Nation Taiwan Ocean University
Department of Harbor and River
April 18, 2023 page 1
A Semi-Analytical Approach for Stress Concentration of Cantilever Beams with Holes under Bending
半解析法求解含圓型孔洞懸臂梁之應力集中問題
Jeng-Tzong Chen
Life-time Distinguished Professor
National Taiwan Ocean University
Keelung, Taiwan
JoMpresent.ppt
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page2
Outline
MotivationPresent method
FormulationExpansions of fundamental solution and boundary densityFlowchart
Numerical examplesA circular bar with three circular holes (torsion)A circular beam with four circular holes (bending)
Conclusions
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page3
MotivationMotivation
TorsionTorsion problem: problem: CaulkCaulk (1983) said that the (1983) said that the Ling’sLing’s result (1947) result (1947) may be not correct (three holes)may be not correct (three holes)
Bending problem: Steele (1992) said that the Naghdi’s result (1991) may be not correct (yes) (four holes)
Who is correct ?
T
Q
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page4
MotivationMotivation
Numerical methods for engineering problemsNumerical methods for engineering problems
FDM / FEM / BEM / BIEM / Meshless methodFDM / FEM / BEM / BIEM / Meshless method
BEM / BIEMBEM / BIEM
Treatment of siTreatment of singularity and hyngularity and hypersingularitypersingularity
Boundary-layer Boundary-layer effecteffect
Ill-posed modelIll-posed modelConvergence Convergence raterate
CPV & HPV Nearly-singular Linear algebraic order Fictitious BEMNull-field BIE
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page5
哲人日已遠 典型在宿昔 (1909-1993)
省立中興大學第一任校長
林致平校長(民國五十年 ~民國五十二年 )
林致平所長 (中研院數學所 )
林致平院士 (中研院 )
數學力學家 (挖洞專家 )
全解析 半解析 全數值
2
3
7
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page6
Motivation
BEM / BIEMBEM / BIEM
Improper integralImproper integral
Singularity & hypersingularitySingularity & hypersingularity RegularityRegularity
Bump contourBump contourFictitious Fictitious boundaryboundary
Collocation Collocation pointpoint
Fictitious BEMFictitious BEM
Null-field approachNull-field approach
CPV and HPVCPV and HPVIll-posedIll-posed
Guiggiani (1995)Guiggiani (1995)
Waterman (1965)Waterman (1965)
Achenbach Achenbach et al.et al. (1988) (1988)interior
exterior
Main idea
山不轉 路轉
路不轉 分內外核函數
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page7
Present approach
(s, x)iK
(s, x)eK
(s, x(x) (s) (s))B
dBKj y=ò
Fundamental solutionFundamental solution
(s, x), s x
(s, x), x s
i
i
K
K
ìï ³ïíï >ïîln x s-
No principal valueNo principal value
Advantages of degenerate kernel1. No principal value2. Well-posed3. Exponential convergence4. Free of boundary-layer effect5. Mesh free
Degenerate kernelDegenerate kernel
CPV and HPVCPV and HPV
路不轉 分內外核函數
interior
exterior
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page8
Outline
MotivationPresent method
FormulationExpansions of fundamental solution and boundary densityFlowchart
Numerical examplesA circular bar with three circular holes (torsion)A circular beam with four circular holes (bending)
Conclusions
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page9
Conventional BIEM and current method
s
s
(s, x) ln x s ln
(s, x)(s, x)
n
(s)(s)
n
U r
UT
jy
= - =
¶=
¶
¶=
¶
D
cx DÎ
0 (s, x) (s) (s) (s, x) (s) (s), x c
B BT dB U dB Dj y= - Îò ò
(x) . . . (s, x) (s) (s) . . . (s, x) (s) (s), xB B
C PV T dB R PV U dB Bpj j y= - Îò ò
2 (x) (s, x) (s) (s) (s, x) (s) (s), xB BT dB U dB Dpj j y= - Îò ò
2 (x) (s, x) (s) (s) (s, x) (s) (s), xB BT dB U dB D Bpj j y= - Î Èò ò
0 (s, x) (s) (s) (s, x) (s) (s), x c
B BT dB U d D BBj y= - Î Èò ò
1969-2005
Current Degenerate kernelDegenerate kernel
interiorexterior
Main idea
路不轉 分內外核函數
x DÎx BÎ
Conventional BEM
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page10
Convergence rate between the present method and conventional BEM
Degenerate kernelDegenerate kernel
Fourier series expansionFourier series expansion
Fundamental Fundamental solutionsolution
Boundary Boundary densitydensity
Convergence Convergence raterate
Present methodPresent method Conventional BEM (1969-2005)Conventional BEM (1969-2005)
Two-point function (closed-form)Two-point function (closed-form)
(s, x) ln ln x sU r= = -
Constant, linear, Constant, linear, quadratic elementsquadratic elements
Exponential convergenceExponential convergenceLinear algebraic convergenceLinear algebraic convergence
(s, x) (s) (x), s x
(s, x)(s, x) (x) (s), x s
ij j
j
ej j
j
U A B
UU A B
ìï = ³ïïï=íï = >ïïïî
å
å
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page11
Degenerate (separate) form of fundamental solution (2-D)
s( , )R q
R
r
rx( , )r f
x( , )r f
1
1
1( , ; , ) ln ( ) cos ( ),
(s, x) ln1
( , ; , ) ln ( ) cos ( ),
i m
m
e m
m
U R R m Rm R
U rR
U R m Rm
rq r f q f r
q r f r q f rr
¥
=
¥
=
ìïï = - - ³ïïïï= =íïï = - - >ïïïïî
å
å
o
iU
eU
s
x
2
s x
(s, x)(s, x)
n
(s, x)(s, x)
n
(s, x)(s, x)
n n
UT
UL
UM
¶º
¶
¶º
¶
¶º
¶ ¶
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page12
Outline
MotivationPresent method
FormulationExpansions of fundamental solution and boundary densityFlowchart
Numerical examplesA circular bar with three circular holes (torsion)A circular beam with four circular holes (bending)
Conclusions
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page13
Flowchart of the present method
0 [ (s, x) (s) (s, x) (s)] (s)B
T u U t dB= -òDegenerate kernel Fourier series
Collocation point and matching B.C.
Adaptive observer system
Linear algebraic equation
Fourier coefficientsPotential of
domain point
Stress field
Vector decompositio
n
Numerical
Analytical
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page14
Outline
MotivationPresent method
FormulationExpansions of fundamental solution and boundary densityFlowchart
Numerical examplesA circular bar with three circular holes (torsion)A circular beam with four circular holes (bending)
Conclusions
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page15
Torsional rigidity (Ling’s problem)
Caulk (First-order Approximate)
0.8739 0.8741 0.7261
Caulk (BIE formulation)
0.8713 0.8732 0.7261
Ling’s results 0.8809 0.8093 0.7305
Present method (L=10)
0.8712 0.8732 0.7244
Because there is no apparent reason for the unusually large difference in the second example, Ling’s rather lengthy calculations are probably in error here. --ASME JAM
?
TT
T
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page16
Bending problem for a cantilever beam
b
a
R
2Y 1Y
ABCD
O
E
Q
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page17
Stress concentration at point B
Present Present methodmethod
0.4 0.5 0.6 0.7
a
1.5
2
2.5
3
3.5
Sc
0.4 0.5 0.6 0.7
a
1.5
2
2.5
3
3.5
Sc
Steele & Steele & BirdBird
The two approaches disagree by as much 11%. The grounds for this discrepancy have not yet been identified.
--ASME Applied Mechanics Review
Θ=π/8 Θ=3π/8
a
B
Θ
a a
Steele Steele
Q
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page18
Outlines
MotivationPresent method
FormulationExpansions of fundamental solution and boundary densityFlowchart
Numerical examplesA circular bar with three circular holes (torsion)A circular beam with four circular holes (bending)
Conclusions
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page19
Conclusions
Null-field integral equation in conjunction with Null-field integral equation in conjunction with degenerate kernels and Fourier seriesdegenerate kernels and Fourier series
Singularity free, boundary-layer effect free, Singularity free, boundary-layer effect free, exponential convergence, mesh free and well-exponential convergence, mesh free and well-posed modelposed model
Arbitrary numberArbitrary number of holes, various radii of holes, various radii and positionsand positions ( 三任意 : 數目 大小 與 位置 )
( 五優點 )
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page20
Conclusions
Torsion problem: Torsion problem: CaulkCaulk, 1983 (yes) Ling, 1947 (?) , 1983 (yes) Ling, 1947 (?)
(three holes)(three holes)
Bending problem: Steele, 1992 (?) Naghdi, 1991 (yes)
(four holes)
T
Q
Nation Taiwan Ocean University
Department of Harbor and River
April 18, 2023 page 21
Thanks for your kind attentions.
You can get more information from our website.
http://msvlab.hre.ntou.edu.tw/
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page22
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page23URL: http://ind.ntou.edu.tw/~msvlab E-mail: [email protected] 海洋大學工學院河工所力學聲響振動實驗室 nullsystem2008-9.ppt`
Elasticity & Crack Problem
Laplace Equation
Research topics of NTOU / MSV LAB on null-field BIEMs (2003-2008)
Navier Equation
Null-field BIEM
Biharmonic Equation
Previous research and project
Current work
Plate with circular holes
BiHelmholtz EquationHelmholtz Equation
(Potential flow)(Torsion)
(Anti-plane shear)(Degenerate scale)
(Inclusion)(Piezoleectricity) (Beam bending)
Torsion bar (Inclusion)Imperfect interface
Image method(Green function)
Green function of half plane (Hole and inclusion)
Interior and exteriorAcoustics
SH wave (exterior acoustics)(Inclusions)
Free vibration of plateIndirect BIEM李為民
ASME JAM 2006 蕭嘉俊MRC 2007,CMES 2006EABE 2006
ASME 2007 EABE 2006 CMAME 2007
SDEE 2008
JCA 2008
NUMPDE 2008
JSV 2007
SH wave
Impinging canyons
Degenerate kernel for ellipse
ICOME 2006
Added mass陳義麟
李應德 CFD 14 Water wave impinging circul
ar cylinders
Screw dislocation(addition theorem)
周克勳
Green function foran annular plate
SH wave Impinging hill
Green function of`circular inclusion (special case: statics)
Effective conductivity
CMC 2008
Stokes flow
Free vibration of plate Direct BIEM 李為民
Flexural wave of plate with one and two holes 李為民
CMES 2008 柯佳男
ASME JAM 2008 JoM 2008 康康Comp. Mech. 2008
IJNME 2008 蕭嘉俊
ModifiedHelmholtz Equation
CSSV 2008
Dynamic Green’s function for an infinite plate with a hole
Flexural wave of plate with two inclusions 李為民
Source 林羿州(two cylinders)
Concentric sphere高聖凱
Two spheres radiation
李應德
Annular Green’s function
(Trefftz method and MFS) 祥志與小島
JoM 2007 陳柏源
APCOM 2007
Free vibration of plateReal-part BIEM
李為民
EABE 2007
EABE 2008 rev.
Green function of`circular boundary (statics:superposition)
MRC 2008 rev. 周克勳
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page24
Top 25 scholars on BEM/BIEM
北京清華姚振漢教授提供
USA 劉毅軍教授
NTOU/MSV Taiwan 海洋大學 陳正宗終身特聘教授
北京清華大學工程力學系 -姚振漢教授
高海大造船系 -陳義麟博士
台大土木系 -楊德良終身特聘教授
宜蘭大學土木系陳桂鴻博士
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page25
Some researchers on BEM (1012)Chen (1986) 565 citings in total
Hong and Chen (1988 ) 78 citings ASCE EM
Portela and Aliabadi (1992) 212 citings IJNME
Mi and Aliabadi (1994)
Wen and Aliabadi (1995)
Chen and Chen (1995) 新竹清華 Yao (2005) 北京清華 黎在良等 --- 斷裂力學邊界數值方法 (1996) 周慎杰 (1999)
Chen and Hong (1999) 88 citings ASME AMR
Niu and Wang (2001)
Kuhn G, Wrobel L C, Mukherjee S, Tuhkuri J, Gray L J
Yu D H, Zhu J L, Chen Y Z, Tan R J …
NTUCE
cite
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page26
Engineering problem with holes, inclusions and cracks
Straight boundaryStraight boundary
Degenerate boundaryDegenerate boundary
Circular inclusionCircular inclusion
Elliptic holeElliptic hole
[Mathieu [Mathieu function]function]
[Legendre [Legendre polynomial]polynomial]
[Chebyshev polynomial][Chebyshev polynomial]
[Fourier series][Fourier series]
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page27
Literature review – analytical solutions for problems with circular boundaries
Key pointKey point Main applicationMain application AuthorAuthor
Conformal mappingConformal mapping Torsion problemTorsion problemIn-plane electrostaticsIn-plane electrostaticsAnti-plane elasticityAnti-plane elasticity
Chen & Weng (2001)Chen & Weng (2001)Emets & Onofrichuk (1996)Emets & Onofrichuk (1996)Budiansky & Carrier (1984)Budiansky & Carrier (1984)Steif (1989)Steif (1989)Wu & Funami (2002)Wu & Funami (2002)Wang & Zhong (2003)Wang & Zhong (2003)
Bi-polar coordinateBi-polar coordinate Electrostatic potentialElectrostatic potentialElasticityElasticity
Lebedev Lebedev et al.et al. (1965) (1965)Howland & Knight (1939)Howland & Knight (1939)
MMööbius transformatibius transformationon
Anti-plane piezoelectricity Anti-plane piezoelectricity & elasticity& elasticity
Honein Honein et al.et al. (1992) (1992)
Complex potential Complex potential approachapproach
Anti-plane piezoelectricityAnti-plane piezoelectricity Wang & Shen (2001)Wang & Shen (2001)
Those Those analytical methodsanalytical methods are only limited to are only limited to doubly connected regionsdoubly connected regions even to even toconformal connected regionsconformal connected regions..
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page28
Literature review - Fourier series approximation
AuthorAuthor Main applicationMain application Key pointKey point
LingLing
(1943)(1943)
Torsion of a circular tubeTorsion of a circular tube
Caulk Caulk et al.et al.
(1983)(1983)
Steady heat conduction with Steady heat conduction with circular holescircular holes
Special BIEMSpecial BIEM
Bird and SteeleBird and Steele
(1992)(1992)
Harmonic and biharmonic probleHarmonic and biharmonic problems with circular holesms with circular holes
Trefftz methodTrefftz method
Mogilevskaya Mogilevskaya et al.et al.
(2002)(2002)
Elasticity problems with circular Elasticity problems with circular holes holes oror inclusions inclusions
Galerkin methodGalerkin method
However, no one employed the However, no one employed the null-field approachnull-field approach and and degenerate degenerate kernelkernel to fully capture the circular boundary. to fully capture the circular boundary.
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page29
Vector decomposition technique for potential gradient
x
z
z x-
nt
t
n
True normal vectorTrue normal vector
(s, x) 1 (s, x)(s, x) cos( ) cos( )
2
U ULr
pz x z x
r r f¶ ¶
= - + - +¶ ¶
(s, x) 1 (s, x)(s, x) cos( ) cos( )
2
T TM r
pz x z x
r r f¶ ¶
= - + - +¶ ¶
Special case (concentric case) :Special case (concentric case) : z x=
(s, x)(s, x)
ULr r
¶=
¶(s, x)
(s, x)T
M r r¶
=¶
Non-concentric case:Non-concentric case:
(x)2 (s, x) (s) (s) (s, x) (s) (s), x
n(x)
2 (s, x) (s) (s) (s, x) (s) (s), xt
B B
B B
M dB L dB D
M BdB L d D
B
B
r r
ff
jp j y
jp j y
¶= - Î
¶¶
= - ζ
È
È
ò ò
ò ò
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page30
Explicit form of each submatrix and vector
0 1 11 1 1 1 1
0 1 12 2 2 2 2
0 1 13 3 3 3 3
0 1 12 2 2 2
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
c c s Lc Lsjk jk jk jk jkc c s Lc Ls
jk jk jk jk jkc c s Lc Ls
jk jk jk jk jkjk
c c s Lc Lsjk L jk L jk L jk L jk
U U U U U
U U U U U
U U U U U
U U U U U
ff ff f
ff ff f
ff ff f
ff ff
é ù=ê úë ûU
L
L
L
M M M O M M
L 20 1 1
2 1 2 1 2 1 2 1 2 1
( )
( ) ( ) ( ) ( ) ( )L
c c s Lc Lsjk L jk L jk L jk L jk LU U U U U
f
ff ff f+ + + + +
é ùê úê úê úê úê úê úê úê úê úê úê úê úë ûL
{ } { }0 1 1
Tk k k k kk L Lp p q p q= Ly
1f
2f
3f
2Lf
2 1Lf +
Fourier coefficientsFourier coefficients
Truncated terms of Truncated terms of Fourier seriesFourier series
Number of collocation pointsNumber of collocation points
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page31
Outlines
Motivation and literature reviewPresent method
Expansions of fundamental solution and boundary densityAdaptive observer systemVector decomposition techniqueLinear algebraic equation
Numerical examplesA circular beam with two circular holesA circular beam with four circular holes
Conclusions
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page32
Two holes problem
0 1 2 3 4 5
D /2a
2
3
4
5
6
7
8
9
Sc
Tw o ho les
O ne hole
Present methodPresent method
Steele & Bird’s result [6]Steele & Bird’s result [6]
Point P
Sc
of p
oint
PS
c of
poi
nt PD: Distance between two holes
a: radius of holes
R: radius of circular beam
D
a
R
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page33
Contour of stress concentration Steele & Bird’s result [6]Steele & Bird’s result [6] Present methodPresent method
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page34
Expansions of fundamental solution and boundary density
(s, x) (s) (x), s x
(s, x)(s, x) (x) (s), x s
ij j
j
ej j
j
U A B
UU A B
ìï = ³ïïï=íï = >ïïïî
å
å
01
01
(s) ( cos sin ), s
(s) ( cos sin ), s
L
n nn
L
n nn
a a n b n B
p p n q n B
j q q
y q q
=
=
= + + Î
= + + Î
å
å
Degenerate kernel – fundamental solutionDegenerate kernel – fundamental solution
Fourier series expansion – boundary densityFourier series expansion – boundary density
場源點分離
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page35
Adaptive observer system
collocation pointcollocation point
0 , 01 , 1k , k2 , 2
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page36
Outlines
Motivation and literature reviewPresent method
Expansions of fundamental solution and boundary densityAdaptive observer systemVector decomposition techniqueLinear algebraic equation
Numerical examplesA circular beam with two circular holesA circular beam with four circular holes
Conclusions
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page37
Outlines
Motivation and literature reviewPresent method
Expansions of fundamental solution and boundary densityAdaptive observer systemVector decomposition techniqueLinear algebraic equation
Numerical examplesA circular beam with two circular holesA circular beam with four circular holes
Conclusions
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page38
Linear algebraic equation
{ }
0
1
2
N
ì üï ïï ïï ïï ïï ïï ïï ïï ï=í ýï ïï ïï ïï ïï ïï ïï ïï ïî þ
M
y
y
y y
y
[ ]
00 01 0
10 11 1
0 1
N
N
N N NN
é ùê úê úê ú= ê úê úê úê úë û
U U U
U U UU
U U U
L
L
M M O M
L
Column vector of Fourier coefficientsColumn vector of Fourier coefficients((NthNth routing circle) routing circle)
0B
1B
Index of collocation circleIndex of collocation circle
Index of routing circle Index of routing circle
2B
NB
[ ]{ } { }[ ]=U Ty j
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page39
Advantages of the present method
- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0
Log|Y 1 |
0.8
1.2
1.6
2
2.4
Sc
P o in t B
0 4 8 12 16 20
Fourier term s(L)
2.42
2.422
2.424
2.426
2.428
2.43
Sc
Elimination of boundary-layer effectElimination of boundary-layer effect Convergence test of Fourier seriesConvergence test of Fourier series
Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page40
Compare with Naghdi’s results
0.4 0.5 0.6 0.7
b
1.5
2
2.5
3
3.5
Sc
Theta=3*pi/8
Theta=pi/8
Theta=pi/4
a
Present method Naghdi’s method