Research Article A Corotational Formulation for Large

Research ArticleA Corotational Formulation for Large Displacement Analysis ofFunctionally Graded Sandwich Beam and Frame Structures

Dinh Kien Nguyen and Thi Thom Tran

Department of Solid Mechanics Institute of Mechanics Vietnam Academy of Science and Technology18 Hoang Quoc Viet Hanoi Vietnam

Correspondence should be addressed to Dinh Kien Nguyen ndkienimechacvn

Received 15 March 2016 Accepted 26 May 2016

Academic Editor Zhiqiang Hu

Copyright copy 2016 D K Nguyen and T T TranThis 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

A corotational finite element formulation for large displacement analysis of planar functionally graded sandwich (FGSW) beamand frame structures is presented The beams and frames are assumed to be formed from a metallic soft core and two symmetricfunctionally graded skin layers The Euler-Bernoulli beam theory and von Karman nonlinear strain-displacement relationship areadopted for the local strain Exact solution of nonlinear equilibrium equations for a beam segment is employed to interpolate thedisplacement field for avoiding the membrane locking An incremental-iterative procedure is used in combination with the arc-length control method to compute the equilibrium paths Numerical examples show that the proposed formulation is capable ofevaluating accurately the large displacement response with just several elements A parametric study is carried out to highlight theeffect of the material distribution the core thickness to height ratio on the large displacement behaviour of the FGSW beam andframe structures

1 Introduction

Large displacement analysis of structures is an importanttopic in the field of structural mechanics This topic growsin importance due to the development of new materialswhich enable structures to undergo large deformation Manyinvestigations on the large displacement analysis of structuresusing both analytical and numerical methods are reportedin the literature Numerical methods especially the finiteelement method with its versatility in spatial discretizationare an effective tool for the large displacement analysis ofstructures

In order to analyse beam and frame structures under-going large displacements by the finite element method anonlinear beam element which can model accurately thenonlinear behaviour of a structure is necessary to formulateVarious nonlinear beam elements for analysis of planar beamand frame structures are available in the literature Depend-ing on the choice of reference configuration the nonlinearbeam elements can be classified into three types the total

Lagrangian formulation the updated Lagrangian formula-tion and the corotational formulation In the corotationalformulation which will be discussed herein the kinematicsare described in an element attached local coordinate systemThe finite element formulation is firstly formulated in thelocal system and then transformed into a global system withthe aid of transformation matrices The elements proposedby Hsiao et al [1 2] Meek and Xue [3] Pacoste and Eriksson[4] and Nguyen [5] are some amongst the other corotationalbeam elements for large displacement analysis of planarbeams and frames

Analyses of structures made of functionally gradedmate-rials (FGMs) have been extensively carried out since thesematerials were created by Japanese scientists in 1984 [6]The finite element analysis is often employed to handlethe complexities arising from the material inhomogeneityChakraborty et al [7] proposed a finite element formulationfor studying the thermoelastic behaviour of shear deformableFGM beams The formulation using the exact solutionof an equilibrium FGM Timoshenko beam to interpolate

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 5698351 12 pageshttpdxdoiorg10115520165698351

2 Mathematical Problems in Engineering

the displacement field is free of shear locking Based on thehigher-order shear deformation beam theory Kadoli et al[8] investigated the static behaviour of FGM beams underambient temperature The 119896119901-Ritz method was employed byLee et al [9] in studying the postbuckling response of FGMplates subject to compressive and thermal loads Almeida etal [10] extended the total Lagrange formulation proposedby Pacoste and Eriksson in [4] to investigate the geometri-cally nonlinear behaviour of FGM beams A finite elementformulation based on the linear exact shape functions wasemployed by Taeprasartsit [11] for buckling analysis of perfectand imperfect FGM columns Recently the first author andhis coworkers [12ndash15] derived the corotational finite elementformulations for large displacement analysis of FGM beamand frame structures In these works in order to prevent theformulations from the membrane locking the average strainhas been introduced to replace the membrane strain

Sandwich structures with height strength-to-weight ratioarewidely used in aerospace application such as skin ofwingsaileron and spoilers In order to improve the performance ofthe structures in high thermal environment FGMs could beincorporated in the sandwich construction Several analysesmainly the vibration and buckling of functionally gradedsandwich (FGSW) beams have been carried out in recentyears [16ndash18] To maintain minimum weight for a givenmechanical loading condition FGSW beams and framesare often designed to be slender and they might undergolarge displacements during service Investigation on the largedisplacement behaviour of FGSW beams and frames tothe authorsrsquo best knowledge has not been reported in theliterature and it is studied in this paper for the first timeTo this end a finite element formulation based on Euler-Bernoulli beam theory is derived in the context of thecorotational approach and used in the investigation Differentfrom previous works the interpolation functions for thedisplacement field are obtained by solving the equilibriumnonlinear equations of a beam segment The exact inter-polation functions lead to a balance between the axial andtransverse displacements and the finite element formulationbased on these functions is free of the membrane locking[19] The concept of the average membrane strain mentionedabove is therefore not necessary to use in the present workUsing the derived formulation the equilibrium equationsare constructed and they are solved by an incremental-iterative procedure in combination with the arc-length con-trol method Numerical examples are given to show theaccuracy of the derived formulation and to illustrate the effectof thematerial distribution and the core thickness ratio on thelarge displacement behaviour of the FGSW beam and framestructures

2 Finite Element Formulation

21 FGSWBeam Figure 1 shows a FGSWbeam formed froma metallic soft core and two symmetric FGM skin layers in aCartesian coordinate system The 119909-axis is chosen to be onthe midplane of the beam and the 119911-axis is directed upwardsDenoting the beam height and the soft core thickness by

zz

yxh

b

h0FGM

FGMMetal core

LCeramic

Ceramic

Metal

Figure 1 A FGSW beam in a Cartesian coordinate system

ℎ and ℎ0 respectively the FGM is assumed to be made of

metal and ceramic with the volume fraction of ceramic 119881119888

varying in the thickness direction by the following power-lawdistribution

119881119888

=

(2119911 + ℎ

0

ℎ0

minus ℎ)

119899

for 119911 isin [minusℎ

2 minus

ℎ0

2]

0 for 119911 isin [minusℎ0

2ℎ0

2]

(2119911 minus ℎ

0

ℎ minus ℎ0

)

119899

for 119911 isin [ℎ0

2ℎ

2]

(1)

where 119899 is the nonnegative grading index The volumefraction of metal is 119881

119898= 1 minus 119881

119888 As seen from (1) the top

and bottom surfaces contain only ceramic whereas the core ispure metal The effective Youngrsquos modulus of the beam 119864(119911)evaluated by Voigt model reads

119864 (119911)

=

(119864119888

minus 119864119898

) (2119911 + ℎ

0

ℎ0

minus ℎ)

119899

+ 119864119898

for minusℎ

2le 119911 le minus

ℎ0

2

119864119898

for minusℎ0

2le 119911 le

ℎ0

2

(119864119888

minus 119864119898

) (2119911 minus ℎ

0

ℎ minus ℎ0

)

119899

+ 119864119898

forℎ0

2le 119911 le minus

ℎ

2

(2)

where 119864119888and 119864

119898are Youngrsquos moduli of ceramic and metal

respectively

22 Corotational Approach The corotational approach isa convenient way to derive geometrically nonlinear finiteelement formulations The central idea of the approach isto introduce a local coordinate system that continuouslymoves and rotates with the element during its deformationprocessThe element formulation is firstly derived in the localcoordinate system and then transferred into the global one byusing transformation matrices Depending on the definitionof local coordinate system various corotational beam formu-lations are available in the literatureThe formulation adoptedin the present work is closely related to the ones described in[4 22] and its main points are summarized in the following

Figure 2 shows a planar two-node beam element (119894 119895)and its kinematics in two coordinate systems a local (119909 119911)

and a global (119909 119911) The element is initially inclined to the 119909-axis an angle 120579

0 The global system is fixed while the local

one continuously moves and rotates with the element duringits deformation The local system is chosen with its origin

Mathematical Problems in Engineering 3

wj

x

z

j

j

1205790

1205790

i

i

120579r 120579j

120579i

120579120579i

120579j

ujx

z

wi

ui

uj

Deformed configuration

Initial configuration

Figure 2 A planar corotational beam element and its kinematics

at node 119894 and with the 119909-axis directed towards node 119895 Insuch a local system the axial displacement at node 119894 and thetransverse displacements at the two nodes 119894 and 119895 are alwayszero 119906

119894= 119908119894

= 119908119895

= 0 The element vector of local nodaldisplacements (d) thus contains only three components

d = 119906119895

120579119894

120579119895119879

(3)

where119906119895is the local axial displacement at node 119895 and 120579

119894and 120579119895

are the local rotations at nodes 119894 and 119895 respectively In (3) andhereafter a superscript ldquo119879rdquo denotes the transpose of a vectoror a matrix and the bar suffix is used to indicate a variablewith respect to the local system

The global nodal displacements in general are nonzeroand the global element vector of nodal displacements (d) hassix components as

d = 119906119894 119908119894

120579119894

119906119895

119908119895

120579119895119879

(4)

where 119906119894 119908119894 120579119894are respectively the global axial and trans-

verse displacements and rotation at node 119894 and 119906119895 119908119895 120579119895are

the corresponding quantities at node 119895 The local displace-ment and rotations in (3) are related to the global ones by

119906119895

= 119897119899

minus 119897

120579119894= 120579119894minus 120579119903

120579119895

= 120579119895

minus 120579119903

(5)

The initial and current lengths 119897 and 119897119899 in (5) can be calcu-

lated as

119897 = radic(119909119895

minus 119909119894)2

+ (119911119895

minus 119911119894)2

119897119899

= radic(119909119895

+ 119906119895

minus 119909119894minus 119906119894)2

+ (119911119895

+ 119908119895

minus 119911119894minus 119908119894)2

(6)

with (119909119894 119911119894) and (119909

119895 119911119895) being respectively the coordinates

of the nodes 119894 and 119895 and 120579119903in (5) is the rigid rotation

of the element which can be computed from the elementcoordinates and the current global nodal displacements [22]

Differentiating the local displacements in (5) with respectto the global displacements leads to the relation between thevirtual vector of local displacements and that of global ones

120575d = T120575d (7)

where the transformation matrix (T) has the following form

T =1

119897119899

[[

[

minus119888119897119899

minus119904119897119899

0 119888119897119899

119904119897119899

0

minus119904 119888 119897119899

119904 minus119888 0

minus119904 119888 0 119904 minus119888 119897119899

]]

]

(8)

with

119888 = cos 120579 =119909119895

+ 119906119895

minus 119909119894minus 119906119894

119897119899

119904 = sin 120579 =119911119895

+ 119908119895

minus 119911119894minus 119908119894

119897119899

(9)

Equating the internal virtual work for the element in bothtwo systems leads to the relation between the global and localvectors of nodal forces as

fin = T119879f in (10)

where f in and fin denote the local and global vectors ofinternal forces for the element respectively

The global element tangent stiffness matrix k119905is obtained

by differentiation of the global force vector fin with respect tothe global nodal displacements having the form

k119905

= T119879k119905T +

119891119906

119897119899

zz119879 +

(119891120579119894

+ 119891120579119895

)

1198972119899

(rz119879 + zr119879) (11)

In the above equation119891119906 119891120579119894 and 119891

120579119895are components of the

vector f in k119905 is the local tangent stiffness matrix and

r = minus119888 minus119904 0 119888 119904 0119879

z = 119904 minus119888 0 minus119904 119888 0119879

(12)

Detail derivations of the matrix T and vectors r and z aregiven in [22] Equations (10) and (11) completely definethe global nodal force vector and tangent stiffness matrixprovided that f in and k

119905are known

23 Local Formulation Based on Euler-Bernoulli beam the-ory the displacements in the 119909 and 119911 directions at any pointwith respect to the local system 119906

1and 119906

3 respectively are

given by

1199061

(119909 119911) = 119906 (119909) minus 119911 119908 (119909)119909

1199063 (119909 119911) = 119908 (119909)

(13)

where119906(119909) and119908(119909) are respectively the axial and transversedisplacements of the point on themidplane and a suffix coma


is used to denote the derivative with respect to the followedvariable

The vonKaman nonlinear relationship can be used for thelocal axial strain in large displacement analysis [13 15]

120576119909

= 119906119909

+1

21199082

119909minus 119911 119908119909 119909

(14)

Assuming linearly elastic behaviour for the beam materialthe axial stress 120590

119909is related to the axial strain 120576 by

120590119909

= 119864 (119911) 120576119909 (15)

where 119864(119911) is the effective Young modulus defined by (2)with 119911 replaced by 119911

The strain energy for an element with initial length of 119897

reads

119880 =1

2int

119897

0

120590119909120576119909119889119909

=1

2int

119897

0

[11986011

(119906119909

+1

21199082

119909)

2

+ 11986022

1199082

119909119909] 119889119909

(16)

where 11986011

and 11986022

are respectively the extensional andbending rigidities and they are defined as

(11986011

11986022

) = int119860

119864 (119911) (1 1199112) 119889119860 (17)

with 119860 being the cross-sectional area Substituting (2) into(17) one gets the explicit forms for 119860

11and 119860

22as

11986011

=119887ℎ0

12119864119898

+119887 (ℎ minus ℎ

0)

119899 + 1(119864119888

minus 119899119864119898

)

11986022

=119887ℎ3

0

12119864119898

+(ℎ minus ℎ

0) (ℎ2

+ ℎ2

0+ ℎℎ0)

12119864119898

+(ℎ minus ℎ

0) [(1198992

+ 3119899 + 2) ℎ2

+ 2 (119899 + 1) ℎℎ0

+ 2ℎ2

0]

4 (119899 + 1) (119899 + 2) (119899 + 3)(119864119888

minus 119864119898

)

(18)

where 119887 is the beam width The interpolation functions forthe displacements 119906 and 119908 in the present work are found bysolving the homogeneous equilibrium equations of a beamelement which resulted from the strain energy (16) as

11986011

(119906119909 119909

+ 119908119909

119908119909 119909

) = 0

11986011

(119906119909 119909

119908119909

+ 119906119909

119908119909 119909

+3

21199082

119909119908119909 119909

) + 11986022

119908119909 119909 119909 119909

= 0

(19)

The element boundary conditions for the element are asfollows

119906 (119909 = 0) = 119908 (119909 = 0) = 119908 (119909 = 119897) = 0

119906 (119909 = 119897) = 119906119895

120579 (119909 = 0) = 120579119894

120579 (119909 = 119897) = 120579119895

(20)

The solution in series form of (19) can be easily derived byusing the command dsolve of Maple and by keeping thethird-order terms for 119908(119909) the solution has the followingforms

119906 (119909) = 1198622

+21198621

minus 1198624

2119909 minus

11986241198625

21199092

minus11986241198626

+ 1198622

5

61199093

minus11986251198626

81199094

minus1198622

6

401199095

119908 (119909) = 1198623

+ 1198624119909 +

1198625

2119909 +

1198626

6119909

(21)

where 1198621 1198622 119862

6are constants which can be determined

from element end condition (20) Finally one can expressthe displacements 119906(119909) and 119908(119909) in terms of the local nodaldisplacements as

119906 =119909

119897119906119895

minus (13119909

30minus

21199092

119897+

111199093

31198972minus

31199094

1198973+

91199095

101198974) 1205792

119894

minus (119909

30minus

1199092

119897+

111199093

31198972minus

91199094

21198973+

91199095

51198974) 120579119894120579119895

+ (119909

15minus

21199093

31198972+

31199094

21198973minus

91199095

1198974) 1205792

119895

119908 = (119909 minus2119909

119897+

119909

1198972) 120579119894+ (minus

1199092

119897+

1199093

1198972) 120579119895

(22)

The finite element formulation based on the quintic functionfor the axial displacement and cubic polynomials for thetransverse displacement does not encounter any membranelocking problem [19 22] The average strain used by the firstauthor and his coworkers [13 15] for avoiding the membranelocking when a linear function is adopted for 119906 is notnecessary to use herein

Substituting (22) into (16) one can express the strainenergy 119880 in terms of the local nodal displacements 119906

119895 120579119894

and 120579119895 The local nodal force vector f in and tangent stiffness

matrix k119905for the element are obtained by differentiating the

strain energy with respect to the local nodal force vector as[4]

f in =120597119880

120597d

k119905

=1205972119880

120597d2

(23)

Equation (23) together with (10) and (11) completely definesthe finite element formulation of the element A Maplecode for derivation of the interpolation functions and thelocal element formulation as described above is given in theAppendixThe code also generates a Fortran code for the localinternal force vector and tangent stiffness matrix


Table 1 Comparison of normalized tip deflections 119908(119871)119871 of isotropic beam under transverse tip load

1198751198712119864119898

119868 Mattiasson [20] Nanakorn and Vu [21] Nguyen et al [15] Present1 030172 029946 030247 0301722 049346 048748 049537 0493493 060325 059534 060584 0603314 066996 066126 067291 0670045 071379 070479 071694 0713906 074457 073550 074783 0744707 076737 075831 077069 0767538 078498 077597 078834 0785179 079906 079011 080243 07992610 081061 080173 081331 081085

3 Numerical Procedures

The derived element internal force vector and tangent stiff-nessmatrix are assembled to construct nonlinear equilibriumequations for the structures which can be written in thefollowing form [22]

g (p 120582) = qin (p) minus 120582qef = 0 (24)

where p and qin are the structural vectors of nodal displace-ments and nodal internal forces respectively qef is the fixedexternal loading vector and the scalar 120582 is a load parameterVector g in (24) is known as the residual force vector

The system of (24) can be solved by an incremental-iterative procedure The procedure results in a predictor-corrector algorithm in which a new solution is firstly pre-dicted from a previous converged solution and then succes-sive corrections are added until some chosen convergencecriterion is satisfied In the present work a convergencecriterion based onEuclidean normof the residual force vectoris used for the iterative procedure as

1003817100381710038171003817g1003817100381710038171003817 le 120576

1003817100381710038171003817120582qef1003817100381710038171003817 (25)

where 120576 is the tolerance chosen by 10minus4 for all numericalexamples given in Section 4

In order to deal with the limit point the snap-throughand snap-back situations at which the structure tangentstiffness matrix ceases to be positive definite the arc-lengthconstraint method developed by Crisfield [23 24] is adoptedherewith Numerical procedure in the present work is imple-mented by using the spherical arc-length constraint methodwith the details described in [22]

4 Numerical Examples

Numerical examples are given in this section to show theaccuracy and efficiency of the proposed formulation as wellas illustrate the effect of thematerial distribution and the corethickness-to-height ratio ℎ

0ℎ on the large displacement

behaviour of the FGSW beams and frames

41 Cantilever Beam under Tip Load A FGSW beam com-posed of aluminum (Al) and zirconia (ZrO

2) subjected to

a transverse load 119875 at its free end is considered Youngrsquosmodulus of Al is 70GPa and that of ZrO

2is 151 GPa [25] An

aspect ratio 119871ℎ = 50 is assumed for the beamThe validation of the derived formulation firstly needs

to be confirmed From the literature review it is clear thatthere is no result available on the large displacement of FGSWbeams and frames the validation therefore is carried out ona pure metal cantilever beam In Table 1 the normalizedtip deflection of the isotropic cantilever beam obtained inthe present work is compared to the analytical solution ofMattiasson [20] and the finite element results of Nguyen etal [15] and Nanakorn and Vu [21] In the table (and in thefollowing also) 119864

119898denotes Youngrsquos modulus of the metal

As seen from the table the present formulation is moreaccurate than the two finite element formulations of [15 21]which have been derived by using the corotational and totalLagrangian approaches respectively Thus in addition toavoiding using the average strain the exact interpolationadopted in the present work is capable of improving the accu-racy also The convergence of the proposed formulation isillustrated in Table 2 where the normalized tip displacementsat various load amplitudes are given for different numberof the elements Irrespective of the load amplitude theconvergence is achieved by using just ten elements which isvery fast In this regard ten elements are used to discretize thecantilever beam in the computation reported in the following

Figures 3 and 4 respectively illustrate the effect of thegrading index 119899 and the core thickness to the beam heightℎ0ℎ on the large displacement response of the FGSW

cantilever beam At a given value of the applied load asseen from Figure 3 the tip displacements increase as thegrading index increases The increase of the displacementscan be explained by the fact that as seen from (1) thebeam associated with a higher index 119899 contains less ceramicSince Youngrsquos modulus of the ceramic is considerable higherthan that of the metal the rigidities of the beam with lessceramic percentage are smaller and this leads to the largerdisplacements The influence of the core thickness to thebeam height ratio on the large displacement response of thebeam as seen from Figure 4 is similar to that of the gradingindex 119899 and the tip displacements of the beam are increasedby the increase of the ℎ

0ℎ ratio This phenomenon can also


Table 2 Convergence of formulation in evaluating tip displacements of isotropic beam

nELElowast

1198751198712119864119898

119868 5 6 7 8 9 10

119906(119871)119871

4 032889 032890 032891 032891 032891 0328916 043451 043453 043454 043454 043454 0434548 050471 050475 050477 050478 050478 05047810 055485 055491 055492 055494 055495 055495

119908(119871)119871

4 067000 067002 067004 067004 067004 0670046 074462 074467 074469 074470 074470 0744708 078505 078512 078515 078516 078517 07851710 081068 081077 081082 081084 081085 081085

lowastNote nELE is the number of elements

Pure Aln = 03n = 2

n = 10

0

5

10

15

Appl

ied

load

PL2

minus04 0 04 08 1minus08Tip displacements

w(L)Lu(L)L

EmI

Figure 3 Effect of index 119899 on large displacement response of FGSWcantilever beam (ℎ

0ℎ = 02)

be explained by the decrease of the rigidities of the beamwith a higher ℎ

0ℎ ratio Furthermore the large displacement

curves in Figures 3 and 4 gradually approach the curves ofa homogeneous beam obtained in [15 21] when the index 119899

grows to infinity or the core thicknessℎ0approaches the beam

thickness This is reasonable since the rigidities of the FGSWbeam gradually decrease when raising the index 119899 and theℎ0ℎ ratio and as seen from (2) the beam is fully aluminum

when 119899 = infin or ℎ0

= ℎ It is worth mentioning that a loadincrement Δ119875 = 119864

11989811986821198712 has been used in the numerical

procedure in this example and the maximum number ofiterations is 8

In Figure 5 the thickness distribution of the axial stress atthe clamped end of the FGSWcantilever beam correspondingto a transverse load 119875 = 10119864

1198981198681198712 is illustrated for various

values of the grading index and the core thickness-to-heightratio The effect of the material inhomogeneity and thecore thickness-to-height ratio on the stress distribution isclearly seen from the figure again As seen from the figure

h0h = 0

h0h = 02h0h = 05h0h = 08

0

5

10

15

Appl

ied

load

PL2


w(L)Lu(L)L

EmI

Figure 4 Effect of core thickness-to-height ratio on large displace-ment response of FGSW cantilever beam (119899 = 5)

different from homogeneous and functionally graded beamsthe curves for stress distribution of the FGSW beam arecomposed of three distinct parts in which the stress inthe two functionally graded layers is not straight due tothe power-law variation of the effective modulus The stressdistribution is influenced by both thematerial inhomogeneityand the core thickness-to-height ratio and the maximumstress is increased by the increase of the index 119899 and theℎ0ℎ ratio As the initial yield stress of aluminum is just

2 times 109Nm2 the plastic deformation may be involved when

the beam undergoes the large deformation In order to takethe effect of plastic deformation into account an elastoplasticanalysis should be employed instead of the elastic analysisused herein

42 Asymmetric Frame An asymmetric frame under adownward load 119875 as depicted in the right-hand side ofFigure 6 is analysed The frame as also known as Leeframe in the literature is widely used by researchers to test


n = 03n = 2

n = 10

minus05

minus025

0

025

zh

05

h0h = 05

minus3 0 3 6minus6120590x10

9 (Nm)

(a)

minus05

minus025

0

025

zh

05

h0h = 0

h0h = 04

h0h = 08

n = 3


9 (Nm)

(b)

Figure 5 Thickness distribution of axial stress at clamped end of FGSW cantilever beam with 119875 = 10119864119898

1198681198712

n = 03n = 2

n = 10

20 40 60 80 1000w (cm)

minus2

minus1

0

1

2

3

4

P (t

on)

(a)

n = 03n = 2

n = 10

minus2

minus1

0

1

2

3

4

20 40 60 80 1000u (cm)

P(to

n)

L5

P

u

wL

(b)

Figure 6 Load-displacement curves for asymmetric frame with different grading indexes (ℎ0ℎ = 02)


h0h = 02

h0h = 05

h0h = 08

20 40 60 80 1000w (cm)

minus2

minus1

0

1

2

3P

(ton

)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 80 1000u (cm)

minus2

minus1

0

1

2

3

P (t

on)

(b)

Figure 7 Load-displacement curves for asymmetric frame with different core thickness-to-height ratio (119899 = 5)

the nonlinear finite element formulations and numeri-cal algorithms because of its snap-through and snap-backbehaviour [1 4 15] The frame is formed from two FGSWbeams with length 119871 = 120 cm width 119887 = 3 cm and heightℎ = 2 cm The FGSW material is also assumed to be formedfrom aluminum and zirconia as the above cantilever beamThe frame is discretized by ten elements five for each beam

In Figure 6 the load-displacement curves for the frameare depicted for various values of the grading index 119899 Thetransverse displacement 119908 and axial displacement 119906 in thefigures have been computed at the loaded point The load-displacement curves for the frame with different values ofthe core thickness-to-height ratio are shown in Figure 7 Thesnap-through and snap-back behaviour of the FGSW frameconsidered herein is similar to that of the isotropic frame[1 4] The grading index remarkably affects the limit loadof the frame and the limit load decreases as the index 119899

increasesThe core thickness to the beam height ratio as seenfrom Figure 7 also changes the limit load of the frame andthe limit load decreases when the ℎ

0ℎ ratio increases The

effect of the grading index 119899 and the core thickness-to-heightratio on the limit load of the frame can also be explained bythe decrease of the frame rigidities as in case of the cantileverbeam

The relation between the applied load and axial stress atintersection point of the loaded line and interface surfaceof the soft core with lower functionally graded layer of theasymmetric frame is depicted in Figure 8 for various values ofthe index 119899 and the ℎ

0ℎ ratioThe stress at the point gradually

grows when raising the applied load and it then reaches alimit point Again the plastic deformationmay have occurredduring the frame undergoing the large deformation as thecases 119899 = 3 and ℎ

0ℎ = 08 in the figure and as mentioned

above an elastoplastic analysis should be adopted to handlethe effect of plastic deformation

43 Portal Frame In this last example a FGSW portal framemade of steel and alumina under a download 119875 as depictedin Figure 9 is considered Youngrsquos modulus of steel is 210GPaand that of alumina is 390GPa [25]The beam is formed fromthree beams with 119871 = 120 cm 119887 = 3 cm and ℎ = 2 cm

In Figure 9 the load-displacement curves of the frameare shown for various values of the index 119899 and for ℎ

0ℎ =

05 The load-displacement curves of the frame with differentvalues of the core to thickness ratio are depicted in Figure 10for an index 119899 = 2 Six elements two for each beam havebeen used in the analysis The frame undergoes very largedisplacements before it reaches a limit pointThe effect of thegrading index and the core thickness to beam height on thelarge displacement response of the portal frame is similar tothat of the asymmetric frame

5 Conclusions

A nonlinear corotational finite element formulation forlarge displacement analysis of FGSW beam and framestructures has been derived The FGSW beams and framesare assumed to be composed of a metallic soft core and


n = 03n = 2

n = 10

h0h = 05

05 1 150120590x10

4 (kgfcm2)

minus2

minus1

0

1

2

3

P (t

on)

(a)

h0h = 02h0h = 04h0h = 08

n = 3

05 10 15 222120590x10

4 (kgfcm2)

minus15

minus1

minus05

0

05

1

15

2

25

3

P (t

on)

(b)

Figure 8 Relation between applied load and axial stress at intersection point of loaded line and interface surface of core and lower layer ofasymmetric frame

n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

P (t

on)

20 40 60 80 1000w (cm)

(a)

n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

20 40 60 800u (cm)

P(to

n)

P

uw

L

(b)

Figure 9 Load-displacement curves for portal frame with various values of index 119899 (ℎ0ℎ = 05)


local axial and traverse displacements u(x) and w(x)

local displacement and rotations ujL tiL tjL

local element vector feL tangent stiffness matrix keL

with(linalg)

deq1flA11(diff(u(x)xx) + diff(w(x)x)diff(w(x)xx))=0

deq2flA11(diff(u(x)xx)diff(w(x)x)

+ diff(u(x)x)diff(w(x)xx)

+ 32(diff(w(x)x))and2diff(w(x)xx))

+ A22diff(w(x)x$4)=0dsolve(eq1u(x))



+ 32(diff((x)x))and2diff(w(x)xx))

+ A22diff(w(x)x$4)=0dsolve(eq2w(x)series)

w(x) flsubs(w(0)=C3D(w)(0)=C4D(D(w))(0)=C5

D(D(D(w)))(0)=C6w(x))

w(x) flC3+coeff(w(x)x)x + coeff(w(x)xand2)xand2

+ coeff(w(x)xand3)xand3

u(x) fleval(u(x))

wxfldiff(w(x)x)

eq1fleval(u(x)x=0)eq2fleval(w(x)x=0)eq3fleval(wxx=0)

eq4fleval(u(x)x=l)eq5fleval(w(x)x=l)eq6fleval(wxx=l)

solve(eq1=0eq2=0eq3=tiLeq4=ujLeq5=0eq6=tjL

C1C2C3C4C5C6)

C1fl(130)(2rL1and2LminusLrL1rL2+2LrL2and2+30uL)LC2fl0

C3fl0C4flrL1C5flminus(2(2rL1+rL2))LC6fl(6(rL1+rL2))Land2

uflminus140C6and2xand5 minus 18C5C6xand4 minus 16(C4C6+C5and2)xand3

minus 12C4C5xand2 minus 12C4and2x+C1x+C2

wflC3+C4x+12C5xand2+16C6xand3

uflcollect(u[ujLtiLtjL])

wflcollect(w[ujLtiLtjL])

uxfldiff(ux)

wxfldiff(wx) wxxfldiff(wxx)

Ufl12int(A11(ux+12wxand2)and2+A22wxxand2x=0L)

Uflsimplify(U)

feLflgrad(U[ujLtiLtjL])

keLflhessian(U[uLrL1rL2])

with(codegen)

fortran(feLoptimized)

fortran(keLoptimized)

Algorithm 1

two symmetric functionally grade layers Based on Euler-Bernoulli beam theory the nonlinear equilibrium equationsfor a beam element have been solved and the obtainedsolution was employed to interpolate the displacement fieldAn incremental-iterative method was used in combinationwith the arc-length control method to compute the largedisplacement response of the structures Numerical resultsshow that the convergence of the proposed formulation is fastand the large displacement response of the structures can beaccurately evaluated by just several elements A parametricstudy has been carried out to highlight the effect of thematerial distribution and the core thickness to the beamheight ratio on the large displacement behaviour of theFGSW beam and frame structures As demonstrated in thenumerical examples the stress at some parts of the structures

may exceed the yield stress and it is important to take theeffect of plastic deformation into consideration in the largedisplacement analysis of FGSW beams and frames To thisend a nonlinear beam element which is capable of modelingboth the large displacement and elastoplastic behaviour of theFGSW beams and frames is necessary to develop and thiswork requires more efforts

Appendix

This Appendix lists the Maple code necessary to derive theexact interpolation functions and to generate a Fortran codefor the local internal force vector and tangent stiffness matrixdescribed in Section 23 (see Algorithm 1)


h0h = 02

h0h = 05

h0h = 08

0

05

1

15

2

25

3

35

4P

(ton

)

20 40 60 80 1000w (cm)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 800u (cm)

0

05

1

15

2

25

3

35

4

P (t

on)

(b)

Figure 10 Load-displacement curves for portal frame with different core thickness-to-height ratios (119899 = 2)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) underGrant no 10702-201502

References

[1] K M Hsiao and F Y Hou ldquoNonlinear finite element analysis ofelastic framesrdquo Computers amp Structures vol 26 no 4 pp 693ndash701 1987

[2] K M Hsiao F Y Hou and K V Spiliopoulos ldquoLarge displace-ment analysis of elasto-plastic framesrdquo Computers amp Structuresvol 28 no 5 pp 627ndash633 1988

[3] J L Meek and Q Xue ldquoA study on the instability problemfor 2D-framesrdquo Computer Methods in Applied Mechanics andEngineering vol 136 no 3-4 pp 347ndash361 1996

[4] C Pacoste and A Eriksson ldquoBeam elements in instability prob-lemsrdquoComputerMethods inAppliedMechanics and Engineeringvol 144 no 1-2 pp 163ndash197 1997

[5] D K Nguyen ldquoA Timoshenko beam element for large dis-placement analysis of planar beams and framesrdquo InternationalJournal of Structural Stability and Dynamics vol 12 no 6Article ID 1250048 9 pages 2012

[6] M Koizumi ldquoFGM activities in Japanrdquo Composites Part BEngineering vol 28 no 1-2 pp 1ndash4 1997

[7] A Chakraborty S Gopalakrishnan and J N Reddy ldquoA newbeam finite element for the analysis of functionally gradedmaterialsrdquo International Journal of Mechanical Sciences vol 45no 3 pp 519ndash539 2003

[8] R Kadoli K Akhtar and N Ganesan ldquoStatic analysis of func-tionally graded beams using higher order shear deformationtheoryrdquo Applied Mathematical Modelling vol 32 no 12 pp2509ndash2525 2008

[9] Y Y Lee X Zhao and J N Reddy ldquoPostbuckling analysis offunctionally graded plates subject to compressive and thermalloadsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 199 no 25ndash28 pp 1645ndash1653 2010

[10] C A Almeida J C R Albino I F M Menezes and G HPaulino ldquoGeometric nonlinear analyses of functionally gradedbeams using a tailored Lagrangian formulationrdquo MechanicsResearch Communications vol 38 no 8 pp 553ndash559 2011

[11] S Taeprasartsit ldquoA buckling analysis of perfect and imperfectfunctionally graded columnsrdquo Proceedings of the Institution ofMechanical Engineers Part L Journal of Materials Design andApplications vol 226 no 1 pp 16ndash33 2012

[12] D K Nguyen ldquoLarge displacement response of tapered can-tilever beams made of axially functionally graded materialrdquoComposites Part B Engineering vol 55 pp 298ndash305 2013

[13] D K Nguyen ldquoLarge displacement behaviour of taperedcantilever Euler-Bernoulli beams made of functionally gradedmaterialrdquo Applied Mathematics and Computation vol 237 pp340ndash355 2014

[14] D K Nguyen and B S Gan ldquoLarge deflections of taperedfunctionally graded beams subjected to end forcesrdquo AppliedMathematical Modelling vol 38 no 11-12 pp 3054ndash3066 2014

[15] D K Nguyen B S Gan and T-H Trinh ldquoGeometricallynonlinear analysis of planar beam and frame structures made


of functionally graded materialrdquo Structural Engineering andMechanics vol 49 no 6 pp 727ndash743 2014

[16] T Q Bui A Khosravifard Ch Zhang M R Hematiyanand M V Golub ldquoDynamic analysis of sandwich beams withfunctionally graded core using a truly meshfree radial pointinterpolation methodrdquo Engineering Structures vol 47 pp 90ndash104 2013

[17] T P Vo H-TThai T-K Nguyen A Maheri and J Lee ldquoFiniteelement model for vibration and buckling of functionallygraded sandwich beams based on a refined shear deformationtheoryrdquo Engineering Structures vol 64 pp 12ndash22 2014

[18] T P Vo H-T Thai T-K Nguyen F Inam and J Lee ldquoA quasi-3D theory for vibration and buckling of functionally gradedsandwich beamsrdquo Composite Structures vol 119 pp 1ndash12 2015

[19] L Yunhua ldquoExplanation and elimination of shear locking andmembrane locking with field consistence approachrdquo ComputerMethods in AppliedMechanics and Engineering vol 162 no 1ndash4pp 249ndash269 1998

[20] K Mattiasson ldquoNumerical results from large deection beamand frame problems analysed by means of elliptic integralsrdquoInternational Journal for Numerical Methods in Engineering vol17 no 1 pp 145ndash153 1981

[21] P Nanakorn and L N Vu ldquoA 2D field-consistent beam elementfor large displacement analysis using the total Lagrangianformulationrdquo Finite Elements in Analysis andDesign vol 42 no14-15 pp 1240ndash1247 2006

[22] M A CrisfieldNon-Linear Finite Element Analysis of Solids andStructures Volume 1 Essentials JohnWiley amp Sons ChichesterUK 1991

[23] M A Crisfield ldquoA fast incrementaliterative solution procedurethat handles lsquosnap-throughrsquordquo Computers and Structures vol 13no 1ndash3 pp 55ndash62 1981

[24] M A Crisfield ldquoArc-lengthmethod including line searches andaccelerationsrdquo International Journal for Numerical Methods inEngineering vol 19 no 9 pp 1269ndash1289 1983

[25] G N Praveen and J N Reddy ldquoNonlinear transient ther-moelastic analysis of functionally graded ceramic-metal platesrdquoInternational Journal of Solids and Structures vol 35 no 33 pp4457ndash4476 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the displacement field is free of shear locking Based on thehigher-order shear deformation beam theory Kadoli et al[8] investigated the static behaviour of FGM beams underambient temperature The 119896119901-Ritz method was employed byLee et al [9] in studying the postbuckling response of FGMplates subject to compressive and thermal loads Almeida etal [10] extended the total Lagrange formulation proposedby Pacoste and Eriksson in [4] to investigate the geometri-cally nonlinear behaviour of FGM beams A finite elementformulation based on the linear exact shape functions wasemployed by Taeprasartsit [11] for buckling analysis of perfectand imperfect FGM columns Recently the first author andhis coworkers [12ndash15] derived the corotational finite elementformulations for large displacement analysis of FGM beamand frame structures In these works in order to prevent theformulations from the membrane locking the average strainhas been introduced to replace the membrane strain

Sandwich structures with height strength-to-weight ratioarewidely used in aerospace application such as skin ofwingsaileron and spoilers In order to improve the performance ofthe structures in high thermal environment FGMs could beincorporated in the sandwich construction Several analysesmainly the vibration and buckling of functionally gradedsandwich (FGSW) beams have been carried out in recentyears [16ndash18] To maintain minimum weight for a givenmechanical loading condition FGSW beams and framesare often designed to be slender and they might undergolarge displacements during service Investigation on the largedisplacement behaviour of FGSW beams and frames tothe authorsrsquo best knowledge has not been reported in theliterature and it is studied in this paper for the first timeTo this end a finite element formulation based on Euler-Bernoulli beam theory is derived in the context of thecorotational approach and used in the investigation Differentfrom previous works the interpolation functions for thedisplacement field are obtained by solving the equilibriumnonlinear equations of a beam segment The exact inter-polation functions lead to a balance between the axial andtransverse displacements and the finite element formulationbased on these functions is free of the membrane locking[19] The concept of the average membrane strain mentionedabove is therefore not necessary to use in the present workUsing the derived formulation the equilibrium equationsare constructed and they are solved by an incremental-iterative procedure in combination with the arc-length con-trol method Numerical examples are given to show theaccuracy of the derived formulation and to illustrate the effectof thematerial distribution and the core thickness ratio on thelarge displacement behaviour of the FGSW beam and framestructures

2 Finite Element Formulation

21 FGSWBeam Figure 1 shows a FGSWbeam formed froma metallic soft core and two symmetric FGM skin layers in aCartesian coordinate system The 119909-axis is chosen to be onthe midplane of the beam and the 119911-axis is directed upwardsDenoting the beam height and the soft core thickness by

zz

yxh

b

h0FGM

FGMMetal core

LCeramic

Ceramic

Metal

Figure 1 A FGSW beam in a Cartesian coordinate system

ℎ and ℎ0 respectively the FGM is assumed to be made of

metal and ceramic with the volume fraction of ceramic 119881119888

varying in the thickness direction by the following power-lawdistribution

119881119888

=

(2119911 + ℎ

0

ℎ0

minus ℎ)

119899

for 119911 isin [minusℎ

2 minus

ℎ0

2]

0 for 119911 isin [minusℎ0

2ℎ0

2]

(2119911 minus ℎ

0

ℎ minus ℎ0

)

119899

for 119911 isin [ℎ0

2ℎ

2]

(1)

where 119899 is the nonnegative grading index The volumefraction of metal is 119881

119898= 1 minus 119881

119888 As seen from (1) the top

and bottom surfaces contain only ceramic whereas the core ispure metal The effective Youngrsquos modulus of the beam 119864(119911)evaluated by Voigt model reads

119864 (119911)

=

(119864119888

minus 119864119898

) (2119911 + ℎ

0

ℎ0

minus ℎ)

119899

+ 119864119898

for minusℎ

2le 119911 le minus

ℎ0

2

119864119898

for minusℎ0

2le 119911 le

ℎ0

2

(119864119888

minus 119864119898

) (2119911 minus ℎ

0

ℎ minus ℎ0

)

119899

+ 119864119898

forℎ0

2le 119911 le minus

ℎ

2

(2)

where 119864119888and 119864

119898are Youngrsquos moduli of ceramic and metal

respectively

22 Corotational Approach The corotational approach isa convenient way to derive geometrically nonlinear finiteelement formulations The central idea of the approach isto introduce a local coordinate system that continuouslymoves and rotates with the element during its deformationprocessThe element formulation is firstly derived in the localcoordinate system and then transferred into the global one byusing transformation matrices Depending on the definitionof local coordinate system various corotational beam formu-lations are available in the literatureThe formulation adoptedin the present work is closely related to the ones described in[4 22] and its main points are summarized in the following

Figure 2 shows a planar two-node beam element (119894 119895)and its kinematics in two coordinate systems a local (119909 119911)

and a global (119909 119911) The element is initially inclined to the 119909-axis an angle 120579

0 The global system is fixed while the local

one continuously moves and rotates with the element duringits deformation The local system is chosen with its origin


wj

x

z

j

j

1205790

1205790

i

i

120579r 120579j

120579i

120579120579i

120579j

ujx

z

wi

ui

uj





119894= 119908119894

= 119908119895


d = 119906119895

120579119894

120579119895119879

(3)


119894and 120579119895



d = 119906119894 119908119894

120579119894

119906119895

119908119895

120579119895119879

(4)




119906119895

= 119897119899

minus 119897

120579119894= 120579119894minus 120579119903

120579119895

= 120579119895

minus 120579119903

(5)


lated as

119897 = radic(119909119895

minus 119909119894)2

+ (119911119895

minus 119911119894)2

119897119899

= radic(119909119895

+ 119906119895

minus 119909119894minus 119906119894)2

+ (119911119895

+ 119908119895

minus 119911119894minus 119908119894)2

(6)

with (119909119894 119911119894) and (119909





120575d = T120575d (7)


T =1

119897119899

[[

[

minus119888119897119899

minus119904119897119899

0 119888119897119899

119904119897119899

0

minus119904 119888 119897119899

119904 minus119888 0

minus119904 119888 0 119904 minus119888 119897119899

]]

]

(8)

with

119888 = cos 120579 =119909119895

+ 119906119895

minus 119909119894minus 119906119894

119897119899

119904 = sin 120579 =119911119895

+ 119908119895

minus 119911119894minus 119908119894

119897119899

(9)


fin = T119879f in (10)




k119905

= T119879k119905T +

119891119906

119897119899

zz119879 +

(119891120579119894

+ 119891120579119895

)

1198972119899

(rz119879 + zr119879) (11)




r = minus119888 minus119904 0 119888 119904 0119879

z = 119904 minus119888 0 minus119904 119888 0119879

(12)


119905are known


1and 119906

3 respectively are

given by

1199061

(119909 119911) = 119906 (119909) minus 119911 119908 (119909)119909

1199063 (119909 119911) = 119908 (119909)

(13)





120576119909

= 119906119909

+1

21199082

119909minus 119911 119908119909 119909

(14)



120590119909

= 119864 (119911) 120576119909 (15)



reads

119880 =1

2int

119897

0

120590119909120576119909119889119909

=1

2int

119897

0

[11986011

(119906119909

+1

21199082

119909)

2

+ 11986022

1199082

119909119909] 119889119909

(16)

where 11986011

and 11986022


(11986011

11986022

) = int119860

119864 (119911) (1 1199112) 119889119860 (17)


11and 119860

22as

11986011

=119887ℎ0

12119864119898

+119887 (ℎ minus ℎ

0)

119899 + 1(119864119888

minus 119899119864119898

)

11986022

=119887ℎ3

0

12119864119898

+(ℎ minus ℎ

0) (ℎ2

+ ℎ2

0+ ℎℎ0)

12119864119898

+(ℎ minus ℎ

0) [(1198992

+ 3119899 + 2) ℎ2

+ 2 (119899 + 1) ℎℎ0

+ 2ℎ2

0]

4 (119899 + 1) (119899 + 2) (119899 + 3)(119864119888

minus 119864119898

)

(18)


11986011

(119906119909 119909

+ 119908119909

119908119909 119909

) = 0

11986011

(119906119909 119909

119908119909

+ 119906119909

119908119909 119909

+3

21199082

119909119908119909 119909

) + 11986022

119908119909 119909 119909 119909

= 0

(19)


119906 (119909 = 0) = 119908 (119909 = 0) = 119908 (119909 = 119897) = 0

119906 (119909 = 119897) = 119906119895

120579 (119909 = 0) = 120579119894

120579 (119909 = 119897) = 120579119895

(20)


119906 (119909) = 1198622

+21198621

minus 1198624

2119909 minus

11986241198625

21199092

minus11986241198626

+ 1198622

5

61199093

minus11986251198626

81199094

minus1198622

6

401199095

119908 (119909) = 1198623

+ 1198624119909 +

1198625

2119909 +

1198626

6119909

(21)

where 1198621 1198622 119862



119906 =119909

119897119906119895

minus (13119909

30minus

21199092

119897+

111199093

31198972minus

31199094

1198973+

91199095

101198974) 1205792

119894

minus (119909

30minus

1199092

119897+

111199093

31198972minus

91199094

21198973+

91199095

51198974) 120579119894120579119895

+ (119909

15minus

21199093

31198972+

31199094

21198973minus

91199095

1198974) 1205792

119895

119908 = (119909 minus2119909

119897+

119909

1198972) 120579119894+ (minus

1199092

119897+

1199093

1198972) 120579119895

(22)



119895 120579119894




f in =120597119880

120597d

k119905

=1205972119880

120597d2

(23)




1198751198712119864119898




g (p 120582) = qin (p) minus 120582qef = 0 (24)



1003817100381710038171003817g1003817100381710038171003817 le 120576

1003817100381710038171003817120582qef1003817100381710038171003817 (25)








2) subjected to


2is 151 GPa [25] An










nELElowast

1198751198712119864119898

119868 5 6 7 8 9 10

119906(119871)119871

4 032889 032890 032891 032891 032891 0328916 043451 043453 043454 043454 043454 0434548 050471 050475 050477 050478 050478 05047810 055485 055491 055492 055494 055495 055495

119908(119871)119871

4 067000 067002 067004 067004 067004 0670046 074462 074467 074469 074470 074470 0744708 078505 078512 078515 078516 078517 07851710 081068 081077 081082 081084 081085 081085


Pure Aln = 03n = 2

n = 10

0

5

10

15

Appl

ied

load

PL2


w(L)Lu(L)L

EmI


0ℎ = 02)













h0h = 0

h0h = 02h0h = 05h0h = 08

0

5

10

15

Appl

ied

load

PL2


w(L)Lu(L)L

EmI







n = 03n = 2

n = 10

minus05

minus025

0

025

zh

05

h0h = 05


9 (Nm)

(a)

minus05

minus025

0

025

zh

05

h0h = 0

h0h = 04

h0h = 08

n = 3


9 (Nm)

(b)


1198681198712

n = 03n = 2

n = 10

20 40 60 80 1000w (cm)

minus2

minus1

0

1

2

3

4

P (t

on)

(a)

n = 03n = 2

n = 10

minus2

minus1

0

1

2

3

4

20 40 60 80 1000u (cm)

P(to

n)

L5

P

u

wL

(b)



h0h = 02

h0h = 05

h0h = 08

20 40 60 80 1000w (cm)

minus2

minus1

0

1

2

3P

(ton

)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 80 1000u (cm)

minus2

minus1

0

1

2

3

P (t

on)

(b)














0ℎ =


5 Conclusions



n = 03n = 2

n = 10

h0h = 05

05 1 150120590x10

4 (kgfcm2)

minus2

minus1

0

1

2

3

P (t

on)

(a)

h0h = 02h0h = 04h0h = 08

n = 3

05 10 15 222120590x10

4 (kgfcm2)

minus15

minus1

minus05

0

05

1

15

2

25

3

P (t

on)

(b)


n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

P (t

on)

20 40 60 80 1000w (cm)

(a)

n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

20 40 60 800u (cm)

P(to

n)

P

uw

L

(b)






with(linalg)











D(D(D(w)))(0)=C6w(x))



u(x) fleval(u(x))

wxfldiff(w(x)x)




C1C2C3C4C5C6)








uxfldiff(ux)



Uflsimplify(U)



with(codegen)



Algorithm 1



Appendix



h0h = 02

h0h = 05

h0h = 08

0

05

1

15

2

25

3

35

4P

(ton

)

20 40 60 80 1000w (cm)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 800u (cm)

0

05

1

15

2

25

3

35

4

P (t

on)

(b)


Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










wj

x

z

j

j

1205790

1205790

i

i

120579r 120579j

120579i

120579120579i

120579j

ujx

z

wi

ui

uj





119894= 119908119894

= 119908119895


d = 119906119895

120579119894

120579119895119879

(3)


119894and 120579119895



d = 119906119894 119908119894

120579119894

119906119895

119908119895

120579119895119879

(4)




119906119895

= 119897119899

minus 119897

120579119894= 120579119894minus 120579119903

120579119895

= 120579119895

minus 120579119903

(5)


lated as

119897 = radic(119909119895

minus 119909119894)2

+ (119911119895

minus 119911119894)2

119897119899

= radic(119909119895

+ 119906119895

minus 119909119894minus 119906119894)2

+ (119911119895

+ 119908119895

minus 119911119894minus 119908119894)2

(6)

with (119909119894 119911119894) and (119909





120575d = T120575d (7)


T =1

119897119899

[[

[

minus119888119897119899

minus119904119897119899

0 119888119897119899

119904119897119899

0

minus119904 119888 119897119899

119904 minus119888 0

minus119904 119888 0 119904 minus119888 119897119899

]]

]

(8)

with

119888 = cos 120579 =119909119895

+ 119906119895

minus 119909119894minus 119906119894

119897119899

119904 = sin 120579 =119911119895

+ 119908119895

minus 119911119894minus 119908119894

119897119899

(9)


fin = T119879f in (10)




k119905

= T119879k119905T +

119891119906

119897119899

zz119879 +

(119891120579119894

+ 119891120579119895

)

1198972119899

(rz119879 + zr119879) (11)




r = minus119888 minus119904 0 119888 119904 0119879

z = 119904 minus119888 0 minus119904 119888 0119879

(12)


119905are known


1and 119906

3 respectively are

given by

1199061

(119909 119911) = 119906 (119909) minus 119911 119908 (119909)119909

1199063 (119909 119911) = 119908 (119909)

(13)





120576119909

= 119906119909

+1

21199082

119909minus 119911 119908119909 119909

(14)



120590119909

= 119864 (119911) 120576119909 (15)



reads

119880 =1

2int

119897

0

120590119909120576119909119889119909

=1

2int

119897

0

[11986011

(119906119909

+1

21199082

119909)

2

+ 11986022

1199082

119909119909] 119889119909

(16)

where 11986011

and 11986022


(11986011

11986022

) = int119860

119864 (119911) (1 1199112) 119889119860 (17)


11and 119860

22as

11986011

=119887ℎ0

12119864119898

+119887 (ℎ minus ℎ

0)

119899 + 1(119864119888

minus 119899119864119898

)

11986022

=119887ℎ3

0

12119864119898

+(ℎ minus ℎ

0) (ℎ2

+ ℎ2

0+ ℎℎ0)

12119864119898

+(ℎ minus ℎ

0) [(1198992

+ 3119899 + 2) ℎ2

+ 2 (119899 + 1) ℎℎ0

+ 2ℎ2

0]

4 (119899 + 1) (119899 + 2) (119899 + 3)(119864119888

minus 119864119898

)

(18)


11986011

(119906119909 119909

+ 119908119909

119908119909 119909

) = 0

11986011

(119906119909 119909

119908119909

+ 119906119909

119908119909 119909

+3

21199082

119909119908119909 119909

) + 11986022

119908119909 119909 119909 119909

= 0

(19)


119906 (119909 = 0) = 119908 (119909 = 0) = 119908 (119909 = 119897) = 0

119906 (119909 = 119897) = 119906119895

120579 (119909 = 0) = 120579119894

120579 (119909 = 119897) = 120579119895

(20)


119906 (119909) = 1198622

+21198621

minus 1198624

2119909 minus

11986241198625

21199092

minus11986241198626

+ 1198622

5

61199093

minus11986251198626

81199094

minus1198622

6

401199095

119908 (119909) = 1198623

+ 1198624119909 +

1198625

2119909 +

1198626

6119909

(21)

where 1198621 1198622 119862



119906 =119909

119897119906119895

minus (13119909

30minus

21199092

119897+

111199093

31198972minus

31199094

1198973+

91199095

101198974) 1205792

119894

minus (119909

30minus

1199092

119897+

111199093

31198972minus

91199094

21198973+

91199095

51198974) 120579119894120579119895

+ (119909

15minus

21199093

31198972+

31199094

21198973minus

91199095

1198974) 1205792

119895

119908 = (119909 minus2119909

119897+

119909

1198972) 120579119894+ (minus

1199092

119897+

1199093

1198972) 120579119895

(22)



119895 120579119894




f in =120597119880

120597d

k119905

=1205972119880

120597d2

(23)




1198751198712119864119898




g (p 120582) = qin (p) minus 120582qef = 0 (24)



1003817100381710038171003817g1003817100381710038171003817 le 120576

1003817100381710038171003817120582qef1003817100381710038171003817 (25)








2) subjected to


2is 151 GPa [25] An










nELElowast

1198751198712119864119898

119868 5 6 7 8 9 10

119906(119871)119871

4 032889 032890 032891 032891 032891 0328916 043451 043453 043454 043454 043454 0434548 050471 050475 050477 050478 050478 05047810 055485 055491 055492 055494 055495 055495

119908(119871)119871

4 067000 067002 067004 067004 067004 0670046 074462 074467 074469 074470 074470 0744708 078505 078512 078515 078516 078517 07851710 081068 081077 081082 081084 081085 081085


Pure Aln = 03n = 2

n = 10

0

5

10

15

Appl

ied

load

PL2


w(L)Lu(L)L

EmI


0ℎ = 02)













h0h = 0

h0h = 02h0h = 05h0h = 08

0

5

10

15

Appl

ied

load

PL2


w(L)Lu(L)L

EmI







n = 03n = 2

n = 10

minus05

minus025

0

025

zh

05

h0h = 05


9 (Nm)

(a)

minus05

minus025

0

025

zh

05

h0h = 0

h0h = 04

h0h = 08

n = 3


9 (Nm)

(b)


1198681198712

n = 03n = 2

n = 10

20 40 60 80 1000w (cm)

minus2

minus1

0

1

2

3

4

P (t

on)

(a)

n = 03n = 2

n = 10

minus2

minus1

0

1

2

3

4

20 40 60 80 1000u (cm)

P(to

n)

L5

P

u

wL

(b)



h0h = 02

h0h = 05

h0h = 08

20 40 60 80 1000w (cm)

minus2

minus1

0

1

2

3P

(ton

)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 80 1000u (cm)

minus2

minus1

0

1

2

3

P (t

on)

(b)














0ℎ =


5 Conclusions



n = 03n = 2

n = 10

h0h = 05

05 1 150120590x10

4 (kgfcm2)

minus2

minus1

0

1

2

3

P (t

on)

(a)

h0h = 02h0h = 04h0h = 08

n = 3

05 10 15 222120590x10

4 (kgfcm2)

minus15

minus1

minus05

0

05

1

15

2

25

3

P (t

on)

(b)


n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

P (t

on)

20 40 60 80 1000w (cm)

(a)

n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

20 40 60 800u (cm)

P(to

n)

P

uw

L

(b)






with(linalg)











D(D(D(w)))(0)=C6w(x))



u(x) fleval(u(x))

wxfldiff(w(x)x)




C1C2C3C4C5C6)








uxfldiff(ux)



Uflsimplify(U)



with(codegen)



Algorithm 1



Appendix



h0h = 02

h0h = 05

h0h = 08

0

05

1

15

2

25

3

35

4P

(ton

)

20 40 60 80 1000w (cm)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 800u (cm)

0

05

1

15

2

25

3

35

4

P (t

on)

(b)


Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120576119909

= 119906119909

+1

21199082

119909minus 119911 119908119909 119909

(14)



120590119909

= 119864 (119911) 120576119909 (15)



reads

119880 =1

2int

119897

0

120590119909120576119909119889119909

=1

2int

119897

0

[11986011

(119906119909

+1

21199082

119909)

2

+ 11986022

1199082

119909119909] 119889119909

(16)

where 11986011

and 11986022


(11986011

11986022

) = int119860

119864 (119911) (1 1199112) 119889119860 (17)


11and 119860

22as

11986011

=119887ℎ0

12119864119898

+119887 (ℎ minus ℎ

0)

119899 + 1(119864119888

minus 119899119864119898

)

11986022

=119887ℎ3

0

12119864119898

+(ℎ minus ℎ

0) (ℎ2

+ ℎ2

0+ ℎℎ0)

12119864119898

+(ℎ minus ℎ

0) [(1198992

+ 3119899 + 2) ℎ2

+ 2 (119899 + 1) ℎℎ0

+ 2ℎ2

0]

4 (119899 + 1) (119899 + 2) (119899 + 3)(119864119888

minus 119864119898

)

(18)


11986011

(119906119909 119909

+ 119908119909

119908119909 119909

) = 0

11986011

(119906119909 119909

119908119909

+ 119906119909

119908119909 119909

+3

21199082

119909119908119909 119909

) + 11986022

119908119909 119909 119909 119909

= 0

(19)


119906 (119909 = 0) = 119908 (119909 = 0) = 119908 (119909 = 119897) = 0

119906 (119909 = 119897) = 119906119895

120579 (119909 = 0) = 120579119894

120579 (119909 = 119897) = 120579119895

(20)


119906 (119909) = 1198622

+21198621

minus 1198624

2119909 minus

11986241198625

21199092

minus11986241198626

+ 1198622

5

61199093

minus11986251198626

81199094

minus1198622

6

401199095

119908 (119909) = 1198623

+ 1198624119909 +

1198625

2119909 +

1198626

6119909

(21)

where 1198621 1198622 119862



119906 =119909

119897119906119895

minus (13119909

30minus

21199092

119897+

111199093

31198972minus

31199094

1198973+

91199095

101198974) 1205792

119894

minus (119909

30minus

1199092

119897+

111199093

31198972minus

91199094

21198973+

91199095

51198974) 120579119894120579119895

+ (119909

15minus

21199093

31198972+

31199094

21198973minus

91199095

1198974) 1205792

119895

119908 = (119909 minus2119909

119897+

119909

1198972) 120579119894+ (minus

1199092

119897+

1199093

1198972) 120579119895

(22)



119895 120579119894




f in =120597119880

120597d

k119905

=1205972119880

120597d2

(23)




1198751198712119864119898




g (p 120582) = qin (p) minus 120582qef = 0 (24)



1003817100381710038171003817g1003817100381710038171003817 le 120576

1003817100381710038171003817120582qef1003817100381710038171003817 (25)








2) subjected to


2is 151 GPa [25] An










nELElowast

1198751198712119864119898

119868 5 6 7 8 9 10

119906(119871)119871

4 032889 032890 032891 032891 032891 0328916 043451 043453 043454 043454 043454 0434548 050471 050475 050477 050478 050478 05047810 055485 055491 055492 055494 055495 055495

119908(119871)119871

4 067000 067002 067004 067004 067004 0670046 074462 074467 074469 074470 074470 0744708 078505 078512 078515 078516 078517 07851710 081068 081077 081082 081084 081085 081085


Pure Aln = 03n = 2

n = 10

0

5

10

15

Appl

ied

load

PL2


w(L)Lu(L)L

EmI


0ℎ = 02)













h0h = 0

h0h = 02h0h = 05h0h = 08

0

5

10

15

Appl

ied

load

PL2


w(L)Lu(L)L

EmI







n = 03n = 2

n = 10

minus05

minus025

0

025

zh

05

h0h = 05


9 (Nm)

(a)

minus05

minus025

0

025

zh

05

h0h = 0

h0h = 04

h0h = 08

n = 3


9 (Nm)

(b)


1198681198712

n = 03n = 2

n = 10

20 40 60 80 1000w (cm)

minus2

minus1

0

1

2

3

4

P (t

on)

(a)

n = 03n = 2

n = 10

minus2

minus1

0

1

2

3

4

20 40 60 80 1000u (cm)

P(to

n)

L5

P

u

wL

(b)



h0h = 02

h0h = 05

h0h = 08

20 40 60 80 1000w (cm)

minus2

minus1

0

1

2

3P

(ton

)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 80 1000u (cm)

minus2

minus1

0

1

2

3

P (t

on)

(b)














0ℎ =


5 Conclusions



n = 03n = 2

n = 10

h0h = 05

05 1 150120590x10

4 (kgfcm2)

minus2

minus1

0

1

2

3

P (t

on)

(a)

h0h = 02h0h = 04h0h = 08

n = 3

05 10 15 222120590x10

4 (kgfcm2)

minus15

minus1

minus05

0

05

1

15

2

25

3

P (t

on)

(b)


n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

P (t

on)

20 40 60 80 1000w (cm)

(a)

n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

20 40 60 800u (cm)

P(to

n)

P

uw

L

(b)






with(linalg)











D(D(D(w)))(0)=C6w(x))



u(x) fleval(u(x))

wxfldiff(w(x)x)




C1C2C3C4C5C6)








uxfldiff(ux)



Uflsimplify(U)



with(codegen)



Algorithm 1



Appendix



h0h = 02

h0h = 05

h0h = 08

0

05

1

15

2

25

3

35

4P

(ton

)

20 40 60 80 1000w (cm)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 800u (cm)

0

05

1

15

2

25

3

35

4

P (t

on)

(b)


Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198751198712119864119898




g (p 120582) = qin (p) minus 120582qef = 0 (24)



1003817100381710038171003817g1003817100381710038171003817 le 120576

1003817100381710038171003817120582qef1003817100381710038171003817 (25)








2) subjected to


2is 151 GPa [25] An










nELElowast

1198751198712119864119898

119868 5 6 7 8 9 10

119906(119871)119871

4 032889 032890 032891 032891 032891 0328916 043451 043453 043454 043454 043454 0434548 050471 050475 050477 050478 050478 05047810 055485 055491 055492 055494 055495 055495

119908(119871)119871

4 067000 067002 067004 067004 067004 0670046 074462 074467 074469 074470 074470 0744708 078505 078512 078515 078516 078517 07851710 081068 081077 081082 081084 081085 081085


Pure Aln = 03n = 2

n = 10

0

5

10

15

Appl

ied

load

PL2


w(L)Lu(L)L

EmI


0ℎ = 02)













h0h = 0

h0h = 02h0h = 05h0h = 08

0

5

10

15

Appl

ied

load

PL2


w(L)Lu(L)L

EmI







n = 03n = 2

n = 10

minus05

minus025

0

025

zh

05

h0h = 05


9 (Nm)

(a)

minus05

minus025

0

025

zh

05

h0h = 0

h0h = 04

h0h = 08

n = 3


9 (Nm)

(b)


1198681198712

n = 03n = 2

n = 10

20 40 60 80 1000w (cm)

minus2

minus1

0

1

2

3

4

P (t

on)

(a)

n = 03n = 2

n = 10

minus2

minus1

0

1

2

3

4

20 40 60 80 1000u (cm)

P(to

n)

L5

P

u

wL

(b)



h0h = 02

h0h = 05

h0h = 08

20 40 60 80 1000w (cm)

minus2

minus1

0

1

2

3P

(ton

)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 80 1000u (cm)

minus2

minus1

0

1

2

3

P (t

on)

(b)














0ℎ =


5 Conclusions



n = 03n = 2

n = 10

h0h = 05

05 1 150120590x10

4 (kgfcm2)

minus2

minus1

0

1

2

3

P (t

on)

(a)

h0h = 02h0h = 04h0h = 08

n = 3

05 10 15 222120590x10

4 (kgfcm2)

minus15

minus1

minus05

0

05

1

15

2

25

3

P (t

on)

(b)


n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

P (t

on)

20 40 60 80 1000w (cm)

(a)

n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

20 40 60 800u (cm)

P(to

n)

P

uw

L

(b)






with(linalg)











D(D(D(w)))(0)=C6w(x))



u(x) fleval(u(x))

wxfldiff(w(x)x)




C1C2C3C4C5C6)








uxfldiff(ux)



Uflsimplify(U)



with(codegen)



Algorithm 1



Appendix



h0h = 02

h0h = 05

h0h = 08

0

05

1

15

2

25

3

35

4P

(ton

)

20 40 60 80 1000w (cm)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 800u (cm)

0

05

1

15

2

25

3

35

4

P (t

on)

(b)


Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











nELElowast

1198751198712119864119898

119868 5 6 7 8 9 10

119906(119871)119871

4 032889 032890 032891 032891 032891 0328916 043451 043453 043454 043454 043454 0434548 050471 050475 050477 050478 050478 05047810 055485 055491 055492 055494 055495 055495

119908(119871)119871

4 067000 067002 067004 067004 067004 0670046 074462 074467 074469 074470 074470 0744708 078505 078512 078515 078516 078517 07851710 081068 081077 081082 081084 081085 081085


Pure Aln = 03n = 2

n = 10

0

5

10

15

Appl

ied

load

PL2


w(L)Lu(L)L

EmI


0ℎ = 02)













h0h = 0

h0h = 02h0h = 05h0h = 08

0

5

10

15

Appl

ied

load

PL2


w(L)Lu(L)L

EmI







n = 03n = 2

n = 10

minus05

minus025

0

025

zh

05

h0h = 05


9 (Nm)

(a)

minus05

minus025

0

025

zh

05

h0h = 0

h0h = 04

h0h = 08

n = 3


9 (Nm)

(b)


1198681198712

n = 03n = 2

n = 10

20 40 60 80 1000w (cm)

minus2

minus1

0

1

2

3

4

P (t

on)

(a)

n = 03n = 2

n = 10

minus2

minus1

0

1

2

3

4

20 40 60 80 1000u (cm)

P(to

n)

L5

P

u

wL

(b)



h0h = 02

h0h = 05

h0h = 08

20 40 60 80 1000w (cm)

minus2

minus1

0

1

2

3P

(ton

)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 80 1000u (cm)

minus2

minus1

0

1

2

3

P (t

on)

(b)














0ℎ =


5 Conclusions



n = 03n = 2

n = 10

h0h = 05

05 1 150120590x10

4 (kgfcm2)

minus2

minus1

0

1

2

3

P (t

on)

(a)

h0h = 02h0h = 04h0h = 08

n = 3

05 10 15 222120590x10

4 (kgfcm2)

minus15

minus1

minus05

0

05

1

15

2

25

3

P (t

on)

(b)


n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

P (t

on)

20 40 60 80 1000w (cm)

(a)

n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

20 40 60 800u (cm)

P(to

n)

P

uw

L

(b)






with(linalg)











D(D(D(w)))(0)=C6w(x))



u(x) fleval(u(x))

wxfldiff(w(x)x)




C1C2C3C4C5C6)








uxfldiff(ux)



Uflsimplify(U)



with(codegen)



Algorithm 1



Appendix



h0h = 02

h0h = 05

h0h = 08

0

05

1

15

2

25

3

35

4P

(ton

)

20 40 60 80 1000w (cm)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 800u (cm)

0

05

1

15

2

25

3

35

4

P (t

on)

(b)


Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










n = 03n = 2

n = 10

minus05

minus025

0

025

zh

05

h0h = 05


9 (Nm)

(a)

minus05

minus025

0

025

zh

05

h0h = 0

h0h = 04

h0h = 08

n = 3


9 (Nm)

(b)


1198681198712

n = 03n = 2

n = 10

20 40 60 80 1000w (cm)

minus2

minus1

0

1

2

3

4

P (t

on)

(a)

n = 03n = 2

n = 10

minus2

minus1

0

1

2

3

4

20 40 60 80 1000u (cm)

P(to

n)

L5

P

u

wL

(b)



h0h = 02

h0h = 05

h0h = 08

20 40 60 80 1000w (cm)

minus2

minus1

0

1

2

3P

(ton

)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 80 1000u (cm)

minus2

minus1

0

1

2

3

P (t

on)

(b)














0ℎ =


5 Conclusions



n = 03n = 2

n = 10

h0h = 05

05 1 150120590x10

4 (kgfcm2)

minus2

minus1

0

1

2

3

P (t

on)

(a)

h0h = 02h0h = 04h0h = 08

n = 3

05 10 15 222120590x10

4 (kgfcm2)

minus15

minus1

minus05

0

05

1

15

2

25

3

P (t

on)

(b)


n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

P (t

on)

20 40 60 80 1000w (cm)

(a)

n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

20 40 60 800u (cm)

P(to

n)

P

uw

L

(b)






with(linalg)











D(D(D(w)))(0)=C6w(x))



u(x) fleval(u(x))

wxfldiff(w(x)x)




C1C2C3C4C5C6)








uxfldiff(ux)



Uflsimplify(U)



with(codegen)



Algorithm 1



Appendix



h0h = 02

h0h = 05

h0h = 08

0

05

1

15

2

25

3

35

4P

(ton

)

20 40 60 80 1000w (cm)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 800u (cm)

0

05

1

15

2

25

3

35

4

P (t

on)

(b)


Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










h0h = 02

h0h = 05

h0h = 08

20 40 60 80 1000w (cm)

minus2

minus1

0

1

2

3P

(ton

)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 80 1000u (cm)

minus2

minus1

0

1

2

3

P (t

on)

(b)














0ℎ =


5 Conclusions



n = 03n = 2

n = 10

h0h = 05

05 1 150120590x10

4 (kgfcm2)

minus2

minus1

0

1

2

3

P (t

on)

(a)

h0h = 02h0h = 04h0h = 08

n = 3

05 10 15 222120590x10

4 (kgfcm2)

minus15

minus1

minus05

0

05

1

15

2

25

3

P (t

on)

(b)


n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

P (t

on)

20 40 60 80 1000w (cm)

(a)

n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

20 40 60 800u (cm)

P(to

n)

P

uw

L

(b)






with(linalg)











D(D(D(w)))(0)=C6w(x))



u(x) fleval(u(x))

wxfldiff(w(x)x)




C1C2C3C4C5C6)








uxfldiff(ux)



Uflsimplify(U)



with(codegen)



Algorithm 1



Appendix



h0h = 02

h0h = 05

h0h = 08

0

05

1

15

2

25

3

35

4P

(ton

)

20 40 60 80 1000w (cm)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 800u (cm)

0

05

1

15

2

25

3

35

4

P (t

on)

(b)


Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










n = 03n = 2

n = 10

h0h = 05

05 1 150120590x10

4 (kgfcm2)

minus2

minus1

0

1

2

3

P (t

on)

(a)

h0h = 02h0h = 04h0h = 08

n = 3

05 10 15 222120590x10

4 (kgfcm2)

minus15

minus1

minus05

0

05

1

15

2

25

3

P (t

on)

(b)


n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

P (t

on)

20 40 60 80 1000w (cm)

(a)

n = 02n = 1

n = 3n = 10

0

05

1

15

2

25

3

35

4

45

20 40 60 800u (cm)

P(to

n)

P

uw

L

(b)






with(linalg)











D(D(D(w)))(0)=C6w(x))



u(x) fleval(u(x))

wxfldiff(w(x)x)




C1C2C3C4C5C6)








uxfldiff(ux)



Uflsimplify(U)



with(codegen)



Algorithm 1



Appendix



h0h = 02

h0h = 05

h0h = 08

0

05

1

15

2

25

3

35

4P

(ton

)

20 40 60 80 1000w (cm)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 800u (cm)

0

05

1

15

2

25

3

35

4

P (t

on)

(b)


Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













with(linalg)











D(D(D(w)))(0)=C6w(x))



u(x) fleval(u(x))

wxfldiff(w(x)x)




C1C2C3C4C5C6)








uxfldiff(ux)



Uflsimplify(U)



with(codegen)



Algorithm 1



Appendix



h0h = 02

h0h = 05

h0h = 08

0

05

1

15

2

25

3

35

4P

(ton

)

20 40 60 80 1000w (cm)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 800u (cm)

0

05

1

15

2

25

3

35

4

P (t

on)

(b)


Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










h0h = 02

h0h = 05

h0h = 08

0

05

1

15

2

25

3

35

4P

(ton

)

20 40 60 80 1000w (cm)

(a)

h0h = 02

h0h = 05

h0h = 08

20 40 60 800u (cm)

0

05

1

15

2

25

3

35

4

P (t

on)

(b)


Competing Interests


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

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