MATSUOKA  Kenichi

NOMACHI  Sumio

Muroran-IT Academic Resource Archive

On Three-dimensional Stress Distribution dueto Displacement of a Cylindrical Inclusion著者 MATSUOKA  Kenichi, NOMACHI  Sumiojournal orpublication titleMemoirs of the Muroran Institute ofTechnology. Science and engineeringvolume 8number 2page range 269-277year 1974-10-15URL http://hdl.handle.net/10258/3588On Three-dimensional Stress Distribution due to Displacement of a Cylindrical Inclusion Kenichi G. MA TSUOKA and Sumio G. NOMACI-王子Abstract An interaction between the五niteelastic body and the cylindrical inclusion is handled by solving a three-dimensional stress problem written in the cylindrical coordinate system. The problem is analyzed by means of五niteFourier-Hankel transforms， on the assumption that the elastic body is very thick cylinder and the solid core as the inclusion keeps its sectional area unchanged duping the interaction. The numerical calculations wer巴 carriedon for the cases with the various ratio between outer and inner radi， as well as the di丘erentratio between elastic moduli of the outer body and the inclusion 1. General expression of displacement Three-dimensional stress problems were solve by means of五niteFourier-Hankel transforms1l，2l， and as an application of it， the correct solution concerning the bending of the thick hollow cylinder， has been obtained by the outhors with the expressions of the displacements3l， and replacements of sine for cosine and cosine for sine， into these expressions yield another set of displacements. Thus obtained displacements will be taken for the problem now considered. The origin of coordinate is placed as shown in Fig. 1， in which a and b denote the inner and outer radii， and c denotes the height of the cylinder. Let u， v and w be the displacement components in the r， {} and z directions. The boundary conditions satisfying that the shearing stress vanishes and w is zero for z=O and c， give the displacement vector as follows: t ∞ u=: 1: cν (Ay•，，+ 尻町.)cos v(} “ ν~O E 田り=三1:(ム悶-Byz，.) sin v(} ∞∞ I II-l- J I 町=主1完芸云す一，豆芹P十A川 -B叫+札止}]会s山∞日ν。* Department of Civl Engineering， Hokkaido University. (53) Z Fig. 1. ( 1) ( 2 ) ( 3 ) 270 where Kenichi G. Matsuoka and Sumio G. Nomachi ~ r a2νl'" r-(叶 1) 1 仇rI 1 AMJ511FヲZ 不可~ (_ly-1 a~-'12 α.0"+玄ß.Ok2uμ一(μ+え)え 1， _F(k) I ¥ f ν+1 AuM一工てB，o"J +f.<;) (r) ¥ 2~p a，Ok 十 1 ハ +3ρ十え A"n.+μ十え RJl一一一日一一2(2μ+A) 1"，0kT 2μ+えι 川 2μ+A .，0") c 2 ∞ 1 ( 1 + ;n~ N~2戸時)(Nr){α川+ん正+4μA川十μD叫}-4品門誌)(Nr){2~s，nk+A叫 -B，nk十九})cosNz] (4) ~r 1 r，-1 1. ... ."f1 1 _ .u+A B山 =511Fヲ Z 可+A; (-1)ト 142i?批ー が岬+tfA批2l.!μ+(μ+え)n i 1'1 k) / ¥ 1 f 1 + _. l.!二了一九~-f.~k)(r) ・;i2~.u 一一一一一月J J，s ¥'/ cl・α，Ok 2(2μ+λ)νOk +tiAo汁生+~B"J2μ+え νUIC' 2μ+え州j2 ∞ 1 ( 1 +; ~1 N¥一石狩)(N:め{α叫ーん+4μB，nkー μD川}-__I!_土ι ~:) (Nr) ~ ')1. ß山正 +A.nk-B.nk+D.n1c~)cos Nzl (5) 2(2μ+A)ω Jl2μ fcos 1YZJ nπ N=7，(n=IA…) ;ν=0，1，2，… ， a]=b，α2=a， ao=a， 1 μ，A=Lame's constants， ci= 2 for i=O and Ci= 1， for 伴 0，α叫止=8νOn[r']r=η ，s叫=ぜ;，On[σ'r]r=匂A 「 t い川+刊1川 [0，引仰ν」ル[μ仲μD叫叫止 =NO杭プ工ν，$札ず凡匁~[即]戸一α句止， ん=1，2札 [f]= 1:f(x) cos子xdx，札[fIIj;f(r)sin?zdzand the functions : 引所)一旦担 X~~)(Nr) ー鳴斗担， ufW)一主主岨-R~~C: (Na，) ， 叩 - R~:パ4叫kj-Rr(NG正) ， F丹即y川1c)判(仰州N附め= 而お恭品己記i矛子沖[トトいいR民町;叩出川可引(川州N一寸R~貯配附;r九叩出仰川?tヴ引川νJパ川川)刊明 (仰伽Nルr)十(トa向ω州kR~り41 1バ川訓品仰仰仇N泊恥G仇叶必Stress Distribution due to Displacement of a Cylindrical lnclusion 271 ω;;)(J.;[r) = {R川おa，)Y[R~:'J仇){rR~~2(州一向 1Rm+1(州}-R~tJh+1(州(州出 1バ川品品州(仰仙仇州N幼仇刷α向ω止，)トH一寸G仇リ止ト日J一→バ川1ぷRωば立伊)附=問為九偽A叩引川(仰州N一R尺即m机J巳β児此k)凸九山〕Lし1.vνg~Tc )(r) =時{(日-(与)}，刀βj，<:)(川;f刊内刊巾(収ト7子走LTぷ叫当z{出出詰了戸門戸問4叫仏zZ1h十詐品百(α一仲恥い…一一→引七1幻に刊)_一→b一2仙ν一1)}β) (め口詰品1(a~-J叶1+dTI7(与)ν1}2νανbνr a;:~1 ，，，'，..~ . n/ ，.~， a~_" n .. '" 1 一一一L~正L(G2(叶1)_b2(叶1いJKL(dーが) ~ ， (bZ'-a2? l4(ν+ 1) ¥u， v; 1 4(ν_1)¥<<- v;J Rg>(Nrう=Ii(Nr) Kj(Na止ー 1)ー (-1/十jIj(Na"ー1)Ki(Nrう.which may correspond to the bending behaviour of the beam on the elastic subgrade as well that of the pile struck into the earth. 2. The formulas of stress components The stress components are related to the displacement components by the well-known Hooke's law: ou σ，.=2μ百戸+2.e ( 6 ) σ，= 2μ作+忽)+2.e ( 7 ) ow σ.= 2μoz十1・2 ( 8 ) OU U OV oz幻e=一一+一+一一一+一一 ( 9 ) or . rγ.0 . oz (議 auu) 7:dJ =μ 十 or-7ご (10) 8z = fl(1反~+1d;) (11) (55) 272 Kenichi G. Matsuoka and Sumio G. Nomachi T剖 =μ(乞+3Z) (12) where σ，.: the normal stress in the r・ direction内:the normal stress in the θdirection 仏:the normal stress in the z direction 1:，'0: the shearing stress around the z axis 'Oz: the shearing stress around the r axis 'zr: the shearing stress around the θaXls The eqs. (1)，-_，(5)， through the eqs. (6)，._(11)， lead to the stress components as followes: 2μ ぷ (iJA目。B，z'，¥ 口一一 γc.( -;，zr +一一一一 Icos J){} + A • e π 士も ν¥ iJr ' iJr) σ内0=ごご=之=竿=ふ仁ιバ，(王竺壬lA，同山〆〆I!zr一-一 E竺ミ王弘B，旦ι九ν川岬岬Jパ4刷z酎ふ写肘7"ν=0 ¥ I ノ分 2 ∞ 1 A μ). 111 Tl J σz=251ECνll2(2戸可s，01c+不可(A，Ok-Bν0，1，)f g~k)(r) cos νθ/μf 1" ， A T>， 2(μ十A)r. 1 + n~l\ G~円Nr) 不可{写AJAM-BJ-7-D叫止j2μ(μ+え)r，.( k) 11¥ T ¥ f 1 f) 1 Tl. T'¥ ) ¥ ¥. T 1 十ヲ戸I-R什Nr)l万戸川+A川-B'nk+D，川)coslVZJ ~ 2 ∞ 1 1 μi =ニ:6:6 C，I) 0(0.二~ß，Ok+ 一一ーで (A，o止 -BμOk)(g~k)(r) わいー~"'ll2(2μ +À) 雌 2p十A¥..l.LvOk .L.I.，  J 古三dJ平訂訂訂i正A一-{いFη吋川J止汁バ+ι2勾勾山Fμ州件1κ作l(Aレ凶ぷ，nk-B''k + D，札九川叫山川川イz吋Jイ止心k)}いドG(13) (14) (15) (16) H ∞ r iJA._. íJB ν 十Iν一1.~ ) ω=77511fL一ずLーマ-ι-7-B町ドinJ){} (17) リ 2 ∞∞ 1 . 1 . 1 1 λ11νlν1_ _ I TOz = ~主主認司F亙p日1ぷぷ芦主日目15λ完忍引訓J川It炉Xい仰WX説咋材;2r川7引川川)刊引)(Nr仰仲州Nル川fめ)一瓦芯干G;町kJ(代幻吋)川(仰州N州めh川川jドいいトα叱ιhν叫山川2η叫Lι正一7芯Nrポ〆戸G;昨kJ(グ?門幻吋叩明)刊川伽州(仰仲州Nルfめ仙)f'V (l: i 11¥ T__¥ ;1 I 'v (k ~ 11¥ T__¥ T> 1 2 ν 臼叫(μ 十」心) + 叩 J与N;F引川)刊明(凹Nル州T付付)比A，山叫mk+叫zバけ止汁+叫叫χ記応rJ立川:T引川)刊明(仰Nル州f付柑)河B及ν山川イ寸η叫正イナjγ一7芯V貯ポ引正メ戸R町貯即(は川k 叩)×ぺ(ふかかかινn叫け'k+A汁パ叫A川一B'nk+D吋Jふ)]si山 sln (18) ? 2 ∞∞ r" ( ) Tzr' :::=云:6:6 :6 c， 1 一長-::G~k)(NrHα川 +s"'k-4μBmk( ι む ~1ν~Oη::-1 '1 lvr 'l .. . -.， --"1 r _ ._， . ~， 1 .u + A r ，'， _ _ ， J) _，，'， _ _ ，) 一時J(Ni付x{ルー 2μ(A，，正也けD川)]cos J){} sin民 (19) Stress Distribution due to Displacement of a Cylindrical lnclusion 273 3. Boundary conditions As previously discribed， the periodic boundary condition is taken for z=O and z=c， so what we need now is to五ndthe condition on the interface between the elastic body and the inclusion， and the condition on the surface of the outer radius r=b. To simplify the further discussion， the condition is assumed that u and w are zero as well as 77'8 vanishes on the outer surface: u = 0 for r= b， τ0=7r8=0 for r=b， A山 l=-BC同 1= 0 and α岬 1= 0 (20) (21) On the interface r= a， the shearing stress vanishes， the radial displacement is continuous， and the radial stress occurs to hold an equilibrium state with the beam action by the inclusion， so that the boundary conditions are written as follows: 7 r8 = 0 for r = a ， αμn2 = 0 7.r=0 for r=a， E.I d:竿=(10cosod-正7Z' Jo' which is transformed into ? r-C 一一十一 c一寸一-:;羊吾長吾寄与警告玉、守持争舟守#示吊台，:，:-.::-7:¥舟向噌巳丞径が呼砕4・A宇長剖ー が議....a E し E，IFig. 2. ÉJ{N4 的。]川(13:izzo 一(一吋会1~σ)+(叶ZFi-[会lJ}1'2.， = a ¥ Cn[σァ]cos Odθ Z where uo: the displacement of the center of the inclusion， (22) (23) (24) Ei: the elastic modulus of the inclusion， 1: the moment of inartia. Because the inclusion keeps its initial section during strained， the surface of the inclusion displaces by uα=μ。cosθ， 九=- Uo Sln θ， (25) (57) Kenichi G. Matsuoka and Sumio G. Nomachi which are identical with the radial displacement on the inner surface of the elastic body. So that 274 。???? ??????????????????????? ??? (26) and the cosine transformation of u is found as ????????????? ??』? ??「? ????? ??????? ?? (27) Equating the eqs. (26) and (27)， we have Cn[Uo] = A>m2+Bvn2 (28) and ν=1. This means that the displacements and stresses on this case， correspond to the eqs. (1)""(5) and (13)，.，(19) with ν=1. Hence， the right side of eq. (24) becomes αJ:"札 [σr]叫み=as，n2 (29) in which 1. = 1. The inclusion also takes the periodic boundary condition in the z direction as the outer body does. The shearing stress appears for z=O and z=c. Denoting the resultant of the prescribed shearing stress by P， we五nallyget (30) 凡IN4{A1n2+The mathematical de五nitionrequires the following equations : ソ∞AlIzr)r=b =云L;cμA山 1COs Nz (31) ー，曲A附)…=ラL;cνAvn2CoS Nz L 偽~o(32) ソ∞B附 )Tzb=??ECmBmImNZ (33) ソ∞B時ふ =α= 云~^cν及川osNz (34) in which B'nk comes with (1.-1) and the eqs. (33) and (34) vanish in case of ν-1. As a result， the unknown constants s，nb s，仙 A叫 b A凶，2and c叫 2are to be determined and this can be done by the boundary conditions (20)， (23)， (30)， (31) and (32). Numerical examples The numerical calculation was carried on for many cases varying the ratio bJa， the ratio cJa， and the di旺erentratio between the elastic moduli of the outer (58) 4. 275 下 L- G;)同仰向白戸以Jy.A..仙併Stress Distribution due to Displacement of a Cylindrical lnclusion 広句 80号否需IJ()O'Po(~$'批旨 1'4.t.'1>，，~.z5" ().4. /).?J ふ又? ? 、 ? ? ??。、Zt 4.0，・d;to.o 之仰ケぇ.5:ρ Fig.4. 之、主0・1/)・9%. = ~O ちを旨800'ßω61t.~ 臨む。~o・'<S'.y.o ，d.D 三〈一-gy)r'.吋・由司自由以)r叫叫払， ， #' _，" _. 一.....~_"...-，，，，，， F ? ???????? ??????? 、??の)r吐U )r叫Fig.3. f.'?S'<IO-Z 一一私-c叩 :t"a.. t:-ー 弘ー "'C.州 d伽才%胃110?叫s!o~'s 陶，，，.・'þ -~、z5"f.zs叫6三与えFig. 6. 弘.8，ω弘-100・E?? ?{.2$"KJO'・4{).(J f、む? ??????%.甥80弘=(0 ?'t$，帥告 'l'A.t'd・3・2.5"。'.()Z・9，t.ρ うi一-0;)ド九一ーー弘)y.p.，1 仰 l/zo! <0 -<0 T/a. d'P.白10 古之宮1 jp/)o 1 80 Fig.5. ちをFig.7. (59) 276 。{.oトよ「Kenichi G. Matsuoka and Sumio G. Nomachi 一一弘(あ，咋手芸〉lj.，(. U叶ι t).o o.!l 一一一 一-;r;1 1 IO~1I ぺごjml/lZo l ゑ宮10// j/ 11 l 1/ I /! / 11 F 之官40I // 11 // 号f~8o 1/ // !I PO'.S5'心J IJ Yof;:o 台 ρ25"11 一-6訪問似---U)r'，叫Fig. 8. %::40 与~=80 z二O 臥時三936R161L% =80 Poisso昨日!iOご0.25/r 孟Fig.9. body and the inclusion. Poisson's ratio of the elastic body is taken as 0.25. The distributions of the radial displacement u and the radial stressσγ1n the z direction for (}=O， with various value of b/a are shown in Fig. 3 from which we五nedthat the shapes of the displacement curve are quite similar， while the magnitudes are quite different one another. The ratio σ，. and I takes approximately a constant value for each ratio b/a. The connection of the maximum value of u and σ，. with the ratio b/αlS illustrated as in Fig二4，which shows that久.decreases down to a constant value as b/αincreases， while the value of u increases with the increment of b/α. The ratio σ/u円 therefore，tends to zero for b/α→∞. The subgrade coe伍cientwhich is conventionally used in the theory of the beam on elastic subgrade， iseffected not only by the elastic property of the subgrade， but also the size ratio between the subgrade and the beam. The distributions of u and σ，. with the variation of the ratio between the elastic body and the inclusion， are drawn in Fig. 5， which shows that the inclu-sion has the smaller elastic modulus， the distribution has the more prompt v'ariation in the z direction. 80 that the value of σr/U widely changes. Letting the elastic modulus of the inclusion be constant and varying the modulus of the outer body， we can五ndthe maximum values of u and σr as shown in Fig. 6. Fig. 7 show the relation ofσT向 withthe variation of b/αand Ed E respec-tively. We see thatσr/U takes almost the constant value with the variation of EdE， while it gradually decreases as b/a increases. Though σ，.ju tends to zero for b→∞， the rate of decrement is very small. Fig. 8 shows the distribution of u and σ，. in the z direction， with the vari-(60) Stress Distribution due to Displacement of a Cylindrical Inclusion 277 ation of c/a. Fig. 9 shows the isochromatic line concerning σ門 whichpromply decreases w hen r/αincreases. 6. Closing remarks The e旺ectof the displacement of the cylindrical inclusion on the stress distribution in the elastic body is studied by treating the chick elastic cylinder with the co-centered cylindrical inclusion which behaves as a beam. In this manner， we write the problem in the cylindrical co-ordinate system， which can be conveniently handled by means of五nite F ourier凶Hankeltransforms. Carring on the various numerical calculation， we come to the conclusion: a) the interaction between the elastic body and the inclusion， depends not only on the both elastic properties but also on the size ratio between them， b)σ，./u， which coincides with the subgrade coefficientラ approximatelytakes a constant value for a larger bja and zjα<3.0 in spite of any value of EtfE. The numerical results can not lead us to a theoretical judgement， on whether the coeventional theory of beam on elastic subgrade stands for the elastic theory or not. We however， can say the beam on elastic subgrade can practically go for the engineering use. The calculation was carried on by F ACOM 230-60 of the computer center on the campus of Hokkaido University. (Received May. 27， 1974) References 1) S. G. Nomachi: On On巴Methoduf Solving Stress Problems in Cylindrical Cu-ordinates by Means of Fourier Hankel Transforms (Part 1)， the Memoirs of the Muroran Insti-tute of Tecenology， Vo1. 3， No. 3 (1960) 2) S. G. Nomachi・ OnOne Method of Stress Problems in Cylindrical Co-ordinrte by Means of Fourier Hakel Transforms (Part I)， the Memoirs of the Muroran Institute of Teohnology， Vo1. 3， No. 4 (1961). 3) S. G. Nomachi anc1 K G. Matsuoka‘ On a Three Dimensional Stress Analysis of an Annular Cylindrical Body Subjected by Non-axisymmetrical Loading， th巴 Memoirsof the Muroran Institute of Technology， Vo1. 8， No. 1 (1973). (61) 

On Three-dimensional Stress Distribution due to Displacement of a Cylindrical Inclusion

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