nation taiwan ocean university department of harbor and river june 27, 2015 page 1 a semi-analytical...

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


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


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


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


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


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


interior

exterior


Outline




Conclusions


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


x DÎx BÎ

Conventional BEM


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


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

¶ ¶


Outline




Conclusions


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


Outline




Conclusions


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


Bending problem for a cantilever beam

b

a

R

2Y 1Y

ABCD

O

E

Q


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


Outlines




Conclusions


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 )


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 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


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


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


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]


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..


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.


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

¶¶

= - Î¶

È

È

ò ò

ò ò


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


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


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


Contour of stress concentration Steele & Bird’s result [6]Steele & Bird’s result [6] Present methodPresent method


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


Adaptive observer system

collocation pointcollocation point

0 , 01 , 1k , k2 , 2


Outlines




Conclusions


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


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


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


Stress concentration

0 4 8 12 160.6

1.2

1.8

2.4

Sc

0 4 8 12 160.6

1.2

1.8

2.4

Sc

0 4 8 12 160

0.4

0.8

1.2

1.6

Sc

A B EC D O

CDAB

O

E

nation taiwan ocean university department of harbor and river june 27, 2015 page 1 a semi-analytical...

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