Theory of Elasticity. Chapter 9 Two-Dimensional Solution
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1 Theo of Elsticit 9 Two-Dimensionl Solution
2 Content Intoduction Mthemticl Peliminies Stess nd Euiliium Displcements nd Stins Mteil Behvio- Line Elstic Solids Fomultion nd Solution Sttegies Two-Dimensionl Polems Intoduction to Finite Element Method Thee-Dimensionl Polems Bending of Thin Pltes 9 Pge
3 Two-Dimensionl Solution in Pol Coodinte 9. Pol Coodinte Fomultion( 极坐标下的求解 9. Coodinte Tnsfomtion of Stess Components ( 应力分量的坐标变换 9.3 Axismmetil stesses nd coesponding displcements( 轴对称应力和位移 9.4 Hollow clinde sujected to unifom pessues ( 圆环受均布压力 9.5 Stess concenttion of the cicul hole( 圆孔的孔口应力集中 Pge 9
4 Two-Dimensionl Solution in Pol Coodinte Some smmet nd cicul stuctue Aeoengine nd its oto sstem Pcticl Engineeing need solutions in pol coodinte 9 Pge 3
5 Two-Dimensionl Solution in Pol Coodinte Stess Concenttion in Pcticl Engineeing Fist civil iline Comet: M nd,953, London-Johnston Ai disste: Jn.0,954, nd Apil 8,954 9 Pge 4
6 Two-Dimensionl Solution in Pol Coodinte Stess Concenttion is the min eson tht cuse the i disstes 9 Pge 5
7 9. Pol Coodinte Fomultion Pge cos x sin x x tn Pol Coodintes x x x sin cos cos sin 9 6
8 9. Pol Coodinte Fomultion Pge Pol Coodintes x cos sin sin cos cos sin cos sin ( x sin cos cos sin 9 7
9 9. Pol Coodinte Fomultion Pge Bsic Eutions in Pol Coodintes( 极坐标下的基本方程极坐标下的基本方程极坐标下的基本方程极坐标下的基本方程 Stesses Φ Φ Φ Φ Lplce opetos x 4 Bihmonic opetos 0 4 ϕ ϕ ϕ 9 8
10 9. Pol Coodinte Fomultion Euiliium Eutions( 平衡方程 k 0 k 0 O d d B d P C d k k A x ( d d d d d 9 Pge 9
11 9. Pol Coodinte Fomultion Geometicl Eutions 几何方程 u ε u ε u γ u u u 9 Pge 0
12 9. Pol Coodinte Fomultion Pge Phsicl Eutions 物理方程物理方程物理方程物理方程 ( µ ε E ( E µ ε µ γ E G ( ( µ µ µ ε E µ γ E G ( ( E µ µ µ ε µ µ µ ν E E Plin Stess Plin Stin 9
13 9. Pol Coodinte Fomultion Bound Conditions( 边界条件 l 0 o 0 0 o α α 0 0 l 0 9 Pge
14 9. Coodinte Tnsfomtion of Stess Components Pol Coodinte Ctesin Coodinte x cos sin cos sin sin cos x Ctesin Coodinte Pol Coodinte x x cos x sin x x cos x sin x sin x cos 9 Pge 3
15 9.3 Axismmetil stesses nd coesponding displcements( 轴对称应力和位移 xismmetic field untities e independent of the ngul coodinte X 9 Pge 4
16 9.3 Axismmetil stesses nd coesponding displcements ϕ ϕ 0 xismmetic ϕ( Φ Φ Φ Φ xismmetic cse dϕ d ϕ 0 d d 4 ϕ ϕ ϕ 0 xismmetic cse d d d ϕ d 0 9 Pge 5
17 9.3 Axismmetil stesses nd coesponding displcements Michell solution of the ihmonic eution d d d ϕ d 0 ϕ dϕ d ϕ 0 d d A B( ln C A B(3 ln C 0 A ln B ln C Plne stess cse 4B u H I sin E D A u ( µ ( µ B (ln ( 3µ B E (µ C ] I cos K sin K cos H,I,K ssocited with the igid-od motion 9 Pge 6
18 9.3 Axismmetil stesses nd coesponding displcements Function of hole on distiution No hole: A nd B vnish. A B( ln C A B(3 ln C 0 Othewise when 0, stess component ecome infinite A plte without hole with no od foces (xissmmeticl const. If thee is hole t the oigin, we will investigte it next 9 Pge 7
19 9.4 Hollow clinde sujected to unifom pessues plne stess conditions Axismmetic polem A B( ln C A B(3 ln C 0 Bound Conditions Pge 8
20 9.4 Hollow clinde sujected to unifom pessues A A B( ln C B( ln C 3 unknowns, Eutions? Single o multipl connected egion? fo multipl connected egions, the comptiilit eutions e not sufficient to guntee single-vlued displcements. A u ( µ (µ B (ln ( 3µ B E (µ C ] I cos K sin 4B u H I sin E K cos 9 Pge 9 B 0
21 9.4 Hollow clinde sujected to unifom pessues Pge C B A ln ( C B A ln ( B 0, ( A ( C 9 0
22 9.4 Hollow clinde sujected to unifom pessues Pge, 0 0( Demonsttion of Sint-Vennt s Pinciple 9
23 9.5 Stess concenttion of the cicul hole Pge Review: 9
24 9.5 Stess concenttion of the cicul hole Wht is stess concenttion? The stess concenttion ne hole is citicl issue concening the stength of solid stuctue. The stess concenttion cn e mesued the stess concenttion coefficients tht e the tios etween the most sevee stess t the citicl point (o temed hot spot nd the emote stess. 3 K mx 3 9 Pge 3
25 9.5 Stess concenttion of the cicul hole Exmples: 9 Pge 4
26 9.5 Stess concenttion of the cicul hole Solution: Selection of coodinte To nlse stess concenttion ne the hole, it is convenient to use pol coodinte. Polem in pol coodinte Bound conditions in pol coodinte: A A x x cos x sin cos x x sin x cos sin 9 Pge 5
27 9.5 Stess concenttion of the cicul hole Polem in pol coodinte 9 Pge cos sin
28 9.5 Stess concenttion of the cicul hole Polem in pol coodinte Polem Polem 0 0 cos sin sin cos 9 Pge 7
29 9.5 Stess concenttion of the cicul hole Solution of Polem B.C. when >> Pge 8
30 9.5 Stess concenttion of the cicul hole Solution of Polem Polem 由边界条件可假设 : 为 的 某一函数乘以 cos ; 为 的某一函数乘 sin ϕ ϕ ϕ sin cos Assume: ϕ f ( cos ϕ 0 9 Pge 9
31 9.5 Stess concenttion of the cicul hole Pge 0 ϕ 0 cos ( 9 ( 9 ( ( d df d f d d f d d f d 0 ( 9 ( 9 ( ( d df d f d d f d d f d 4 ( D C B A f ϕ ( cos f ϕ cos 4 D C B A 9 30
32 9.5 Stess concenttion of the cicul hole 4 ϕ A B C D cos ϕ ϕ 4C 6D ( B cos 4 ϕ 6D ( A B cos 4 ϕ C 6D ( 6A B sin 4 B.C. 3 ( ( cos 4 3 cos 4 3 ( ( sin sin cos A 0, B, 4 C, D Pge 3
33 9.5 Stess concenttion of the cicul hole Supeposition of Solution nd 3 ( ( ( cos 4 3 cos 4 3 ( ( sin G. Kisch Solution 9 Pge 3
34 9.5 Stess concenttion of the cicul hole x x x 9 Pge 33
35 9.5 Stess concenttion of the cicul hole Stess concenttion of ellipse hole x ( mx ( 9 Pge 34
36 Homewok Pge 9 35
37 期中考试 005 年 月 3 日, 下午 6:00-8:00 地点 :( 三 8 ( 一班 ( 三 0 ( 三班 一班 二班二班 三班前 5 号 三班 6 号以后, 四班, 七班 Pge
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