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Nation Taiwan Ocean University Department of Harbor and River June 27, 2015 page 1 A Semi-Analytical Approach for Stress Concentration of Cantilever Beams.

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1 Nation Taiwan Ocean University Department of Harbor and River June 27, 2015 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

2 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page2 Outline  Motivation  Present method Formulation Expansions of fundamental solution and boundary density Flowchart  Numerical examples A circular bar with three circular holes (torsion) A circular beam with four circular holes (bending)  Conclusions

3 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page3 Motivation Torsion problem: Caulk (1983) said that the Ling’s 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

4 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page4 Motivation Numerical methods for engineering problems FDM / FEM / BEM / BIEM / Meshless method BEM / BIEM Treatment of singularity and hypersingularity Boundary-layer effect Ill-posed model Convergence rate CPV & HPVNearly-singularLinear algebraic orderFictitious BEM Null-field BIE

5 Mechanics Sound Vibration Laboratory HRE. NTOU http://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

6 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page6 Motivation BEM / BIEM Improper integral Singularity & hypersingularity Regularity Bump contour Fictitious boundary Collocation point Fictitious BEM Null-field approach CPV and HPV Ill-posed Guiggiani (1995) Waterman (1965) Achenbach et al. (1988) interior exterior Main idea 山不轉 路轉 路不轉 分內外核函數

7 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page7 Present approach Fundamental solution No principal value Advantages of degenerate kernel 1.No principal value 2.Well-posed 3.Exponential convergence 4.Free of boundary-layer effect 5.Mesh free Degenerate kernel CPV and HPV 路不轉 分內外核函數 interior exterior

8 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page8 Outline  Motivation  Present method Formulation Expansions of fundamental solution and boundary density Flowchart  Numerical examples A circular bar with three circular holes (torsion) A circular beam with four circular holes (bending)  Conclusions

9 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page9 Conventional BIEM and current method 1969-2005 Current Degenerate kernel interior exterior Main idea 路不轉 分內外核函數 Conventional BEM

10 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page10 Convergence rate between the present method and conventional BEM Degenerate kernel Fourier series expansion Fundamental solution Boundary density Convergence rate Present method Conventional BEM (1969-2005) Two-point function (closed-form) Constant, linear, quadratic elements Exponential convergence Linear algebraic convergence

11 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page11 Degenerate (separate) form of fundamental solution (2-D)

12 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page12 Outline  Motivation  Present method Formulation Expansions of fundamental solution and boundary density Flowchart  Numerical examples A circular bar with three circular holes (torsion) A circular beam with four circular holes (bending)  Conclusions

13 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page13 Flowchart of the present method Degenerate kernel Fourier series Collocation point and matching B.C. Adaptive observer system Linear algebraic equation Fourier coefficients Potential of domain point Stress field Vector decomposition Numerical Analytical

14 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page14 Outline  Motivation  Present method Formulation Expansions of fundamental solution and boundary density Flowchart  Numerical examples A circular bar with three circular holes (torsion) A circular beam with four circular holes (bending)  Conclusions

15 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page15 Torsional rigidity (Ling’s problem) Caulk (First-order Approximate) 0.87390.87410.7261 Caulk (BIE formulation) 0.87130.87320.7261 Ling’s results0.88090.80930.7305 Present method (L=10) 0.87120.87320.7244 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 ? T T T

16 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page16 Bending problem for a cantilever beam ABCD O E Q

17 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page17 Stress concentration at point B Present method Steele & Bird The two approaches disagree by as much 11%. The grounds for this discrepancy have not yet been identified. --ASME Applied Mechanics Review Θ=π/8Θ=3π/8 a B Θ a a Steele Q

18 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page18 Outlines  Motivation  Present method Formulation Expansions of fundamental solution and boundary density Flowchart  Numerical examples A circular bar with three circular holes (torsion) A circular beam with four circular holes (bending)  Conclusions

19 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page19 Conclusions  Null-field integral equation in conjunction with degenerate kernels and Fourier series  Singularity free, boundary-layer effect free, exponential convergence, mesh free and well- posed model  Arbitrary number of holes, various radii and positions ( 三任意 : 數目 大小 與 位置 ) ( 五優點 )

20 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page20 Conclusions  Torsion problem: Caulk, 1983 (yes) Ling, 1947 (?) (three holes) (three holes)  Bending problem: Steele, 1992 (?) Naghdi, 1991 (yes) (four holes) T Q

21 Nation Taiwan Ocean University Department of Harbor and River June 27, 2015 page 21 Thanks for your kind attentions. You can get more information from our website. http://msvlab.hre.ntou.edu.tw/

22 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page22

23 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page23 URL: http://ind.ntou.edu.tw/~msvlab E-mail: jtchen@mail.ntou.edu.tw 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 Equation Helmholtz 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 exterior Acoustics SH wave (exterior acoustics) (Inclusions) Free vibration of plate Indirect BIEM 李為民 ASME JAM 2006 蕭嘉俊 MRC 2007,CMES 2006EABE 2006 ASME 2007EABE 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 circular cylinders Screw dislocation (addition theorem) 周克勳 Green function for an 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 Modified Helmholtz 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 plate Real-part BIEM 李為民 EABE 2007 EABE 2008 rev. Green function of`circular boundary (statics:superposition) MRC 2008 rev. 周克勳

24 Mechanics Sound Vibration Laboratory HRE. NTOU http://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

25 Mechanics Sound Vibration Laboratory HRE. NTOU http://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 … Ó NTUCE cite

26 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page26 Engineering problem with holes, inclusions and cracks Straight boundary Degenerate boundary Circular inclusion Elliptic hole [Mathieu function] [Legendre polynomial] [Chebyshev polynomial] [Fourier series]

27 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page27 Literature review – analytical solutions for problems with circular boundaries Key point Main application Author Conformal mapping Torsion problem In-plane electrostatics Anti-plane elasticity Chen & Weng (2001) Emets & Onofrichuk (1996) Budiansky & Carrier (1984) Steif (1989) Wu & Funami (2002) Wang & Zhong (2003) Bi-polar coordinate Electrostatic potential Elasticity Lebedev et al. (1965) Howland & Knight (1939) Möbius transformation Anti-plane piezoelectricity & elasticity Honein et al. (1992) Complex potential approach Anti-plane piezoelectricity Wang & Shen (2001) Those analytical methods are only limited to doubly connected regions even to conformal connected regions.

28 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page28 Literature review - Fourier series approximation Author Main application Key point Ling(1943) Torsion of a circular tube Caulk et al. (1983) Steady heat conduction with circular holes Special BIEM Bird and Steele (1992) Harmonic and biharmonic problems with circular holes Trefftz method Mogilevskaya et al. (2002) Elasticity problems with circular holes or inclusions Galerkin method However, no one employed the null-field approach and degenerate kernel to fully capture the circular boundary.

29 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page29 Vector decomposition technique for potential gradient True normal vector Special case (concentric case) : Non-concentric case:

30 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page30 Explicit form of each submatrix and vector Fourier coefficients Truncated terms of Fourier series Number of collocation points

31 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page31 Outlines  Motivation and literature review  Present method Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation  Numerical examples A circular beam with two circular holes A circular beam with four circular holes  Conclusions

32 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page32 Two holes problem Present method Steele & Bird’s result [6] Point P Sc of point P D: Distance between two holes a: radius of holes R: radius of circular beam D a R

33 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page33 Contour of stress concentration Steele & Bird’s result [6] Present method

34 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page34 Expansions of fundamental solution and boundary density Degenerate kernel – fundamental solution Fourier series expansion – boundary density 場源點分離

35 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page35 Adaptive observer system collocation point r 0, f 0 r 1, f 1 rk, fkrk, fk r2, f2r2, f2

36 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page36 Outlines  Motivation and literature review  Present method Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation  Numerical examples A circular beam with two circular holes A circular beam with four circular holes  Conclusions

37 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page37 Outlines  Motivation and literature review  Present method Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation  Numerical examples A circular beam with two circular holes A circular beam with four circular holes  Conclusions

38 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page38 Linear algebraic equation Column vector of Fourier coefficients ( Nth routing circle) Index of collocation circle Index of routing circle

39 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page39 Advantages of the present method Elimination of boundary-layer effect Convergence test of Fourier series

40 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page40 Compare with Naghdi’s results a Present methodNaghdi’s method

41 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ October 24, 2008 page41 Stress concentration ABE C DO CD AB O E

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