Research Article Semi-Analytical Finite Strip Transfer...

Research ArticleSemi-Analytical Finite Strip Transfer Matrix Method forBuckling Analysis of Rectangular Thin Plates

Li-Ke Yao Bin He Yu Zhang and Wei Zhou

Mechanics Department Nanjing Tech University Nanjing China

Correspondence should be addressed to Bin He hebin123njtecheducn

Received 16 June 2015 Revised 28 September 2015 Accepted 22 October 2015

Academic Editor Alessandro Gasparetto

Copyright copy 2015 Li-Ke Yao et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Plates and shells are main components of modern engineering structures whose buckling analysis has been focused by researchersIn this investigation rectangular thin plates with loaded edges simply supported can be discretized by semi-analytical finite striptechnology Then the control equations of the strip elements of the buckling plate will be rewritten as the transfer equations bytransfermatrixmethod A new approach namely semi-analytical Finite Strip TransferMatrixMethod is developed for the bucklinganalysis of plates This method requires no global stiffness matrix of the system reduces the systemmatrix order and improves thecomputational efficiency Comparing with some theoretical results and FEMrsquos results of two illustrations (the plates and the ribbedplates) under six boundary conditions the method is proved to be reliable and effective

1 Introduction

Buckling analysis is one of the important steps in the designof thin-walled structures which can be applied in differentbranches of engineering including shipbuilding civil archi-tecture and mechanical construction [1] Thin plate a mainkind of thin-walled structure is widely utilized to lightenengineering structures as well as save materials [2] The reli-ability of one single plate lies in its stability chiefly which hasbeen studied by experimental or mathematical means [3]

In the early works the vibration and buckling perfor-mances of rectangular plates loaded by in-plane hydrostaticforces for awide variety of aspect ratios boundary conditionsand loading magnitudes have been analyzed by numericaltechnology [4] The solutions of the differential equations ofthe buckling Mindlin plate are obtained in discrete forms byapplying numerical integrations [5] And extensive numericalresults have been presented for the critical buckling loads ofsimply supported rectangular composite plates subjected tofive types of loading conditions (1) uniaxial (2) hydrostaticbiaxial (3) compression-tension biaxial (4) positive shearand (5) negative shear [6] By introducing an unified ana-lytical solution technique for a multitude of combinations ofboundary conditions an analytical method is presented for

the problem of elastic buckling of orthotropic rectangularplates [7] If the biharmonic operator in the buckling controlequations of rectangular plates is reduced by performingthe Laplacersquos operator and the finite difference method thebuckling load of the plate can be investigated [8] Up to nowmany methods have been used to analyze the buckling prob-lems of rectangular plates such as the extended Kantorovichmethod [9] differential quadrature procedures [10] asymp-totic finite stripmethod [11] blockGMRESmethod [12] first-order shear deformation meshless method [13] radial pointinterpolation method [14] untruncated infinite series tech-nology [15] discrete singular convolution approach [16] andhierarchical Rayleigh-Ritz and finite element method [17]

Among these methods finite element method is a pow-erful tool for engineering analysis while the choices of theelements and the mesh sizes have significant influences onthe results of buckling analysis [18] When calculating thebuckling problems of regular geometry shape structuresfinite strip method can be regarded as an efficient way Andthe arbitrary shaped plate may be discretized as many stripelements by the subparametric mapping concept [19] Toconsider the transverse shear effect the spline strip methodhas been proposed to analyze the buckling behaviors ofrectangular Mindlin plates with linearly tapered thickness

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 485686 11 pageshttpdxdoiorg1011552015485686

2 Mathematical Problems in Engineering

in one direction [20] The buckling stresses and naturalfrequencies of rectangular laminated plateswith arbitrary lay-ups and general boundary conditions can be predicted bythe improved spline finite strip which combines the super-strip concept [21]Then a higher-order shear deformable platefinite strip element is developed and employed to investigatethe critical buckling loads of composite laminated plates [22]The spline finite strip method and multilevel substructuringprocedures are combined for the buckling stress analysis ofcomposite laminated prismatic shell structures with generalboundary conditions [23] Bucklingmode localization in rib-stiffened plates with randomly misplaced stiffeners is studiedby finite strip method [24] Notably a comprehensive reviewof the finite strip method for structural analysis is given[25] A layerwise B-spline finite strip method is developedfor free vibration analysis of truly thick and thin compositelaminated plates [26] And a finite strip Fourier p-element isdeveloped to analyze the natural vibration characteristics ofthe thin plate [27] Based on the concept of the semienergyapproach the finite strip method can be proposed to ana-lyze the buckling [28] postbuckling [29] dynamic bucklingbehaviors of the laminated plates [30] and stability phe-nomena of cylindrical shell structures [31] The harmonicseries satisfying the boundary conditions in the loaded endsof thin-walled members are generally employed in Semi-Analytical Finite Strip Transfer Matrix Method (SAFSM)[32] And the buckling stress of the cold-formed sections canbe analyzed by SAFSM [33] which has been widely usedin computer softwares (such as THIN-WALL [34] CUFSM[35]) to develop the signature curves [36] of the bucklingstress versus buckling half wavelength for thin-walled mem-bers Furthermore the constrained finite strip method inno-vated from SAFSM is developed and applied in cold-formedsteel design [37] Generalized constrained finite strip methodfor thin-walled members with arbitrary cross-sections can beused to analyze secondary modes [38] and primary modes[39] The impacts of basis orthogonalization and normal-ization in constrained finite strip method are discussed forstability solution of open thin-walled members [40]

Transfer matrix method has been developed as an effec-tive tool for vibration analysis of engineering structuresespecially for chain connected system from topological per-spective [41] Then a combined finite element transfer matrixmethod is developed to study the statics [42] and dynamics[43] of structures Later a new transfer matrix method basedon the boundary element and transfer matrix technologyis proposed for the vibration analysis of two-dimensionalplate acted by uniform [44] and concentrated [45] loads Byintroducing the numerical integration nonlinear dynamicsof structures [46] multi-rigid-body system [47] and multi-flexible-body system [48] can be dealt with by transfer matrixmethod And a procedure which combinesmultiport transfermatrices andfinite elements has been developed to resolve theacoustic phenomena of automotive hollow body networks[49] It should be noted that three references enlightenus to start this investigation The buckling analysis of theplate with built-in rectangular delamination has been imple-mented by strip distributed transfer function method [50]And the transfer matrix method can be used to analyze

Y

Z

X

1 2 3 middot middot middot

middot middot middot

n minus 1

1 2 3 nb

wi wj

120579i 120579j

a

Ti Tj

y

x u

z w

Figure 1 Coordinates degree of freedom and loads of a typicalstrip

the instability in unsymmetrical rotor-bearing systems [51]and tall unbraced frames [52]

Here by combining two powerful methods of the semi-analytical finite strip and the transfermatrix an efficient tech-nology named semi-analytical Finite Strip Transfer MatrixMethod (FSTMM) for the buckling analysis of plates isdeveloped The text is organized as follows In Section 2the general theorem of the semi-analytical finite strip forbuckling analysis of plates is shown Section 3 presents semi-analytical Finite Strip Transfer Matrix Method for bucklinganalysis In Section 4 some results calculated by FSTMMandother methods are given which can validate the proposedmethod The conclusions are included in Section 5

2 The Semi-Analytical Finite Strip Analysis

21 Degree of Freedom and Shape Function In classicalKirchhoffplate theory themembrane or in-plane translationsare neglected and only the 119911-axisrsquos translation 119908 and the 119910-axisrsquos rotation 120579 are considered as local displacements [1]Theglobal coordinate system (119883119884 119885) and local coordinate sys-tem (119909 119910 119911) are actually the same in the plate model In finitestrip method (FSM) a thin plate as shown in Figure 1 canbe divided into many strip elements

We introduce a numbering system of finite strip modelshown in Figure 1 The total number of nodal lines is 119899therefore the total number of strips is 119899 minus 1 Generally oneinternal nodal line is always connected with two strips whilethe first and last nodal lines are only connected with onestrip This numbering system will be used to depict the statevector of the nodal line and the transfer matrix of the strip inSection 3

Here the plate is assumed to have two opposite loadededges with the simply supported boundary analytical trigo-nometric functions that satisfy the simply-simply supportedboundary condition of the loaded edges can be given toexpress the transverse deflection of the plate along thelongitudinal direction [35 37]

119884119901 (119910) = sin119901120587119910

119886 119901 = 1 2 3 119899 (1)

where 119901 is the half-wave number which also stands forcertain half wavelength along the longitudinal direction 120587 isthe circumference ratio 119910 is the longitudinal coordinate inlocal system 119886 is the length of the plate

Mathematical Problems in Engineering 3

Four cubic polynomials that satisfy the boundary condi-tion can be selected as the shape function matrix to depictthe out-of-plane displacement of the plate strip along thetransverse direction namely [39 40]

N (119909)

= [1 minus31199092

1198872+21199093

1198873119909 minus

21199092

119887+1199093

1198872

31199092

1198872minus21199093

1198873

1199093

1198872minus1199092

119887

]

(2)

where 119887 is the width of the strip as shown in Figure 1 and 119909 isthe coordinate along the width direction in local system

Then the transverse deflection119908of any points (119909 119910) in thestrip can be denoted by the so-called nodal line displacementvector 120575119901 and shape function matrix N119901(119909 119910)

119908 (119909 119910) =

119898

sum

119901=1

N119901 (119909 119910) 120575119901 (3)

whereN119901(119909 119910) can be denoted as the production of the shapefunctions of (1) and (2)

N119901 (119909 119910) = N (119909) 119884119901 (119910) (4)

120575119901 can be defined as the 119911-axisrsquos translation119908 and the 119910-axisrsquos

rotation 120579 of the two nodal lines in one strip

120575119901= [119908119901

119894120579119901

119894119908119901

119895120579119901

119895 ]119879

(5)

the subscripts 119894 and 119895 denote two edges of one strip whichare connected with other strips or are the boundaries of theplate respectively and 119898 is the maximum half-wave numberas well as order of modal employed in the analysis which is afinite positive integer

22 Fundamental StiffnessMatrix So far many plate theorieshave been developed by engineers and researchers [2] TheKirchhoff plate theory is a classical and simple one whichwillbe used in this investigation It must be noted that other platetheories can be applied in the proposed method if the elasticdeformation energy and the nodal line displacement vectoraremodified accordinglyThebending strain vector ofmiddleplane in Kirchhoff plate can be defined as

120576119861 = minus[1205972119908

1205971199092

1205972119908

120597119910221205972119908

120597119909120597119910]

119879

(6)

We substitute (3) into above equation the bending strainvector can be expressed by the nodal line displacement vectorand shape function matrix namely

120576119861 =

119898

sum

119901=1

N10158401015840119901120575119901 (7)

where N10158401015840119901= minus[120597N119879

1199011205971199092120597N11987911990112059711991022(120597N119879119901120597119909120597119910)]

119879 is a (3 times4)matrix that gives the relationship between the strain vector120576119861 and the nodal line displacement vector 120575119901 As to generallinear elastic material the elastic deformation energy can beexpressed as

119880 =1

2int

119881

120576119879120590 d119881 (8)

where 120576 and 120590 denote strain and stress vectors respectively119881 is the volume of the material According to Kirchhoff platetheory with constant thickness the elastic bending deforma-tion energy of (8) can be rewritten as

119880 =1

2int

119886

0

int

119887

0

120576119879

119861D119861120576119861d119909 d119910 (9)

where D119861 is an elastic constant matrix of the material with adimension of (3 times 3) which is related to the thickness of theplate [1 2] We substitute (4) into (9) the new deformationenergy formula can be obtained as follows

119880 =1

2

119898

sum

119901=1

119898

sum

119902=1

120575119901119879k119901119902119890120575119902 (10)

where the elastic stiffness matrix of the strip can be conciselyexpressed in a form

k119901119902119890= int

119886

0

int

119887

0

N10158401015840119879119901D119861N10158401015840

119902d119909 d119910 (11)

Above elastic stiffness is a (4times4)matrixWhen the parameters119901 119902 vary the elastic stiffness changes accordinglyHowever tothe case of 119901 = 119902 elastic stiffness matrix always equals zerothat can be found by integrating (11) The reason is that theshape function N10158401015840

119901is orthogonal about the elastic constant

matrixD119861 which can be expressed as

k119901119902119890

=

int

119886

0

int

119887

0

N10158401015840119879119901D119861N10158401015840

119902d119909 d119910 119901 = 119902 = 1 2 3 119898

0 119901 = 119902

(12)

As shown in Figure 1 the strip is loaded with linearlyvarying edge tractions The in-plane compressive loads canbe expressed as

119879119909 = 119879119894 minus (119879119894 minus 119879119895)119909

119887 (13)

where 119879119894 119879119895 are the forces in two edges of the strip 119887 is thewidth of the strip as shown in Figure 1 and 119909 is the transversecoordinate Similar to the deduction of the elastic stiffnessmatrix the (4 times 4) geometric stiffness matrix caused by in-plane compressive loads can be obtained [34 35]

k119901119902119892= int

119886

0

int

119887

0

(119879119894 minus (119879119894 minus 119879119895)119909

119887)N1015840119879119901N1015840119902d119909 d119910 (14)

where N1015840119901= N(119909)(120597119884119901(119910)120597119910) is a (1 times 4) matrix It can be

proofed that N1015840119901includes an orthogonal system of functions

namely


k119901119902119892=

int

119886

0

int

119887

0

(119879119894 minus (119879119894 minus 119879119895)119909

119887)N1015840119879119901N1015840119902d119909 d119910 119901 = 119902 = 1 2 3 119898

0 119901 = 119902

(15)

The solution of the buckling problems can be deduced bythe orthotropic condition of (12) and (15) conveniently Forother boundary conditions different analytical functions canbe selected to describe the buckling shape of the strip [53]These will be discussed in the successive related papers

3 Semi-Analytical Finite Strip Transfer MatrixMethod for Buckling Analysis

31 Control Equations of Strip Element In both finite ele-ment transfer matrix method and boundary element transfermatrix method transfer equations of the given substructurecan be deduced by the control equations of this substructurewhich consider the interaction forces between this substruc-ture and other structures As to the proposed Finite StripTransfer Matrix Method the strip element can be regardedas the substructure

The control equations of the buckling strip can beobtained by virtual work principle [1 2]

(k119901119901119890+ k119901119901119892) 120575119901= R119901 (16)

where k119901119901119890

is the elastic stiffness matrix of (11) k119901119901119892

is the geo-metric stiffness matrix as shown in (14) 120575119901 is the nodal linedisplacement vector can be defined by (5) 119901 is the half-wavenumber of the buckling strip shape and R119901 can be regardedas the generalized internal forces acted on the strip that can bedenoted as

R119901 = [119865119901119894 119872119901

119894119865119901

119895119872119901

119895 ]119879

(17)

where 119865119901119894(and 119865119901

119895) is the generalized internal force associated

with the transverse deflection 119908119901119894(and 119908119901

119895) of the nodal line

119894 (and 119895) and119872119901119894(and119872119901

119895) is the generalized internal force

associated with the 119910-axisrsquos rotation 120579119901119894(and 120579119901

119895) of the nodal

line 119894 (and 119895) respectivelyAssuming 1198790(119909) as the initial axial force the real axial

forces in the geometric stiffness matrix can be expressed as

119879 (119909) = 1205821198790 (119909) (18)

where 120582 is the buckling coefficient So the geometric stiffnessmatrix k119901119901

119892can be rewritten as the function of the constant

geometric stiffness matrix k119901119901119892|1198790

caused by initial axial force1198790(119909)

k119901119901119892= 120582 k119901119901119892

100381610038161003816100381610038161198790 (19)

We substitute (19) into (16) the control equations of thestrip can be rewritten as follows

(k119901119901119890+ 120582 k119901119901119892

100381610038161003816100381610038161198790) 120575119901= R119901 (20)

To simplify the equations the coefficient matrix of the nodalline displacement vector 120575119901 in above equation can be markedby

K = k119901119901119890+ 120582 k119901119901119892

100381610038161003816100381610038161198790 (21)

where both coefficient matrices k119901119901119890

and k119901119901119892|1198790

are constantvalues when the loads determined by (18) vary

32 State Vector Transfer Equations and Transfer MatrixDuring the analysis of transfer matrix method the statevector of the strip is an important concept that includes twoparts one part describes the configuration of the strip theother part gives the generalized internal forces acted on thestrip by other members in the system For example the statevector of the nodal line 119894(119895) can be defined as

Z119897119899 = [120575119901119879

119897R119901119879119897]119879

119899(119897 = 119894 119895) (22)

where the first subscript 119897 denotes the number of the nodallines the second subscript 119899 denotes the number of the strips120575119901

119897= [119908119901

119897120579119901

119897]119879 can be regarded as the generalized displace-

ment vector of the nodal line 119897 and R119901119897= [119865119901

119897119872119901

119897]119879 is the

generalized internal force vector acted on the nodal line 119897correspondingly

Using the block forms of (22) the control equations (20)can be rewritten as the form of the transfer equations of thisstrip which are

Z119895119899 = U119890119899Z119894119899 (23)

where the transfer matrix of the strip 119899 is

U119890119899= [

minusKminus112K11 Kminus1

12

K21 minus K22Kminus112K11 K22Kminus112] (24)

where the superscript 119890 denotes the strip element the sub-script 119899 denotes the number of the strips the subscripts 119894 and119895 denote the two nodal lines of the strip 119899 andK119894119895 (119894 119895 = 1 2)are (2 times 2) block submatrices that can be determined by (21)Actually coefficient matrix of (21) can be denoted as

K = [[

K112times2

K122times2

K212times2

K222times2

]

]

(25)

Follow the condition of displacement continuum and thelaw of action and reaction the state vectors of the same nodalline in two connected strips can be obtained

Z119897119899+1 = [1 0

0 minus1

]Z119897119899 (26)


Table 1 State vectors of nodal lines under various boundary conditions

Boundary condition (BC) Simply (s) Clamped (c) Free (f)State vectors of nodal lines 119885s = [0 120579 119865 0]

119879

119885c = [0 0 119865 119872]119879

119885f = [119908 120579 0 0]119879

Table 2 Eigen-matrices under various boundary conditions

Boundary condition (BC) SSss SScc SSsc SSfs SSfc SSff

Eigen-matrix [

[

11988012 11988013

11988042 11988043

]

]

[

[

11988013 11988014

11988023 11988024

]

]

[

[

11988012 11988013

11988022 11988023

]

]

[

[

11988032 11988033

11988042 11988043

]

]

[

[

11988011 11988012

11988021 11988022

]

]

[

[

11988031 11988032

11988041 11988042

]

]

where the subscript 119897 denotes the nodal line number and thesubscripts 119899 and 119899 + 1 denote the order numbers of the twostrips which are connected by the nodal line 119897 So the transferequations between particular nodal lines of conjunctionalstrips can be obtained by multiplying (23) and (26) namely

Z119897+1119899+1 = U119890119897119899Z119897119899 (27)

where the transfer matrix of this substructure is

U119890119897119899= [


12

K22Kminus112K11 minus K21 minusK22Kminus1

12

] (28)

where the superscripts 119890 and 119897 denote the strip element andthe nodal line

Introducing the same procedure in the classical transfermatrix method the overall system transfer equations and theoverall transfer matrix Uall which relate the state vectors attwo edges of the plate can be assembled and calculated as

Z119899119899minus1 = UallZ11

Uall = U119890119897119899U119890119897119899minus1sdot sdot sdotU1198901198972U1198901198971

(29)

Three classical boundary conditions are considered ingeneral engineering problems simply supported clampedsupported and free edge Different boundary conditions ofthe plate can give appropriate limits to specific variables inthe state vectors which are shown in Table 1 For the buck-ling analysis of plates with simply-simply (S-S) supportedboundary condition of loaded edges in this dissertation theunloaded edges may sustain six kinds of boundary condi-tions which can be expressed by SSss SScc SSsc SSfs SSfcand SSff (s c and f mean simply clamped and free boundaryconditions resp) Here the capital letters and lowercaseletters denote the loaded edges and unloaded edges corre-spondingly

Taking the boundary condition SSss for example thetotal transfer equations can be deduced as follows

[0 120579 119865 0]119879

119871= Uall [0 120579 119865 0]

119879

119865 (30)

where subscripts 119865 and 119871 denote the first and last edges of theplate Uall is the overall transfer matrix of the plate Then thenonzero variables in the state vector of the first edge of theplate have the relationship that can be deduced by (30)

[

0

0

]

119879

119871

= [

11988012 11988013

11988042 11988043

][

120579

119865

]

119879

119865

(31)

where11988012119880131198804211988043 are the elements ofUall To make thenonzero solution of (31) possible it must satisfy the followingcondition

det([11988012 11988013

11988042 11988043

]) = 0 (32)

Above equation gives the characteristic equation of thebuckling plate by the proposed semi-analytical Finite StripTransfer Matrix Method which can be used to calculate thebuckling coefficients We combine (31) and (27) the bucklingmode can be obtained Eigen-matrices under various bound-ary conditions are shown in Table 2

4 Examples and Analysis

41 Illustrations of Thin Plates The geometrical and materialproperties of the plate during the research are as followswidth 119887 = 5mm thickness 119905 = 01mm elastic modulus 119864 =2 times 10

5Nmm2 Poissonrsquos ratio ] = 03 shear modulus 119866 =1198642(1 + ]) and initial axial force 1198790 = 120587

21198641199053121198872(1 minus ]2) =

72305NmmThe length of the plate is a variable parameterwhich is selected for each boundary condition

The section is divided into five segments along the loadededge The relationship schemas between buckling coefficient120582 and length-width ratio 120572 of the plate are obtained by theproposed FSTMM finite element method (FEM) and theo-retical analysis FSTMMrsquos results are compared with theoret-ical and FEMrsquos results under the boundary conditions of SSssand SSff as shown in Figures 2(a) and 2(b) For the remain-ing four boundary conditions comparisons with FSTMMrsquosand FEMrsquos results are shown in Figures 2(c)ndash2(f) The sixcomparisons confirm the reliability of FSTMM for the buck-ling analysis of the plate

42 Illustrations of Ribbed Plates Plates with ribs are impor-tant engineering components for the load-carrying Manymethods are available to analyze the buckling of prismatic flatand stiffened shell structure [54] For further application ofFSTMM ribbed plates are discussed below Notably due tothe application condition of the Kirchhoff plate theory thethickness of the ribs must be relatively small Besides com-parisons with FEMrsquos results are used to testify the FSTMM

421 Plate with Three Ribs The material properties are thesame as the plate as stated in Section 41 The geometricalparameters are as follows The width of the plate is 5mm


FSTMM resultsFEM results

Theoretical results

2 3 4 51Length-width ratio 120572

4

5

6

7Bu

cklin

g co

effici

ent120582

(a) Boundary conditions of SSss


Theoretical results

0

1

2

3

4

5

6

7

Buck

ling

coeffi

cien

t120582

2 3 4 5 61Length-width ratio 120572

(b) Boundary conditions of SSff


120582 = 54

21 43Length-width ratio 120572

5

6

7

8

9

Buck

ling

coeffi

cien

t120582

(c) Boundary conditions of SSsc


120582 = 7

1 15 2 25 3 3505Length-width ratio 120572

6

8

10

12

14Bu

cklin

g co

effici

ent120582

(d) Boundary conditions of SScc


120582 = 044

2 3 4 5 6 7 81Length-width ratio 120572

0

1

2

3

4

5

6

7

Buck

ling

coeffi

cien

t120582

(e) Boundary conditions of SSfs


120582 = 128


1

2

3

4

5

6

7

8

Buck

ling

coeffi

cien

t120582

(f) Boundary conditions of SSfc

Figure 2 Relations between 120582 and 120572 of the plate


(a) Strip model in FSTMM (b) Shell model for FEM

(c) Solid model for FEM

Figure 3 Meshed grids of the plate with three ribs

(a) Length-width ratio 120572 = 2 (b) Length-width ratio 120572 = 35

Figure 4 Buckling deformation mode of the plate with three ribs and boundary conditions of SSss

the thickness is 01mm The ribs are 04mm in width and02mm in thickness Three ribs are uniformly spaced alongthe transverse direction the width of the interval space is095mm The length of the plate is a variable parameterranging from 05mm to 25mm In shell models of FSTMMand FEM the initial axial force of the plate is 72305Nmmand the initial axial force of ribs is 14461Nmm In solidmodel of FEM the initial axial pressure in the cross sectionis 72305MPa Then the loading forces of the shell and solidmodels are equivalent Figure 3(a) shows the meshed grid ofthe plate in FSTMM and the meshed plate in FEM using theshell elements and the solid elements can be given in Figures3(b) and 3(c) respectively

As the plate simply supported in four edges Figure 4shows the total buckling deformation of the plate with threeribs when length-width ratios are 2 and 35There are two (orthree) half waves in the buckling mode of the length-widthratio 120572 = 2 (or 120572 = 35) shown in Figures 4(a) and 4(b) corre-spondingly As amatter of fact the bucklingmodes of Figures4(a) and 4(b) can be obtained by setting the computationalparameters of the half-wave number 119901 = 2 and 119901 = 3 in theFSTMM with clearer engineering meaning

The relation curves between buckling coefficient andlength-width ratio of the plate with three ribs and boundaryconditions of SSss can be calculated by the FSTMM the FEM

FSTMM resultsFEM results (shell)

FEM results (solid)

1 15 2 25 3 35 4 45 505Length-width ratio 120572

4

6

8

10

12

14

16

18

20

Buck

ling

coeffi

cien

t120582

120582 = 6

Figure 5 Relations between 120582 and 120572 of the plate with three ribs andboundary conditions of SSss

with the shell elements and solid elements as shown inFigure 5Three curves have similar variation tendency whichproves FSTMM can also be credible in the buckling analysis



Figure 6 Buckling deformation mode of the plate with four ribs and boundary bonditions of SSfc


0

2

4

6

8

10

12

14Bu

cklin

g co

effici

ent120582


FEM results (solid)120582 = 2

Figure 7 Relations between 120582 and 120572 of the plate with four ribs and boundary conditions of SSfc

of the ribbed plate And there is an obvious difference aboutthe computational consumption that depends on the elementnumber among these three methods shown in Figure 3So the proposed method is highly efficient Furthermorethe buckling coefficient of FEM with solid model is alwayssmaller than that of FEM with shell model and the bucklingcoefficient of FEM with shell model is always smaller thanthat of FSTMM the reason is that the shear strain is generallyincluded in the FEM analysis with solid elements and partlyincluded in the analysis with the FEMrsquos shell elements whileneglected in the FSTMM analysis It also indicates that theclassical Kirchhoff assumption in the plate theory will makethe critical buckling loads a bit larger than the true resultsComparing with Figures 2(a) and 5 the buckling coefficientsconverge to 4 and 6 respectively when 120572 increases We canobviously find that the plate can be stiffened by the ribs

422 Plate with Four Ribs The material properties are thesame as stated in the former chapterThe geometrical param-eters are as follows The width of the plate is 5mm and thethickness is 01mmThe ribs are 02mm in width and 02mmin thickness Four ribs are uniformly spaced along the trans-verse direction the width of the internal space is 084mmThe length of the plate is a variable parameter ranged from05mm to 35mm In shell model of FEM and FSTMM theinitial axial force of the plate is 72305Nmm and the initialaxial force of ribs is 14461Nmm In solid model of FEMthe initial axial pressure in the cross section is 72305MPa

Then the loading forces of the shell and solid models areequivalent The plate is simply supported in loaded edgeswhile two unloaded edges are free and clapped separately

Figure 6(a) shows the buckling deformation of a platewith four ribs and boundary conditions of SSfc when thelength is 10mm in other words the length-width ratio is 2Figure 6(b) shows the buckling deformation of a plate withthe length 175mm (length-width ratio 35) As to the caseof Figure 6(a) (or Figure 6(b)) this critical buckling config-uration includes one (two) half-wave(s) model Actually theminimal critical buckling load can be obtained by setting thehalf-wave number 119901 = 1 (or 119901 = 2) in the FSTMM Thevalue of the half-wave number 119901 can be used to distinguishthe buckling models of the plate which is convenient for thesuppression of the instable failure

In Figure 7 three curves obtained by three differentmeth-ods give the relationship between buckling coefficient andlength-width ratio of the four-ribbed plate with boundaryconditions of SSfc The good agreement among the threeresults shown in Figures 5 and 7 indicates that the proposedFSTMM can be used in the buckling analysis of the plate withseveral ribs and different boundary conditions

43 Comparative Analysis As a general rule the computa-tional precision can be improved by increasing the number ofthe elements Here the influence of the element number to thecomputational precision can be analyzed if FEMrsquos shell modeland FSTMM are applied to calculate the buckling problems


(a) 100 square elements (b) 264 square elements

(c) 1600 square elements

Figure 8 Three mesh sizes in FEM for the same plate

05 1 15 2 25 3 35 4 45 5

35

4

45

5

55

6

65

7

75

8

Buck

ling

coeffi

cien

t120582

Length-width ratio 120572

FEM results of 100 elementsFEM results of 264 elementsFEM results of 1600 elementsTheoretical results

(a) Results of FEM

FSTMM results of 1 stripFSTMM results of 2 stripsFSTMM results of 10 stripsTheoretical results

35

4

45

5

55

6

65

7

75

8Bu

cklin

g co

effici

ent120582


(b) Results of FSTMM

Figure 9 Relations between 120582 and 120572 of the plate with boundary conditions of SSss by using three mesh sizes

of thin plate with the boundary conditions of SSss Thebuckling behaviors can be obtained by the FEMrsquos shell modelwith different mesh sizes which are shown in Figures 8(a)8(b) and 8(c) with the element numbers 100 264 and 1600respectively

Figure 9 compares the influence of the element numberto the computational precision in FEMrsquos shell model andFSTMM In order to make the errors between the FEMrsquosand theoretical results acceptable the element number mayincrease unexpectedly from 100 264 to 1600 as shown inFigure 9(a) In contrast with this performance of the FEMas the number of strips 119899 = 2 in FSTMM is selected the com-putational results have good agreement with the theoretical

results as shown in Figure 9(b) So it may be confirmed thatthe proposed FSTMM has good efficiency for the bucklinganalysis of rectangular thin plates under the boundary con-dition of simply supported loaded edges

5 Conclusions

In this investigation the semi-analytical Finite Strip TransferMatrix Method is proposed to analyze the buckling problemsof rectangular thin plates under the boundary conditionof simply supported loaded edges In order to validate themethod the examples of the plate and the plate with ribs canbe designed and analyzed by different methods


It may be found that the method holds several highlights(1) it demands no global stiffness matrix and reduces matrixorder in the system analysis (2) the orthogonality among theanalytical shape functions of the loaded direction of the plateleads to the clear computational mode of the buckling anal-ysis (3) both the semi-analytical finite strip and the transfermatrix technologies ensure the proposed method efficient

Conflict of Interests

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

Acknowledgment

The research is financed by National Key Science FoundationProgram (51624001) Natural Science Foundation of JiangsuProvince China (BK20130911)

References

[1] S P Timoshenko and S Woinowsky-Krieger Theory of Platesand Shells McGraw-Hill New York NY USA 1959

[2] J N ReddyMechanics of LaminatedComposite Plates and ShellsTheory and Analysis CRC Press New York NY USA 2004

[3] J Rhodes ldquoBuckling of thin plates and membersmdashand earlywork on rectangular tubesrdquoThin-Walled Structures vol 40 no2 pp 87ndash108 2002

[4] R E Kielb and L SHan ldquoVibration and buckling of rectangularplates under in-plane hydrostatic loadingrdquo Journal of Sound andVibration vol 70 no 4 pp 543ndash555 1980

[5] S Takeshi and M Hiroshi ldquoElastic buckling of rectangularmindlin plate with mixed boundary conditionsrdquo Computers ampStructures vol 25 no 5 pp 801ndash808 1987

[6] Y Narita and A W Leissa ldquoBuckling studies for simply sup-ported symmetrically laminated rectangular platesrdquo Interna-tional Journal ofMechanical Sciences vol 32 no 11 pp 909ndash9241990

[7] I E Harik and N Balakrishnan ldquoStability of orthotropic rect-angular platesrdquo Applied Mathematical Modelling vol 18 no 7pp 400ndash402 1994

[8] Y ZChen ldquoEvaluation of buckling loading of rectangular bend-ing plate by using an iterative approachrdquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 91ndash991998

[9] S Yuan and Y Jin ldquoComputation of elastic buckling loadsof rectangular thin plates using the extended Kantorovichmethodrdquo Computers amp Structures vol 66 no 6 pp 861ndash8671998

[10] T M Teo and K M Liew ldquoA differential quadrature procedurefor three-dimensional buckling analysis of rectangular platesrdquoInternational Journal of Solids and Structures vol 36 no 8 pp1149ndash1168 1998

[11] C-P Wu Y-M Wang and Y-C Hung ldquoAsymptotic finitestrip analysis of doubly curved laminated shellsrdquoComputationalMechanics vol 27 no 2 pp 107ndash118 2001

[12] C-S Chien S-L Chang and Z Mei ldquoTracing the buckling ofa rectangular plate with the Block GMRES methodrdquo Journal ofComputational and Applied Mathematics vol 136 no 1-2 pp199ndash218 2001

[13] J Wang K M Liew M J Tan and S Rajendran ldquoAnalysisof rectangular laminated composite plates via FSDT meshlessmethodrdquo International Journal of Mechanical Sciences vol 44no 7 pp 1275ndash1293 2002

[14] K M Liew and X L Chen ldquoBuckling of rectangular Mindlinplates subjected to partial in-plane edge loads using the radialpoint interpolation methodrdquo International Journal of Solids andStructures vol 41 no 5-6 pp 1677ndash1695 2004

[15] S Kshirsagar and K Bhaskar ldquoAccurate and elegant free vibra-tion andbuckling studies of orthotropic rectangular plates usinguntruncated infinite seriesrdquo Journal of Sound and Vibration vol314 no 3ndash5 pp 837ndash850 2008

[16] O Civalek A Korkmaz and C Demir ldquoDiscrete singular con-volution approach for buckling analysis of rectangular Kirch-hoff plates subjected to compressive loads on two-oppositeedgesrdquo Advances in Engineering Software vol 41 no 4 pp 557ndash560 2010

[17] O Vaseghi H R Mirdamadi and D Panahandeh-ShahrakildquoNon-linear stability analysis of laminated composite plates onone-sided foundation by hierarchical Rayleigh-Ritz and finiteelementsrdquo International Journal of Non-Linear Mechanics vol57 pp 65ndash74 2013

[18] FMillar andDMora ldquoA finite elementmethod for the bucklingproblem of simply supported Kirchhoff platesrdquo Journal of Com-putational and Applied Mathematics vol 286 pp 68ndash78 2015

[19] YKCheung LGTham andWY Li ldquoFree vibration and staticanalysis of general plate by spline finite striprdquo ComputationalMechanics vol 3 no 3 pp 187ndash197 1988

[20] T Mizusawa ldquoBuckling of rectangular Midlin plates withtapered thickness by the spline strip methodrdquo InternationalJournal of Solids and Structures vol 30 no 12 pp 1663ndash16771993

[21] D J Dawe and SWang ldquoSpline finite strip analysis of the buck-ling and vibration of rectangular composite laminated platesrdquoInternational Journal of Mechanical Sciences vol 37 no 6pp 645ndash667 1995

[22] W J Wang Y P Tseng and K J Lin ldquoStability of laminatedplates using finite strip method based on a higher-order platetheoryrdquo Composite Structures vol 34 no 1 pp 65ndash76 1996

[23] SWang and D J Dawe ldquoBuckling of composite shell structuresusing the spline finite strip methodrdquo Composites Part B Engi-neering vol 30 no 4 pp 351ndash364 1999

[24] W-C Xie and A Ibrahim ldquoBuckling mode localization inrib-stiffened plates with misplaced stiffenersmdasha finite stripapproachrdquo Chaos Solitons amp Fractals vol 11 no 10 pp 1543ndash1558 2000

[25] Y K Cheung and L G Tham ldquoA review of the finite stripmethodrdquoProgress in Structural Engineering andMaterials vol 2no 3 pp 369ndash375 2000

[26] S Wang and Y Zhang ldquoVibration analysis of rectangularcomposite laminated plates using layerwise B-spline finite stripmethodrdquoComposite Structures vol 68 no 3 pp 349ndash358 2005

[27] L Yongqiang ldquoFree vibration analysis of plate using finite stripFourier p-elementrdquo Journal of Sound andVibration vol 294 no4 pp 1051ndash1059 2006

[28] H R Ovesy and H Assaee ldquoAn investigation on the post-buckling behavior of symmetric cross-ply laminated platesusing a semi-energy finite strip approachrdquoComposite Structuresvol 71 no 3-4 pp 365ndash370 2005

[29] H Assaee and H R Ovesy ldquoA multi-term semi-energy finitestrip method for post-buckling analysis of composite platesrdquo


International Journal for Numerical Methods in Engineering vol70 no 11 pp 1303ndash1323 2007

[30] H R Ovesy and J Fazilati ldquoStability analysis of compositelaminated plate and cylindrical shell structures using semi-analytical finite strip methodrdquo Composite Structures vol 89 no3 pp 467ndash474 2009

[31] H R Ovesy A Totounferoush and S A M GhannadpourldquoDynamic buckling analysis of delaminated composite platesusing semi-analytical finite strip methodrdquo Journal of Sound andVibration vol 343 pp 131ndash143 2015

[32] S Tarasovs and J Andersons ldquoBuckling of a coating strip offinite width bonded to elastic half-spacerdquo International Journalof Solids and Structures vol 45 no 2 pp 593ndash600 2008

[33] H C Bui ldquoSemi-analytical finite strip method based on theshallow shell theory in buckling analysis of cold-formed sec-tionsrdquoThin-Walled Structures vol 50 no 1 pp 141ndash146 2012

[34] G JHancock andCH Pham ldquoBuckling analysis of thin-walledsections under localised loading using the semi-analytical finitestrip methodrdquoThin-Walled Structures vol 86 pp 35ndash46 2015

[35] S Adany and B W Schafer ldquoBuckling mode decompositionof single-branched open cross-section members via finite stripmethod derivationrdquo Thin-Walled Structures vol 44 no 5 pp563ndash584 2006

[36] S Adany and B W Schafer ldquoBuckling mode decomposition ofsingle-branched open cross-section members via finite stripmethod application and examplesrdquoThin-Walled Structures vol44 no 5 pp 585ndash600 2006

[37] Z Li J C Batista Abreu J Leng S Adany and B W SchaferldquoReview constrained finite strip method developments andapplications in cold-formed steel designrdquo Thin-Walled Struc-tures vol 81 pp 2ndash18 2014

[38] S Adany andBW Schafer ldquoGeneralized constrained finite stripmethod for thin-walled members with arbitrary cross-sectionsecondary modes orthogonality examplesrdquoThin-Walled Struc-tures vol 84 pp 123ndash133 2014

[39] S Adany andBW Schafer ldquoGeneralized constrained finite stripmethod for thin-walled members with arbitrary cross-sectionprimary modesrdquo Thin-Walled Structures vol 84 pp 150ndash1692014

[40] Z LiM THanna S Adany and BW Schafer ldquoImpact of basisorthogonalization and normalization on the constrained FiniteStrip Method for stability solutions of open thin-walled mem-bersrdquoThin-Walled Structures vol 49 no 9 pp 1108ndash1122 2011

[41] R Uhrig ldquoThe transfer matrix method seen as one methodof structural analysis among othersrdquo Journal of Sound andVibration vol 4 no 2 pp 136ndash148 1966

[42] T J McDaniel and K B Eversole ldquoA combined finite element-transfer matrix structural analysis methodrdquo Journal of Soundand Vibration vol 51 no 2 pp 157ndash169 1977

[43] G Chiatti and A Sestieri ldquoAnalysis of static and dynamic struc-tural problems by a combined finite element-transfer matrixmethodrdquo Journal of Sound and Vibration vol 67 no 1 pp 35ndash42 1979

[44] M Ohga T Shigematsu and T Hara ldquoStructural analysis by acombined boundary element-transfer matrix methodrdquo Com-puters amp Structures vol 24 no 3 pp 385ndash389 1986

[45] M Ohga and T Shigematsu ldquoAnalysis of continuous platesby a combined boundary element-transfer matrix methodrdquoComputers amp Structures vol 36 no 1 pp 81ndash89 1990

[46] C Yuhua ldquoLarge deflection analysis of structures by animproved combined finite element-transfer matrix methodrdquoComputers amp Structures vol 55 no 1 pp 167ndash171 1995

[47] X Rui B He Y LuW Lu andGWang ldquoDiscrete time transfermatrix method for multibody system dynamicsrdquo MultibodySystem Dynamics vol 14 no 3-4 pp 317ndash344 2005

[48] B Rong ldquoEfficient dynamics analysis of large-deformationflexible beams by using the absolute nodal coordinate transfermatrix methodrdquoMultibody System Dynamics vol 32 no 4 pp535ndash549 2014

[49] F Chevillotte and R Panneton ldquoCoupling transfer matrixmethod to finite element method for analyzing the acoustics ofcomplex hollow body networksrdquo Applied Acoustics vol 72 no12 pp 962ndash968 2011

[50] D Li G Tang J Zhou and Y Lei ldquoBuckling analysis of aplate with built-in rectangular delamination by strip distributedtransfer functionmethodrdquoActaMechanica vol 176 no 3-4 pp231ndash243 2005

[51] Y Kang Y-G Lee and S-C Chen ldquoInstability analysis ofunsymmetrical rotor-bearing systems using the transfer matrixmethodrdquo Journal of Sound and Vibration vol 199 no 3 pp 381ndash400 1997

[52] E S Kameshki and S Syngellakis ldquoInelastic stability of rectan-gular frames by transfer matricesrdquo Computers amp Structures vol73 no 1ndash5 pp 373ndash383 1999

[53] MA Bradford andMAzhari ldquoBuckling of plates with differentend conditions using the finite strip methodrdquo Computers ampStructures vol 56 no 1 pp 75ndash83 1995

[54] E Ruocco and M Fraldi ldquoCritical behavior of flat and stiffenedshell structures through different kinematical models a com-parative investigationrdquoThin-Walled Structures vol 60 pp 205ndash215 2012

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

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

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


in one direction [20] The buckling stresses and naturalfrequencies of rectangular laminated plateswith arbitrary lay-ups and general boundary conditions can be predicted bythe improved spline finite strip which combines the super-strip concept [21]Then a higher-order shear deformable platefinite strip element is developed and employed to investigatethe critical buckling loads of composite laminated plates [22]The spline finite strip method and multilevel substructuringprocedures are combined for the buckling stress analysis ofcomposite laminated prismatic shell structures with generalboundary conditions [23] Bucklingmode localization in rib-stiffened plates with randomly misplaced stiffeners is studiedby finite strip method [24] Notably a comprehensive reviewof the finite strip method for structural analysis is given[25] A layerwise B-spline finite strip method is developedfor free vibration analysis of truly thick and thin compositelaminated plates [26] And a finite strip Fourier p-element isdeveloped to analyze the natural vibration characteristics ofthe thin plate [27] Based on the concept of the semienergyapproach the finite strip method can be proposed to ana-lyze the buckling [28] postbuckling [29] dynamic bucklingbehaviors of the laminated plates [30] and stability phe-nomena of cylindrical shell structures [31] The harmonicseries satisfying the boundary conditions in the loaded endsof thin-walled members are generally employed in Semi-Analytical Finite Strip Transfer Matrix Method (SAFSM)[32] And the buckling stress of the cold-formed sections canbe analyzed by SAFSM [33] which has been widely usedin computer softwares (such as THIN-WALL [34] CUFSM[35]) to develop the signature curves [36] of the bucklingstress versus buckling half wavelength for thin-walled mem-bers Furthermore the constrained finite strip method inno-vated from SAFSM is developed and applied in cold-formedsteel design [37] Generalized constrained finite strip methodfor thin-walled members with arbitrary cross-sections can beused to analyze secondary modes [38] and primary modes[39] The impacts of basis orthogonalization and normal-ization in constrained finite strip method are discussed forstability solution of open thin-walled members [40]

Transfer matrix method has been developed as an effec-tive tool for vibration analysis of engineering structuresespecially for chain connected system from topological per-spective [41] Then a combined finite element transfer matrixmethod is developed to study the statics [42] and dynamics[43] of structures Later a new transfer matrix method basedon the boundary element and transfer matrix technologyis proposed for the vibration analysis of two-dimensionalplate acted by uniform [44] and concentrated [45] loads Byintroducing the numerical integration nonlinear dynamicsof structures [46] multi-rigid-body system [47] and multi-flexible-body system [48] can be dealt with by transfer matrixmethod And a procedure which combinesmultiport transfermatrices andfinite elements has been developed to resolve theacoustic phenomena of automotive hollow body networks[49] It should be noted that three references enlightenus to start this investigation The buckling analysis of theplate with built-in rectangular delamination has been imple-mented by strip distributed transfer function method [50]And the transfer matrix method can be used to analyze

Y

Z

X

1 2 3 middot middot middot

middot middot middot

n minus 1

1 2 3 nb

wi wj

120579i 120579j

a

Ti Tj

y

x u

z w

Figure 1 Coordinates degree of freedom and loads of a typicalstrip

the instability in unsymmetrical rotor-bearing systems [51]and tall unbraced frames [52]

Here by combining two powerful methods of the semi-analytical finite strip and the transfermatrix an efficient tech-nology named semi-analytical Finite Strip Transfer MatrixMethod (FSTMM) for the buckling analysis of plates isdeveloped The text is organized as follows In Section 2the general theorem of the semi-analytical finite strip forbuckling analysis of plates is shown Section 3 presents semi-analytical Finite Strip Transfer Matrix Method for bucklinganalysis In Section 4 some results calculated by FSTMMandother methods are given which can validate the proposedmethod The conclusions are included in Section 5

2 The Semi-Analytical Finite Strip Analysis

21 Degree of Freedom and Shape Function In classicalKirchhoffplate theory themembrane or in-plane translationsare neglected and only the 119911-axisrsquos translation 119908 and the 119910-axisrsquos rotation 120579 are considered as local displacements [1]Theglobal coordinate system (119883119884 119885) and local coordinate sys-tem (119909 119910 119911) are actually the same in the plate model In finitestrip method (FSM) a thin plate as shown in Figure 1 canbe divided into many strip elements

We introduce a numbering system of finite strip modelshown in Figure 1 The total number of nodal lines is 119899therefore the total number of strips is 119899 minus 1 Generally oneinternal nodal line is always connected with two strips whilethe first and last nodal lines are only connected with onestrip This numbering system will be used to depict the statevector of the nodal line and the transfer matrix of the strip inSection 3

Here the plate is assumed to have two opposite loadededges with the simply supported boundary analytical trigo-nometric functions that satisfy the simply-simply supportedboundary condition of the loaded edges can be given toexpress the transverse deflection of the plate along thelongitudinal direction [35 37]

119884119901 (119910) = sin119901120587119910

119886 119901 = 1 2 3 119899 (1)

where 119901 is the half-wave number which also stands forcertain half wavelength along the longitudinal direction 120587 isthe circumference ratio 119910 is the longitudinal coordinate inlocal system 119886 is the length of the plate



N (119909)

= [1 minus31199092

1198872+21199093

1198873119909 minus

21199092

119887+1199093

1198872

31199092

1198872minus21199093

1198873

1199093

1198872minus1199092

119887

]

(2)



119908 (119909 119910) =

119898

sum

119901=1

N119901 (119909 119910) 120575119901 (3)


N119901 (119909 119910) = N (119909) 119884119901 (119910) (4)



120575119901= [119908119901

119894120579119901

119894119908119901

119895120579119901

119895 ]119879

(5)



120576119861 = minus[1205972119908

1205971199092

1205972119908

120597119910221205972119908

120597119909120597119910]

119879

(6)


120576119861 =

119898

sum

119901=1

N10158401015840119901120575119901 (7)

where N10158401015840119901= minus[120597N119879

1199011205971199092120597N11987911990112059711991022(120597N119879119901120597119909120597119910)]


119880 =1

2int

119881

120576119879120590 d119881 (8)


119880 =1

2int

119886

0

int

119887

0

120576119879

119861D119861120576119861d119909 d119910 (9)


119880 =1

2

119898

sum

119901=1

119898

sum

119902=1

120575119901119879k119901119902119890120575119902 (10)


k119901119902119890= int

119886

0

int

119887

0

N10158401015840119879119901D119861N10158401015840

119902d119909 d119910 (11)




k119901119902119890

=

int

119886

0

int

119887

0

N10158401015840119879119901D119861N10158401015840

119902d119909 d119910 119901 = 119902 = 1 2 3 119898

0 119901 = 119902

(12)


119879119909 = 119879119894 minus (119879119894 minus 119879119895)119909

119887 (13)


k119901119902119892= int

119886

0

int

119887

0

(119879119894 minus (119879119894 minus 119879119895)119909

119887)N1015840119879119901N1015840119902d119909 d119910 (14)



namely


k119901119902119892=

int

119886

0

int

119887

0

(119879119894 minus (119879119894 minus 119879119895)119909

119887)N1015840119879119901N1015840119902d119909 d119910 119901 = 119902 = 1 2 3 119898

0 119901 = 119902

(15)





(k119901119901119890+ k119901119901119892) 120575119901= R119901 (16)

where k119901119901119890



R119901 = [119865119901119894 119872119901

119894119865119901

119895119872119901

119895 ]119879

(17)

where 119865119901119894(and 119865119901




119894 (and 119895) and119872119901119894(and119872119901






119879 (119909) = 1205821198790 (119909) (18)





k119901119901119892= 120582 k119901119901119892

100381610038161003816100381610038161198790 (19)


(k119901119901119890+ 120582 k119901119901119892

100381610038161003816100381610038161198790) 120575119901= R119901 (20)


K = k119901119901119890+ 120582 k119901119901119892

100381610038161003816100381610038161198790 (21)


and k119901119901119892|1198790



Z119897119899 = [120575119901119879

119897R119901119879119897]119879

119899(119897 = 119894 119895) (22)


119897= [119908119901

119897120579119901



119897119872119901

119897]119879 is the



Z119895119899 = U119890119899Z119894119899 (23)


U119890119899= [


12



K = [[

K112times2

K122times2

K212times2

K222times2

]

]

(25)


Z119897119899+1 = [1 0

0 minus1

]Z119897119899 (26)




119879

119885c = [0 0 119865 119872]119879

119885f = [119908 120579 0 0]119879



Eigen-matrix [

[

11988012 11988013

11988042 11988043

]

]

[

[

11988013 11988014

11988023 11988024

]

]

[

[

11988012 11988013

11988022 11988023

]

]

[

[

11988032 11988033

11988042 11988043

]

]

[

[

11988011 11988012

11988021 11988022

]

]

[

[

11988031 11988032

11988041 11988042

]

]


Z119897+1119899+1 = U119890119897119899Z119897119899 (27)


U119890119897119899= [


12


12

] (28)



Z119899119899minus1 = UallZ11


(29)



[0 120579 119865 0]119879

119871= Uall [0 120579 119865 0]

119879

119865 (30)


[

0

0

]

119879

119871

= [

11988012 11988013

11988042 11988043

][

120579

119865

]

119879

119865

(31)


det([11988012 11988013

11988042 11988043

]) = 0 (32)





21198641199053121198872(1 minus ]2) =







Theoretical results


4

5

6

7Bu

cklin

g co

effici

ent120582



Theoretical results

0

1

2

3

4

5

6

7

Buck

ling

coeffi

cien

t120582




120582 = 54


5

6

7

8

9

Buck

ling

coeffi

cien

t120582



120582 = 7


6

8

10

12

14Bu

cklin

g co

effici

ent120582



120582 = 044


0

1

2

3

4

5

6

7

Buck

ling

coeffi

cien

t120582



120582 = 128


1

2

3

4

5

6

7

8

Buck

ling

coeffi

cien

t120582













FEM results (solid)


4

6

8

10

12

14

16

18

20

Buck

ling

coeffi

cien

t120582

120582 = 6







0

2

4

6

8

10

12

14Bu

cklin

g co

effici

ent120582














05 1 15 2 25 3 35 4 45 5

35

4

45

5

55

6

65

7

75

8

Buck

ling

coeffi

cien

t120582



(a) Results of FEM


35

4

45

5

55

6

65

7

75

8Bu

cklin

g co

effici

ent120582







5 Conclusions






Acknowledgment


References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











N (119909)

= [1 minus31199092

1198872+21199093

1198873119909 minus

21199092

119887+1199093

1198872

31199092

1198872minus21199093

1198873

1199093

1198872minus1199092

119887

]

(2)



119908 (119909 119910) =

119898

sum

119901=1

N119901 (119909 119910) 120575119901 (3)


N119901 (119909 119910) = N (119909) 119884119901 (119910) (4)



120575119901= [119908119901

119894120579119901

119894119908119901

119895120579119901

119895 ]119879

(5)



120576119861 = minus[1205972119908

1205971199092

1205972119908

120597119910221205972119908

120597119909120597119910]

119879

(6)


120576119861 =

119898

sum

119901=1

N10158401015840119901120575119901 (7)

where N10158401015840119901= minus[120597N119879

1199011205971199092120597N11987911990112059711991022(120597N119879119901120597119909120597119910)]


119880 =1

2int

119881

120576119879120590 d119881 (8)


119880 =1

2int

119886

0

int

119887

0

120576119879

119861D119861120576119861d119909 d119910 (9)


119880 =1

2

119898

sum

119901=1

119898

sum

119902=1

120575119901119879k119901119902119890120575119902 (10)


k119901119902119890= int

119886

0

int

119887

0

N10158401015840119879119901D119861N10158401015840

119902d119909 d119910 (11)




k119901119902119890

=

int

119886

0

int

119887

0

N10158401015840119879119901D119861N10158401015840

119902d119909 d119910 119901 = 119902 = 1 2 3 119898

0 119901 = 119902

(12)


119879119909 = 119879119894 minus (119879119894 minus 119879119895)119909

119887 (13)


k119901119902119892= int

119886

0

int

119887

0

(119879119894 minus (119879119894 minus 119879119895)119909

119887)N1015840119879119901N1015840119902d119909 d119910 (14)



namely


k119901119902119892=

int

119886

0

int

119887

0

(119879119894 minus (119879119894 minus 119879119895)119909

119887)N1015840119879119901N1015840119902d119909 d119910 119901 = 119902 = 1 2 3 119898

0 119901 = 119902

(15)





(k119901119901119890+ k119901119901119892) 120575119901= R119901 (16)

where k119901119901119890



R119901 = [119865119901119894 119872119901

119894119865119901

119895119872119901

119895 ]119879

(17)

where 119865119901119894(and 119865119901




119894 (and 119895) and119872119901119894(and119872119901






119879 (119909) = 1205821198790 (119909) (18)





k119901119901119892= 120582 k119901119901119892

100381610038161003816100381610038161198790 (19)


(k119901119901119890+ 120582 k119901119901119892

100381610038161003816100381610038161198790) 120575119901= R119901 (20)


K = k119901119901119890+ 120582 k119901119901119892

100381610038161003816100381610038161198790 (21)


and k119901119901119892|1198790



Z119897119899 = [120575119901119879

119897R119901119879119897]119879

119899(119897 = 119894 119895) (22)


119897= [119908119901

119897120579119901



119897119872119901

119897]119879 is the



Z119895119899 = U119890119899Z119894119899 (23)


U119890119899= [


12



K = [[

K112times2

K122times2

K212times2

K222times2

]

]

(25)


Z119897119899+1 = [1 0

0 minus1

]Z119897119899 (26)




119879

119885c = [0 0 119865 119872]119879

119885f = [119908 120579 0 0]119879



Eigen-matrix [

[

11988012 11988013

11988042 11988043

]

]

[

[

11988013 11988014

11988023 11988024

]

]

[

[

11988012 11988013

11988022 11988023

]

]

[

[

11988032 11988033

11988042 11988043

]

]

[

[

11988011 11988012

11988021 11988022

]

]

[

[

11988031 11988032

11988041 11988042

]

]


Z119897+1119899+1 = U119890119897119899Z119897119899 (27)


U119890119897119899= [


12


12

] (28)



Z119899119899minus1 = UallZ11


(29)



[0 120579 119865 0]119879

119871= Uall [0 120579 119865 0]

119879

119865 (30)


[

0

0

]

119879

119871

= [

11988012 11988013

11988042 11988043

][

120579

119865

]

119879

119865

(31)


det([11988012 11988013

11988042 11988043

]) = 0 (32)





21198641199053121198872(1 minus ]2) =







Theoretical results


4

5

6

7Bu

cklin

g co

effici

ent120582



Theoretical results

0

1

2

3

4

5

6

7

Buck

ling

coeffi

cien

t120582




120582 = 54


5

6

7

8

9

Buck

ling

coeffi

cien

t120582



120582 = 7


6

8

10

12

14Bu

cklin

g co

effici

ent120582



120582 = 044


0

1

2

3

4

5

6

7

Buck

ling

coeffi

cien

t120582



120582 = 128


1

2

3

4

5

6

7

8

Buck

ling

coeffi

cien

t120582













FEM results (solid)


4

6

8

10

12

14

16

18

20

Buck

ling

coeffi

cien

t120582

120582 = 6







0

2

4

6

8

10

12

14Bu

cklin

g co

effici

ent120582














05 1 15 2 25 3 35 4 45 5

35

4

45

5

55

6

65

7

75

8

Buck

ling

coeffi

cien

t120582



(a) Results of FEM


35

4

45

5

55

6

65

7

75

8Bu

cklin

g co

effici

ent120582







5 Conclusions






Acknowledgment


References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










k119901119902119892=

int

119886

0

int

119887

0

(119879119894 minus (119879119894 minus 119879119895)119909

119887)N1015840119879119901N1015840119902d119909 d119910 119901 = 119902 = 1 2 3 119898

0 119901 = 119902

(15)





(k119901119901119890+ k119901119901119892) 120575119901= R119901 (16)

where k119901119901119890



R119901 = [119865119901119894 119872119901

119894119865119901

119895119872119901

119895 ]119879

(17)

where 119865119901119894(and 119865119901




119894 (and 119895) and119872119901119894(and119872119901






119879 (119909) = 1205821198790 (119909) (18)





k119901119901119892= 120582 k119901119901119892

100381610038161003816100381610038161198790 (19)


(k119901119901119890+ 120582 k119901119901119892

100381610038161003816100381610038161198790) 120575119901= R119901 (20)


K = k119901119901119890+ 120582 k119901119901119892

100381610038161003816100381610038161198790 (21)


and k119901119901119892|1198790



Z119897119899 = [120575119901119879

119897R119901119879119897]119879

119899(119897 = 119894 119895) (22)


119897= [119908119901

119897120579119901



119897119872119901

119897]119879 is the



Z119895119899 = U119890119899Z119894119899 (23)


U119890119899= [


12



K = [[

K112times2

K122times2

K212times2

K222times2

]

]

(25)


Z119897119899+1 = [1 0

0 minus1

]Z119897119899 (26)




119879

119885c = [0 0 119865 119872]119879

119885f = [119908 120579 0 0]119879



Eigen-matrix [

[

11988012 11988013

11988042 11988043

]

]

[

[

11988013 11988014

11988023 11988024

]

]

[

[

11988012 11988013

11988022 11988023

]

]

[

[

11988032 11988033

11988042 11988043

]

]

[

[

11988011 11988012

11988021 11988022

]

]

[

[

11988031 11988032

11988041 11988042

]

]


Z119897+1119899+1 = U119890119897119899Z119897119899 (27)


U119890119897119899= [


12


12

] (28)



Z119899119899minus1 = UallZ11


(29)



[0 120579 119865 0]119879

119871= Uall [0 120579 119865 0]

119879

119865 (30)


[

0

0

]

119879

119871

= [

11988012 11988013

11988042 11988043

][

120579

119865

]

119879

119865

(31)


det([11988012 11988013

11988042 11988043

]) = 0 (32)





21198641199053121198872(1 minus ]2) =







Theoretical results


4

5

6

7Bu

cklin

g co

effici

ent120582



Theoretical results

0

1

2

3

4

5

6

7

Buck

ling

coeffi

cien

t120582




120582 = 54


5

6

7

8

9

Buck

ling

coeffi

cien

t120582



120582 = 7


6

8

10

12

14Bu

cklin

g co

effici

ent120582



120582 = 044


0

1

2

3

4

5

6

7

Buck

ling

coeffi

cien

t120582



120582 = 128


1

2

3

4

5

6

7

8

Buck

ling

coeffi

cien

t120582













FEM results (solid)


4

6

8

10

12

14

16

18

20

Buck

ling

coeffi

cien

t120582

120582 = 6







0

2

4

6

8

10

12

14Bu

cklin

g co

effici

ent120582














05 1 15 2 25 3 35 4 45 5

35

4

45

5

55

6

65

7

75

8

Buck

ling

coeffi

cien

t120582



(a) Results of FEM


35

4

45

5

55

6

65

7

75

8Bu

cklin

g co

effici

ent120582







5 Conclusions






Acknowledgment


References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119879

119885c = [0 0 119865 119872]119879

119885f = [119908 120579 0 0]119879



Eigen-matrix [

[

11988012 11988013

11988042 11988043

]

]

[

[

11988013 11988014

11988023 11988024

]

]

[

[

11988012 11988013

11988022 11988023

]

]

[

[

11988032 11988033

11988042 11988043

]

]

[

[

11988011 11988012

11988021 11988022

]

]

[

[

11988031 11988032

11988041 11988042

]

]


Z119897+1119899+1 = U119890119897119899Z119897119899 (27)


U119890119897119899= [


12


12

] (28)



Z119899119899minus1 = UallZ11


(29)



[0 120579 119865 0]119879

119871= Uall [0 120579 119865 0]

119879

119865 (30)


[

0

0

]

119879

119871

= [

11988012 11988013

11988042 11988043

][

120579

119865

]

119879

119865

(31)


det([11988012 11988013

11988042 11988043

]) = 0 (32)





21198641199053121198872(1 minus ]2) =







Theoretical results


4

5

6

7Bu

cklin

g co

effici

ent120582



Theoretical results

0

1

2

3

4

5

6

7

Buck

ling

coeffi

cien

t120582




120582 = 54


5

6

7

8

9

Buck

ling

coeffi

cien

t120582



120582 = 7


6

8

10

12

14Bu

cklin

g co

effici

ent120582



120582 = 044


0

1

2

3

4

5

6

7

Buck

ling

coeffi

cien

t120582



120582 = 128


1

2

3

4

5

6

7

8

Buck

ling

coeffi

cien

t120582













FEM results (solid)


4

6

8

10

12

14

16

18

20

Buck

ling

coeffi

cien

t120582

120582 = 6







0

2

4

6

8

10

12

14Bu

cklin

g co

effici

ent120582














05 1 15 2 25 3 35 4 45 5

35

4

45

5

55

6

65

7

75

8

Buck

ling

coeffi

cien

t120582



(a) Results of FEM


35

4

45

5

55

6

65

7

75

8Bu

cklin

g co

effici

ent120582







5 Conclusions






Acknowledgment


References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Theoretical results


4

5

6

7Bu

cklin

g co

effici

ent120582



Theoretical results

0

1

2

3

4

5

6

7

Buck

ling

coeffi

cien

t120582




120582 = 54


5

6

7

8

9

Buck

ling

coeffi

cien

t120582



120582 = 7


6

8

10

12

14Bu

cklin

g co

effici

ent120582



120582 = 044


0

1

2

3

4

5

6

7

Buck

ling

coeffi

cien

t120582



120582 = 128


1

2

3

4

5

6

7

8

Buck

ling

coeffi

cien

t120582













FEM results (solid)


4

6

8

10

12

14

16

18

20

Buck

ling

coeffi

cien

t120582

120582 = 6







0

2

4

6

8

10

12

14Bu

cklin

g co

effici

ent120582














05 1 15 2 25 3 35 4 45 5

35

4

45

5

55

6

65

7

75

8

Buck

ling

coeffi

cien

t120582



(a) Results of FEM


35

4

45

5

55

6

65

7

75

8Bu

cklin

g co

effici

ent120582







5 Conclusions






Acknowledgment


References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















FEM results (solid)


4

6

8

10

12

14

16

18

20

Buck

ling

coeffi

cien

t120582

120582 = 6







0

2

4

6

8

10

12

14Bu

cklin

g co

effici

ent120582














05 1 15 2 25 3 35 4 45 5

35

4

45

5

55

6

65

7

75

8

Buck

ling

coeffi

cien

t120582



(a) Results of FEM


35

4

45

5

55

6

65

7

75

8Bu

cklin

g co

effici

ent120582







5 Conclusions






Acknowledgment


References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













0

2

4

6

8

10

12

14Bu

cklin

g co

effici

ent120582














05 1 15 2 25 3 35 4 45 5

35

4

45

5

55

6

65

7

75

8

Buck

ling

coeffi

cien

t120582



(a) Results of FEM


35

4

45

5

55

6

65

7

75

8Bu

cklin

g co

effici

ent120582







5 Conclusions






Acknowledgment


References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













05 1 15 2 25 3 35 4 45 5

35

4

45

5

55

6

65

7

75

8

Buck

ling

coeffi

cien

t120582



(a) Results of FEM


35

4

45

5

55

6

65

7

75

8Bu

cklin

g co

effici

ent120582







5 Conclusions






Acknowledgment


References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgment


References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Semi-Analytical Finite Strip Transfer...

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Transcript of Research Article Semi-Analytical Finite Strip Transfer...