平面问题的极坐标解答课件.ppt
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§4-1DifferentialEquationsof
EquilibriuminPolarCoordinates
Dealingwithelasticityproblems,whatformofcoordinate
systemwechoose,whichcan’taffectondescribingproblem
essence,butrelatetothelevelofdifficultyonsolvingproblem
directly。Ifcoordinateissuitable,itcansimplifytheproblem
considerably。Forexample,forcircular、wedgedandsector
andsoon,solvedbyusingpolarcoordinatesaremore
convenientthanusingrectangularcoordinates.
ConsideringandifferentialfieldPACBintheplate
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§4-1極座標中的平衡微分方程
在處理彈性力學問題時,選擇什麼形式的坐標系統,雖
不會影響對問題本質的描繪,但將直接關係到解決問題的難
易程度。如座標選得合適,可使問題大為簡化。例如對於圓
形、楔形、扇形等物體,採用極座標求解比用直角坐標方便
的多。
考慮平面上的一個微分體PACB
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normalstressintherdirectioniscalledradialnormalstress
denotedbyr;normalstressinthedirectioniscalled
tangentialnormalstressdenotedby;shearstressisdenoted
byr,stipulationofsignofeachstresscomponentaresimilarto
onesinrectangularcoordinates.Bodyforcecomponentsofradial
andhooparedenotedbyKrandK,respectively.Fig.4-1.
x
o
Pr
Consideringequilibriumofanunitr
drA
element,therehavethreeequilibrium
Krr
Krdr
equations:Br
dr
r
Fr0,F0,M0rdr
Cr
d