Analytical trial functionelement

Absolutevalue

Relative error()

Absolutevalue

Relative error()

Absolutevalue

Relative error()

Cantilever beam 438875389 438875390 289119864 minus 07 440740 0423 438875391 312119864 minus 07

Simply supportedbeam 1231941891 1231941888 minus859119864 minus 08 1365902 10874 1231941889 minus637119864 minus 08

One end clamped andanother simplysupported beam

1924528355 1924528348 minus150119864 minus 07 2554366 32727 1924528349 minus153119864 minus 07

Clamped-clampedbeam 2792673996 2792673991 minus132119864 minus 07 2834156 1485 2792673992 minus127119864 minus 07

Note calculation of eigenvalue and eigenvector cannot be conducted due to all DOFs of two ends of clamped-clamped beam being restrained If the beam isdivided into one element semistructure is used for vibration simulation of clamped-clamped beam

Table 2 Comparison of different element results (relative error )

Beam typeElement number for analytical

trial function Element number for interpolation trial function

1 1 2 10 20Cantilever beam 312119864 minus 07 04249 minus00003 minus00489 minus00489Simply supported beam minus637119864 minus 08 108739 02909 minus01018 minus01018One end clamped and anothersimply supported beam minus153119864 minus 07 327269 08037 minus01169 minus01202

Clamped-clamped beam minus127119864 minus 07 14854 14854 minus01255 minus01278

vibration governing equation of beam beam natural vibra-tion is harmonic vibration The following assumption can beobtained

Δ (119905) = 119884 sin120596119905 (45)

Substituting (45) into (44) yields

119872minus1

(120596)119870 (120596) 119884 = 1205962

119884 (46)

Assuming 120578 = 1205962 then 119864(120578) = 119872minus1(120578)119870(120578)The following equation can be obtained

119864 (120578) 119884 = 120578119884 (47)

Beamnatural vibration analysis is converted to eigenvalueand eigenvector analysis of matrix 119864(120578) = 119872minus1(120578)119870(120578)

Iteration method is used to calculate base frequency ofbeam vibration Through circular calculations the iterationstops when frequency relative error reaches 120581

119910

= (120596119894+1

minus

120596119894

)120596119894

lt 10minus8

62 Result Comparison and Analysis

621 Element Accuracy Vibration analysis is conducted bydividing beam into one element Table 1 shows base frequencyof beam free vibration and the relative error between calcu-lated solution and theoretical solution

As shown in Table 1 the following results can be obtained(1) By applying analytical trial function element to the

analysis a more accurate solution can be attained when the

relative error is less than 10minus8 The series of relative errorbetween this solution and theoretical solution is about 1 times10minus9 It can be supposed that this error is calculation error ofiteration efficiency

(2) By applying interpolation trial function element tothe analysis the maximum relative error between the solu-tion and theoretical solution reaches 30 It indicates thatdynamic beam element constructed by taking polynomialfunction as displacement trial function has bigger error

(3)The solution of dynamic stiffness matrix method [11]is close to theoretical solution The relative error is almostidentical with that calculated by analytical trial functionmethod The reason is that their displacement curves forvibrating beam are both derived from dynamic equilibriumgoverning equation Element load value and node balancecondition need to be analyzed in dynamic stiffness matrixmethod Dynamic stiffness matrix of beam element can bedirectly obtained in analytical trial function method Thelatter is more simple and intuitive and widely applied

622 Element Efficiency The beam is divided into 1 element2 elements 10 elements and 20 elements respectively Beamvibration numerical simulation is conducted by applyinginterpolation trial function method The relative error ofcalculated base frequency of free vibration for every typicalbeam and theoretical solution is obtained Table 2 presentsthe results

As shown in Table 2 the result of interpolation trialfunction is closer to theoretical solution with the increasing

Mathematical Problems in Engineering 9

of element number But it is obvious that base frequencybecomes smaller and its difference with theoretical solution islargerThe reason is that the more the elements are the lowerthe structure calculation stiffness isThe greater the differencebetween the actual stiffness and calculation stiffness Thenthe calculated base frequency further has large differencewith theoretical solution It indicates that actual deformationcurve for vibrating beam is not polynomial and polynomialfunction cannot be taken as displacement trial function forvibrating beam

7 Conclusions

Based on the results of this investigation the followingconclusions can be drawn

(1) The result of interpolation trial function elementfor simulating beam vibration is not in accordance withtheoretical solution and the relative error is larger Withelement number increasing base frequency becomes smallerand has large difference with theoretical solution It indicatesthat vibrating beam displacement mode is different frompolynomial mode and the precision requirement cannot bemet by taking polynomial function as displacement trialfunction for vibrating beam

(2)The solution of dynamic stiffness matrix method forsimulating beam vibration is close to theoretical solutionBut in this method element load value and node balancecondition need to be specially analyzed and it is not easy toderive nonstructural question Its application in engineeringis limited

(3)Dynamic stiffness matrix of beam element is obtainedby applying analytical trial function put forward in thispaper Base frequencies of four typical beams are attained byanalyzing eigenvalue and eigenvector and are compared withtheoretical solutionThe results show that the series of relativeerror is about 1 times 10minus9 and it is actually calculation error ofiteration efficiency The dynamic beam element in the lightof analytical trial function put forward in this paper is high-precision element

Conflict of Interests

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

Acknowledgments

This research is supported by National Natural ScienceFunds no 51279206 Beijing Natural Science Foundation no3144029 Chinese Universities Scientific Fund no 2011JS126and the Specialized Research Fund for the Doctoral Programof Higher Education of China no 20110008120017

References

[1] A K Clough and J Penzien Dynamic of Structures McGraw-Hill New York NY USA 1995

[2] Q Jin ldquoAn analytical solution of dynamic response for therigid perfectly plastic Timoshenko beamrdquo Chinese Journal ofTheoretical and Applied Mechanics vol 16 no 5 pp 504ndash5111984

[3] X Guo H Chen and Q Zeng ldquoDynamic characteristicsanalysis model of prestressed concrete T-type beamrdquo ChineseJournal of Computational Mechanics vol 17 no 2 pp 176ndash1832000

[4] D-P Fang and Q-F Wang ldquoDynamic behavior analysis of anexternally prestressed beam with energy methodrdquo Journal ofVibration and Shock vol 31 no 1 pp 177ndash181 2012

[5] M Lou and T Hong ldquoAnalytical approach for dynamic charac-teristics of prestressed beam with external tendonsrdquo Journal ofTongji University Natural Science vol 34 no 10 pp 1284ndash12882006

[6] J A Carrer S A Fleischfresser L F Garcia and W J MansurldquoDynamic analysis of Timoshenko beams by the boundary ele-ment methodrdquo Engineering Analysis with Boundary Elementsvol 37 no 12 pp 1602ndash1616 2013

[7] J-J Wu ldquoUse of the elastic-and-rigid-combined beam elementfor dynamic analysis of a two-dimensional frame with arbi-trarily distributed rigid beam segmentsrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 3 pp 1240ndash1251 2011

[8] M A De Rosa C Franciosi and M J Maurizi ldquoOn thedynamic behaviour of slender beams with elastic ends carryinga concentrated massrdquo Computers and Structures vol 58 no 6pp 1145ndash1159 1996

[9] R W Clough ldquoThe finite element method in plane stress anal-ysisrdquo in Proceedings of the 2nd ASCE Conference on ElectronicComputation Pittsburgh Pa USA 1960

[10] V Kolousek ldquoAnwendung des Gesetzes der virtuellen Verschiebungen and des in der Reziprozitatssatzesrdquo Stab weksDynamik Lngenieur Archiv vol 12 pp 363ndash370 1941

[11] Y-Q Long and S-H Bao Structural Mechanics II HigherEducation Press Beijing China 2nd edition 1996

[12] S M Hashemi and M J Richard ldquoFree vibrational analysis ofaxially loaded bending-torsion coupled beams a dynamic finiteelementrdquo Computers amp Structures vol 77 no 6 pp 711ndash7242000

[13] S Chen M Geradin and E Lamine ldquoAn improved dynamicstiffness method and modal analysis for beam-like structuresrdquoComputers amp Structures vol 60 no 5 pp 725ndash731 1996

[14] J R Banerjee H Su and C Jayatunga ldquoA dynamic stiffnesselement for free vibration analysis of composite beams and itsapplication to aircraft wingsrdquo Computers amp Structures vol 86no 6 pp 573ndash579 2008

[15] J R Banerjee S Guo and W P Howson ldquoExact dynamicstiffness matrix of a bending-torsion coupled beam includingwarpingrdquo Computers and Structures vol 59 no 4 pp 613ndash6211996

[16] M Shavezipur and S M Hashemi ldquoFree vibration of triplycoupled centrifugally stiffened nonuniform beams using arefined dynamic finite element methodrdquo Aerospace Science andTechnology vol 13 no 1 pp 59ndash70 2009

[17] A K ChopraDynamics of Structures PrenticeHall EnglewoodCliffs NJ USA 2000

[18] X-L Hu and Z-M Dai ldquoOn the dynamic stress concentrationsin orthotropic plates with an arbitrary cutoutrdquo Chinese Journalof Applied Mechanics vol 15 no 1 pp 12ndash17 1998

10 Mathematical Problems in Engineering

[19] X Wang L Yang and W Gao ldquoDynamic FEM analysis forthe integration ballast structure based on variation principlerdquoJournal of Vibration and Shock vol 24 no 4 pp 99ndash102 2005

[20] S M Hashemi and M J Richard ldquoA dynamic finite element(DFE) method for free vibrations of bending-torsion coupledbeamsrdquo Aerospace Science and Technology vol 4 no 1 pp 41ndash55 2000

[21] A Pagani E Carrera M Boscolo and J R Banerjee ldquoRefineddynamic stiffness elements applied to free vibration analysis ofgenerally laminated composite beams with arbitrary boundaryconditionsrdquo Composite Structures vol 110 no 1 pp 305ndash3162014

[22] S M Nabi and N Ganesan ldquoA generalized element for thefree vibration analysis of composite beamsrdquo Computers andStructures vol 51 no 5 pp 607ndash610 1994

[23] L Zhao and Q Chen ldquoDynamic analysis of the stochasticvariational principle and stochastic finite element methodfor structures with random parametersrdquo Chinese Journal ofComputational Mechanics vol 15 no 3 pp 263ndash274 1998

[24] G-H Wang Y-N Gan and Z-B Wang ldquoEnergy-variationalmethod for the dynamic response of thin-walled I-beams withwide flangerdquo Engineering Mechanics vol 27 no 8 pp 15ndash202010

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 Mathematical Problems in Engineering

Table 1 Base frequency of beam free vibration and its comparison (rads)

Beam type Theoreticalsolution

Dynamic stiffness matrixmethod

Interpolation trialfunction element

Analytical trial functionelement

Absolutevalue

Relative error()

Absolutevalue

Relative error()

Absolutevalue

Relative error()

Cantilever beam 438875389 438875390 289119864 minus 07 440740 0423 438875391 312119864 minus 07

Simply supportedbeam 1231941891 1231941888 minus859119864 minus 08 1365902 10874 1231941889 minus637119864 minus 08

One end clamped andanother simplysupported beam

1924528355 1924528348 minus150119864 minus 07 2554366 32727 1924528349 minus153119864 minus 07

Clamped-clampedbeam 2792673996 2792673991 minus132119864 minus 07 2834156 1485 2792673992 minus127119864 minus 07

Note calculation of eigenvalue and eigenvector cannot be conducted due to all DOFs of two ends of clamped-clamped beam being restrained If the beam isdivided into one element semistructure is used for vibration simulation of clamped-clamped beam

Table 2 Comparison of different element results (relative error )

Beam typeElement number for analytical

trial function Element number for interpolation trial function

1 1 2 10 20Cantilever beam 312119864 minus 07 04249 minus00003 minus00489 minus00489Simply supported beam minus637119864 minus 08 108739 02909 minus01018 minus01018One end clamped and anothersimply supported beam minus153119864 minus 07 327269 08037 minus01169 minus01202

Clamped-clamped beam minus127119864 minus 07 14854 14854 minus01255 minus01278

vibration governing equation of beam beam natural vibra-tion is harmonic vibration The following assumption can beobtained

Δ (119905) = 119884 sin120596119905 (45)

Substituting (45) into (44) yields

119872minus1

(120596)119870 (120596) 119884 = 1205962

119884 (46)

Assuming 120578 = 1205962 then 119864(120578) = 119872minus1(120578)119870(120578)The following equation can be obtained

119864 (120578) 119884 = 120578119884 (47)

Beamnatural vibration analysis is converted to eigenvalueand eigenvector analysis of matrix 119864(120578) = 119872minus1(120578)119870(120578)

Iteration method is used to calculate base frequency ofbeam vibration Through circular calculations the iterationstops when frequency relative error reaches 120581

119910

= (120596119894+1

minus

120596119894

)120596119894

lt 10minus8

62 Result Comparison and Analysis

621 Element Accuracy Vibration analysis is conducted bydividing beam into one element Table 1 shows base frequencyof beam free vibration and the relative error between calcu-lated solution and theoretical solution

As shown in Table 1 the following results can be obtained(1) By applying analytical trial function element to the

analysis a more accurate solution can be attained when the

relative error is less than 10minus8 The series of relative errorbetween this solution and theoretical solution is about 1 times10minus9 It can be supposed that this error is calculation error ofiteration efficiency

(2) By applying interpolation trial function element tothe analysis the maximum relative error between the solu-tion and theoretical solution reaches 30 It indicates thatdynamic beam element constructed by taking polynomialfunction as displacement trial function has bigger error

(3)The solution of dynamic stiffness matrix method [11]is close to theoretical solution The relative error is almostidentical with that calculated by analytical trial functionmethod The reason is that their displacement curves forvibrating beam are both derived from dynamic equilibriumgoverning equation Element load value and node balancecondition need to be analyzed in dynamic stiffness matrixmethod Dynamic stiffness matrix of beam element can bedirectly obtained in analytical trial function method Thelatter is more simple and intuitive and widely applied

622 Element Efficiency The beam is divided into 1 element2 elements 10 elements and 20 elements respectively Beamvibration numerical simulation is conducted by applyinginterpolation trial function method The relative error ofcalculated base frequency of free vibration for every typicalbeam and theoretical solution is obtained Table 2 presentsthe results

As shown in Table 2 the result of interpolation trialfunction is closer to theoretical solution with the increasing

Mathematical Problems in Engineering 9

of element number But it is obvious that base frequencybecomes smaller and its difference with theoretical solution islargerThe reason is that the more the elements are the lowerthe structure calculation stiffness isThe greater the differencebetween the actual stiffness and calculation stiffness Thenthe calculated base frequency further has large differencewith theoretical solution It indicates that actual deformationcurve for vibrating beam is not polynomial and polynomialfunction cannot be taken as displacement trial function forvibrating beam

7 Conclusions

Based on the results of this investigation the followingconclusions can be drawn

(1) The result of interpolation trial function elementfor simulating beam vibration is not in accordance withtheoretical solution and the relative error is larger Withelement number increasing base frequency becomes smallerand has large difference with theoretical solution It indicatesthat vibrating beam displacement mode is different frompolynomial mode and the precision requirement cannot bemet by taking polynomial function as displacement trialfunction for vibrating beam

(2)The solution of dynamic stiffness matrix method forsimulating beam vibration is close to theoretical solutionBut in this method element load value and node balancecondition need to be specially analyzed and it is not easy toderive nonstructural question Its application in engineeringis limited

(3)Dynamic stiffness matrix of beam element is obtainedby applying analytical trial function put forward in thispaper Base frequencies of four typical beams are attained byanalyzing eigenvalue and eigenvector and are compared withtheoretical solutionThe results show that the series of relativeerror is about 1 times 10minus9 and it is actually calculation error ofiteration efficiency The dynamic beam element in the lightof analytical trial function put forward in this paper is high-precision element

Conflict of Interests

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

Acknowledgments

This research is supported by National Natural ScienceFunds no 51279206 Beijing Natural Science Foundation no3144029 Chinese Universities Scientific Fund no 2011JS126and the Specialized Research Fund for the Doctoral Programof Higher Education of China no 20110008120017

References

[1] A K Clough and J Penzien Dynamic of Structures McGraw-Hill New York NY USA 1995

[2] Q Jin ldquoAn analytical solution of dynamic response for therigid perfectly plastic Timoshenko beamrdquo Chinese Journal ofTheoretical and Applied Mechanics vol 16 no 5 pp 504ndash5111984

[3] X Guo H Chen and Q Zeng ldquoDynamic characteristicsanalysis model of prestressed concrete T-type beamrdquo ChineseJournal of Computational Mechanics vol 17 no 2 pp 176ndash1832000

[4] D-P Fang and Q-F Wang ldquoDynamic behavior analysis of anexternally prestressed beam with energy methodrdquo Journal ofVibration and Shock vol 31 no 1 pp 177ndash181 2012

[5] M Lou and T Hong ldquoAnalytical approach for dynamic charac-teristics of prestressed beam with external tendonsrdquo Journal ofTongji University Natural Science vol 34 no 10 pp 1284ndash12882006

[6] J A Carrer S A Fleischfresser L F Garcia and W J MansurldquoDynamic analysis of Timoshenko beams by the boundary ele-ment methodrdquo Engineering Analysis with Boundary Elementsvol 37 no 12 pp 1602ndash1616 2013

[7] J-J Wu ldquoUse of the elastic-and-rigid-combined beam elementfor dynamic analysis of a two-dimensional frame with arbi-trarily distributed rigid beam segmentsrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 3 pp 1240ndash1251 2011

[8] M A De Rosa C Franciosi and M J Maurizi ldquoOn thedynamic behaviour of slender beams with elastic ends carryinga concentrated massrdquo Computers and Structures vol 58 no 6pp 1145ndash1159 1996

[9] R W Clough ldquoThe finite element method in plane stress anal-ysisrdquo in Proceedings of the 2nd ASCE Conference on ElectronicComputation Pittsburgh Pa USA 1960

[10] V Kolousek ldquoAnwendung des Gesetzes der virtuellen Verschiebungen and des in der Reziprozitatssatzesrdquo Stab weksDynamik Lngenieur Archiv vol 12 pp 363ndash370 1941

[11] Y-Q Long and S-H Bao Structural Mechanics II HigherEducation Press Beijing China 2nd edition 1996

[12] S M Hashemi and M J Richard ldquoFree vibrational analysis ofaxially loaded bending-torsion coupled beams a dynamic finiteelementrdquo Computers amp Structures vol 77 no 6 pp 711ndash7242000

[13] S Chen M Geradin and E Lamine ldquoAn improved dynamicstiffness method and modal analysis for beam-like structuresrdquoComputers amp Structures vol 60 no 5 pp 725ndash731 1996

[14] J R Banerjee H Su and C Jayatunga ldquoA dynamic stiffnesselement for free vibration analysis of composite beams and itsapplication to aircraft wingsrdquo Computers amp Structures vol 86no 6 pp 573ndash579 2008

[15] J R Banerjee S Guo and W P Howson ldquoExact dynamicstiffness matrix of a bending-torsion coupled beam includingwarpingrdquo Computers and Structures vol 59 no 4 pp 613ndash6211996

[16] M Shavezipur and S M Hashemi ldquoFree vibration of triplycoupled centrifugally stiffened nonuniform beams using arefined dynamic finite element methodrdquo Aerospace Science andTechnology vol 13 no 1 pp 59ndash70 2009

[17] A K ChopraDynamics of Structures PrenticeHall EnglewoodCliffs NJ USA 2000

[18] X-L Hu and Z-M Dai ldquoOn the dynamic stress concentrationsin orthotropic plates with an arbitrary cutoutrdquo Chinese Journalof Applied Mechanics vol 15 no 1 pp 12ndash17 1998

10 Mathematical Problems in Engineering

[19] X Wang L Yang and W Gao ldquoDynamic FEM analysis forthe integration ballast structure based on variation principlerdquoJournal of Vibration and Shock vol 24 no 4 pp 99ndash102 2005

[20] S M Hashemi and M J Richard ldquoA dynamic finite element(DFE) method for free vibrations of bending-torsion coupledbeamsrdquo Aerospace Science and Technology vol 4 no 1 pp 41ndash55 2000

[21] A Pagani E Carrera M Boscolo and J R Banerjee ldquoRefineddynamic stiffness elements applied to free vibration analysis ofgenerally laminated composite beams with arbitrary boundaryconditionsrdquo Composite Structures vol 110 no 1 pp 305ndash3162014

[22] S M Nabi and N Ganesan ldquoA generalized element for thefree vibration analysis of composite beamsrdquo Computers andStructures vol 51 no 5 pp 607ndash610 1994

[23] L Zhao and Q Chen ldquoDynamic analysis of the stochasticvariational principle and stochastic finite element methodfor structures with random parametersrdquo Chinese Journal ofComputational Mechanics vol 15 no 3 pp 263ndash274 1998

[24] G-H Wang Y-N Gan and Z-B Wang ldquoEnergy-variationalmethod for the dynamic response of thin-walled I-beams withwide flangerdquo Engineering Mechanics vol 27 no 8 pp 15ndash202010

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

Mathematical Problems in Engineering 9

of element number But it is obvious that base frequencybecomes smaller and its difference with theoretical solution islargerThe reason is that the more the elements are the lowerthe structure calculation stiffness isThe greater the differencebetween the actual stiffness and calculation stiffness Thenthe calculated base frequency further has large differencewith theoretical solution It indicates that actual deformationcurve for vibrating beam is not polynomial and polynomialfunction cannot be taken as displacement trial function forvibrating beam

7 Conclusions

Based on the results of this investigation the followingconclusions can be drawn

(1) The result of interpolation trial function elementfor simulating beam vibration is not in accordance withtheoretical solution and the relative error is larger Withelement number increasing base frequency becomes smallerand has large difference with theoretical solution It indicatesthat vibrating beam displacement mode is different frompolynomial mode and the precision requirement cannot bemet by taking polynomial function as displacement trialfunction for vibrating beam

(2)The solution of dynamic stiffness matrix method forsimulating beam vibration is close to theoretical solutionBut in this method element load value and node balancecondition need to be specially analyzed and it is not easy toderive nonstructural question Its application in engineeringis limited

(3)Dynamic stiffness matrix of beam element is obtainedby applying analytical trial function put forward in thispaper Base frequencies of four typical beams are attained byanalyzing eigenvalue and eigenvector and are compared withtheoretical solutionThe results show that the series of relativeerror is about 1 times 10minus9 and it is actually calculation error ofiteration efficiency The dynamic beam element in the lightof analytical trial function put forward in this paper is high-precision element

Conflict of Interests

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

Acknowledgments

This research is supported by National Natural ScienceFunds no 51279206 Beijing Natural Science Foundation no3144029 Chinese Universities Scientific Fund no 2011JS126and the Specialized Research Fund for the Doctoral Programof Higher Education of China no 20110008120017

References

[1] A K Clough and J Penzien Dynamic of Structures McGraw-Hill New York NY USA 1995

[2] Q Jin ldquoAn analytical solution of dynamic response for therigid perfectly plastic Timoshenko beamrdquo Chinese Journal ofTheoretical and Applied Mechanics vol 16 no 5 pp 504ndash5111984

[3] X Guo H Chen and Q Zeng ldquoDynamic characteristicsanalysis model of prestressed concrete T-type beamrdquo ChineseJournal of Computational Mechanics vol 17 no 2 pp 176ndash1832000

[4] D-P Fang and Q-F Wang ldquoDynamic behavior analysis of anexternally prestressed beam with energy methodrdquo Journal ofVibration and Shock vol 31 no 1 pp 177ndash181 2012

[5] M Lou and T Hong ldquoAnalytical approach for dynamic charac-teristics of prestressed beam with external tendonsrdquo Journal ofTongji University Natural Science vol 34 no 10 pp 1284ndash12882006

[6] J A Carrer S A Fleischfresser L F Garcia and W J MansurldquoDynamic analysis of Timoshenko beams by the boundary ele-ment methodrdquo Engineering Analysis with Boundary Elementsvol 37 no 12 pp 1602ndash1616 2013

[7] J-J Wu ldquoUse of the elastic-and-rigid-combined beam elementfor dynamic analysis of a two-dimensional frame with arbi-trarily distributed rigid beam segmentsrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 3 pp 1240ndash1251 2011

[8] M A De Rosa C Franciosi and M J Maurizi ldquoOn thedynamic behaviour of slender beams with elastic ends carryinga concentrated massrdquo Computers and Structures vol 58 no 6pp 1145ndash1159 1996

[9] R W Clough ldquoThe finite element method in plane stress anal-ysisrdquo in Proceedings of the 2nd ASCE Conference on ElectronicComputation Pittsburgh Pa USA 1960

[10] V Kolousek ldquoAnwendung des Gesetzes der virtuellen Verschiebungen and des in der Reziprozitatssatzesrdquo Stab weksDynamik Lngenieur Archiv vol 12 pp 363ndash370 1941

[11] Y-Q Long and S-H Bao Structural Mechanics II HigherEducation Press Beijing China 2nd edition 1996

[12] S M Hashemi and M J Richard ldquoFree vibrational analysis ofaxially loaded bending-torsion coupled beams a dynamic finiteelementrdquo Computers amp Structures vol 77 no 6 pp 711ndash7242000

[13] S Chen M Geradin and E Lamine ldquoAn improved dynamicstiffness method and modal analysis for beam-like structuresrdquoComputers amp Structures vol 60 no 5 pp 725ndash731 1996

[14] J R Banerjee H Su and C Jayatunga ldquoA dynamic stiffnesselement for free vibration analysis of composite beams and itsapplication to aircraft wingsrdquo Computers amp Structures vol 86no 6 pp 573ndash579 2008

[15] J R Banerjee S Guo and W P Howson ldquoExact dynamicstiffness matrix of a bending-torsion coupled beam includingwarpingrdquo Computers and Structures vol 59 no 4 pp 613ndash6211996

[16] M Shavezipur and S M Hashemi ldquoFree vibration of triplycoupled centrifugally stiffened nonuniform beams using arefined dynamic finite element methodrdquo Aerospace Science andTechnology vol 13 no 1 pp 59ndash70 2009

[17] A K ChopraDynamics of Structures PrenticeHall EnglewoodCliffs NJ USA 2000

[18] X-L Hu and Z-M Dai ldquoOn the dynamic stress concentrationsin orthotropic plates with an arbitrary cutoutrdquo Chinese Journalof Applied Mechanics vol 15 no 1 pp 12ndash17 1998

10 Mathematical Problems in Engineering

[19] X Wang L Yang and W Gao ldquoDynamic FEM analysis forthe integration ballast structure based on variation principlerdquoJournal of Vibration and Shock vol 24 no 4 pp 99ndash102 2005

[20] S M Hashemi and M J Richard ldquoA dynamic finite element(DFE) method for free vibrations of bending-torsion coupledbeamsrdquo Aerospace Science and Technology vol 4 no 1 pp 41ndash55 2000

[21] A Pagani E Carrera M Boscolo and J R Banerjee ldquoRefineddynamic stiffness elements applied to free vibration analysis ofgenerally laminated composite beams with arbitrary boundaryconditionsrdquo Composite Structures vol 110 no 1 pp 305ndash3162014

[22] S M Nabi and N Ganesan ldquoA generalized element for thefree vibration analysis of composite beamsrdquo Computers andStructures vol 51 no 5 pp 607ndash610 1994

[23] L Zhao and Q Chen ldquoDynamic analysis of the stochasticvariational principle and stochastic finite element methodfor structures with random parametersrdquo Chinese Journal ofComputational Mechanics vol 15 no 3 pp 263ndash274 1998

[24] G-H Wang Y-N Gan and Z-B Wang ldquoEnergy-variationalmethod for the dynamic response of thin-walled I-beams withwide flangerdquo Engineering Mechanics vol 27 no 8 pp 15ndash202010

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 Mathematical Problems in Engineering

[19] X Wang L Yang and W Gao ldquoDynamic FEM analysis forthe integration ballast structure based on variation principlerdquoJournal of Vibration and Shock vol 24 no 4 pp 99ndash102 2005

[20] S M Hashemi and M J Richard ldquoA dynamic finite element(DFE) method for free vibrations of bending-torsion coupledbeamsrdquo Aerospace Science and Technology vol 4 no 1 pp 41ndash55 2000

[21] A Pagani E Carrera M Boscolo and J R Banerjee ldquoRefineddynamic stiffness elements applied to free vibration analysis ofgenerally laminated composite beams with arbitrary boundaryconditionsrdquo Composite Structures vol 110 no 1 pp 305ndash3162014

[22] S M Nabi and N Ganesan ldquoA generalized element for thefree vibration analysis of composite beamsrdquo Computers andStructures vol 51 no 5 pp 607ndash610 1994

[23] L Zhao and Q Chen ldquoDynamic analysis of the stochasticvariational principle and stochastic finite element methodfor structures with random parametersrdquo Chinese Journal ofComputational Mechanics vol 15 no 3 pp 263ndash274 1998

[24] G-H Wang Y-N Gan and Z-B Wang ldquoEnergy-variationalmethod for the dynamic response of thin-walled I-beams withwide flangerdquo Engineering Mechanics vol 27 no 8 pp 15ndash202010

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