第八章_柱体的扭转

lγz0uyzvxzwκκ=−⎫⎪=⎬⎪=⎭8.1/lκα=(,)uyzvxzwxyκκκξ=−⎫⎪=⎬⎪=⎭8.2(,)xyξ0xyzxyzxzywuyxzxwvxyzyεεεγξγκξγκ⎫⎪====⎪⎪∂∂∂⎪⎛⎞=+=−⎬⎜∂∂∂⎝⎠⎟⎪⎪⎛⎞∂∂∂⎪=+=+⎜⎟∂∂∂⎪⎝⎠⎭8.30()()xyzxyzxzyGyxGxyσσστξτκξτκ⎫⎪====⎪∂⎪=−⎬∂⎪∂⎪=+⎪∂⎭8.4(,)xyξϕzxτzyτ0XYZ===120(0(0zxzxzyzyzyzx,),)fxyzfxyzxyττττττ⎫∂=⇒=⎪∂⎪∂⎪=⇒=⎬∂⎪∂⎪∂+=⎪∂∂⎭aazxτzyτ,xyaz(,)xyϕzxzyyxϕτϕτ∂⎫=⎪∂⎪⎬∂⎪=−⎪∂⎭8.5(,)xyϕ8.5(,)xyϕ(,)xyϕ2200zxzyττ⎫∇=⎪⎬∇=⎪⎭b8.5b22()0()0xyϕϕ∂⎫∇=⎪∂⎪⎬∂⎪∇=∂⎪⎭c2Cϕ∇=dCC8.4(8.5)()()zxzyGyxyGxyxξϕτκξϕτκ∂∂⎫=−=⎪∂∂⎪⎬∂∂⎪=+=−⎪∂∂⎭eeyx22222Gxyϕϕκ∂∂+=−∂∂8.6Poisson()2CGκ=−vG(,,0Tlm)()zxzySlmττ0+=acos,sindydxlmdsdsαα====−0dy0dxa()SSdydxdydsxdsdsϕϕϕ∂∂0+==∂∂bSCϕ=czxτzyτCC=0c0Sϕ=8.7ϕ(vG)0,0,1T−0xzxyzyzPPPττ⎫=−⎪=−⎬⎪=⎭d,,xyPPPz,xyPP0:00:00:()xzxDDyzyDDoxyzxzyDDXPdxdydxdyYPdxdydxdyMPydxdyPxdxdyyxdxdyMττττ⎫==−=⎪⎪⎪==−=⎬⎪⎪=−=−+⎪⎭∑∫∫∫∫∑∫∫∫∫∑∫∫∫∫=(e)(j)Green0Sϕ=0xzxDDDSPdxdydxdydxdydxyϕτϕ∂=−=−==∂∫∫∫∫∫∫∫v(f)(e)(e)()()()()2()()222xyzxzyDDDDDSDDPydxdyPxdxdyyxdxdyyxdxdyyxDxydxdyxydxdydxdyxyxyydxxdydxdydxdyMϕϕττϕϕϕϕϕϕϕϕϕϕ∂∂−=−+=−+∂∂⎡⎤⎡⎤∂∂∂∂=−+−=−−+⎢⎥⎢⎥∂∂∂∂⎣⎦⎣⎦−+==∫∫∫∫∫∫∫∫∫∫∫∫∫∫∫∫∫v(g)Green0Sϕ=2DdxdyMϕ=∫∫8.82222202,SDzxzyGxydxdyMyxϕϕκϕϕϕϕττ⎫∂∂+=−⎪∂∂⎪⎪=⎪⎬=⎪⎪∂∂⎪==−⎪∂∂⎭∫∫8.9Poisson()(,)xyϕq(,)wwxy=w2qwT∇=−8.100Sw=8.11q(,)wwxy=22()10,0,11:()10,0,222DSSDTTwwVwdxdyqqMdxdyGGϕϕϕκκ⎫∇+===⎪⎪⎬⎪∇+===⎪⎭∫∫∫∫8.12Vxoy1//Tq2Gκ2zxzywVMwyywxxϕϕτϕτ⇔⎫⎪⇔⎪⎪∂∂⇔=⎬∂∂⎪⎪∂∂⇔=−⎪∂∂⎭8.131wϕxoyMyϕ∂∂xϕ∂∂zxτzyτ−22221xyab+=(a)0Sϕ=2222(xymabϕ1)=+−(b)2222Gabmabκ=−+(c)Poisson()22222222(Gabxyababκϕ=−+−+1)(d)κκd222222222(1)DGabxydxdyMababκ−+−+∫∫=(e)2233()MabGabκπ+=(f)2222(MxyGababϕ1)=−+−8.143322zxzyMyyabMxxabϕτπϕτπ∂⎫==−⎪∂⎪⎬∂⎪=−=⎪∂⎭8.151122222442()(zxzy2)Mxyababτττπ=+=+8.16max22Mabτπ=8.17baab/2xa=±x00(,)w()xyxxyϕϕϕ∂∂=⇒=⇒=∂∂(a)(a)(8.9),:222202byDdGdydxdyMϕκϕϕ=±⎫=−⎪⎪⎪=⎬⎪⎪=⎪⎭∫∫8.1822333()43MbyabMabGϕκ⎫=−⎪⎪⎬⎪=⎪⎭8.19360zxzyMyyabxϕτϕτ∂⎫==−⎪∂⎪⎬∂⎪=−=⎪∂⎭8.20/2xa=±/2xa=±max223bzxyMabττ=±==±8.21(a)(b)2222002byaxDGdxdyMϕκϕϕϕ=±=±⎫∇=−⎪=⎪⎪⎬=⎪⎪=⎪⎭∫∫8.22Poisson()*oϕϕϕ=+8.23oϕ(Laplace):20oϕ∇=8.24*ϕPoissonPoisson*ϕ22*233()(44MbbyGyabϕκ=−=−2)8.2522(4obGϕϕκ=+−)y8.262222200()4oboyaoxbGyϕϕϕκ=±=±⎫⎪∇=⎪⎪=⎬⎪⎪=−⎪⎭8.27()()oXxYyϕ=⋅20oϕ∇=()()()()XxYyXxYy′′′′=−(b)xy2λb2200XXYYλλ′′⎫−=⎪⎬′′+=⎪⎭(c)1212()cosh()sinh()()cos()sin()XxAxAxYyByByλλλλ=+⎫⎬=+⎭(d)w*oϕϕϕ=+220AB==cosh()cos()oCxyϕλλ=(e)C8.2720boyϕ=±=cos()02bλ=(f)(21)(1,2,3,....)mmbπλ−==(g)1(21)(21)coshcosommmmCxbbππϕ∞=−−⎡⎤⎡=⋅⎢⎥⎢⎣⎦⎣∑y⎤⎥⎦(h)8.27mC222()4aoxbGyϕκ=±=−221(21)(21)coshcos()4mmmambCybbππκ∞=−−⎡⎤⎡⎤⋅=⎢⎥⎢⎥⎣⎦⎣⎦∑Gy−(i)(21)cosnybπ−⎡⎤⎢⎥⎣⎦y2222122(21)(21)(21)(21)coshcoscos()cos4bbmmbbmamnbnCyydyGybbbbπππκ∞=−−−−−−⋅⋅=−⋅⎡⎤⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦⎣⎦∑∫∫ydyπ(j)22(21)cos0(21)cos2bbnydybmnmybbmnππ−−=≠⎧−⎪⎡⎤⎡⎤⎨⎢⎥⎢⎥=⎣⎦⎣⎦⎪⎩∫(k)1233(1)8(1,2,3,....)(21)(21)cosh2nnbGCnnanbκππ−−==−⎡⎤−⎢⎥⎣⎦(l)h8.2622222331()48(1)(21)(21)coshcos()(21)4(21)cosh2ommbGyGbmmbxyGymabbmbϕϕκκπκππ∞==+−−−−⎡⎤⎡⎤=⋅⋅⎢⎥⎢⎥−⎡⎤⎣⎦⎣⎦−⎢⎥⎣⎦∑π+−8.28:2DdxdyMϕ=∫∫3MabGκβ=8.2951(21)tanh16423(2mmabbam51)πβπ∞=−=−⋅−∑8.30122112218(1)(21)(21)coshsin2(21)(21)cosh28(1)(21)(21)sinhcos(21)(21)cosh2mzxmmzymGbmmxyGymaybmbGbmmxymaxbmbϕκππbbτκππϕκππτππ+∞=+∞=⎫∂−−−⎡⎤⎡⎤==⋅⋅−⎪⎢⎥⎢⎥−∂⎡⎤⎣⎦⎣⎦⎪−⎢⎥⎪⎣⎦⎬∂−−−⎡⎤⎡⎤⎪=−=⋅⋅⎢⎥⎢⎥⎪−∂⎡⎤⎣⎦⎣⎦−⎢⎥⎣⎦⎭∑∑⎪8.31:max20,221811(21)(21)cosh2bzxxymGbmambττκππ∞==−=⎡⎤⎢⎥==−⎢⎥−⎢⎥−⎣⎦∑8.328.318.15/ab→∞Tqi,iiabiMiτκ23iiiiMabτ=a33iiiMabGκ=b33iiiabGMκ=cκiMM=∑dcd33iiMGabκ=∑8.33ca33iiiiMbabτ=∑8.34iτiτ1S1S10Sz=10Sϕ=2S2S2Szconst=2Sconstϕ=tanhταδ==ehδ222MdxdyzdxdyAhϕ===∫∫∫∫f2MhA=g2MAτδ=8.35τκqsinqATdsα=⋅∫vhsintanhαααδ≈≈=iih12qhGTAκdsδ==∫vjgκ24MdsGAκδ=∫v8.3624MSGAκδ=8.37SM8.35(8.34)123222332()iiiMMAabMbMababτδδτδ⎫==⎪⎪⎬⎪==+⎪⎭∑8.38213()ababττδ=+8.394.5,2.0,0.4ambmmδ===21/10.4ττ=10.44.5,2.0,0.02ambmmδ===21/207ττ=0uyvxzwκκ=−⎫z⎪=⎬⎪=⎭()DSQPdxdyPdxQdyxy∂∂−=+∂∂∫∫∫v(,),(,)PxyQxy