1. Chapter 9 Two-Dimensional Solution
Theory of Elasticity
Chapter 9
Two-Dimensional Solution

2. Content Theory of Elasticity Introduction Mathematical Preliminaries
Chapter
Page
Content
Introduction
Mathematical Preliminaries
Stress and Equilibrium
Displacements and Strains
Material Behavior- Linear Elastic Solids
Formulation and Solution Strategies
Two-Dimensional Problems
Introduction to Finite Element Method
Three-Dimensional Problems
Bending of Thin Plates 9 1

3. Two-Dimensional Solution in Polar Coordinate
Theory of Elasticity
Chapter
Page
Two-Dimensional Solution in Polar Coordinate
9.1 Polar Coordinate Formulation(极坐标下的求解)
9.2 Coordinate Transformation of Stress Components （应力分量的坐标变换）
9.3 Axisymmetrial stresses and corresponding displacements（轴对称应力和位移）
9.4 Hollow cylinder subjected to uniform pressures （圆环受均布压力）
9.5 Stress concentration of the circular hole（圆孔的孔口应力集中） 9 2

4. Two-Dimensional Solution in Polar Coordinate
Theory of Elasticity
Chapter
Page
Two-Dimensional Solution in Polar Coordinate
Some symmetry and circular structure
Aeroengine and its rotor system
Practical Engineering need solutions in polar coordinate 9 3

5. Two-Dimensional Solution in Polar Coordinate
Theory of Elasticity
Chapter
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Two-Dimensional Solution in Polar Coordinate
Stress Concentration in Practical Engineering
First civil airline Comet: May 2nd,1953 , London-Johnston
Air disaster: Jan.10,1954, and April 8,1954 9 4

6. Two-Dimensional Solution in Polar Coordinate
Theory of Elasticity
Chapter
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Two-Dimensional Solution in Polar Coordinate
Stress Concentration is the main reason that cause the air disasters 9 5

7. 9.1 Polar Coordinate Formulation
Theory of Elasticity
Chapter
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9.1 Polar Coordinate Formulation
Polar Coordinates 9 6

8. 9.1 Polar Coordinate Formulation
Theory of Elasticity
Chapter
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9.1 Polar Coordinate Formulation
Polar Coordinates 9 7

9. 9.1 Polar Coordinate Formulation
Theory of Elasticity
Chapter
Page
9.1 Polar Coordinate Formulation
Basic Equations in Polar Coordinates（极坐标下的基本方程）
Stresses
Laplace operators
Biharmonic operators 9 8

10. 9.1 Polar Coordinate Formulation
Theory of Elasticity
Chapter
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9.1 Polar Coordinate Formulation
Equilibrium Equations(平衡方程) x y O P A B C 9 9

11. 9.1 Polar Coordinate Formulation
Theory of Elasticity
Chapter
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9.1 Polar Coordinate Formulation
Geometrical Equations几何方程 9 10

12. 9.1 Polar Coordinate Formulation
Theory of Elasticity
Chapter
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9.1 Polar Coordinate Formulation
Physical Equations物理方程
Plain Stress
Plain Strain 9 11

13. 9.1 Polar Coordinate Formulation
Theory of Elasticity
Chapter
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9.1 Polar Coordinate Formulation
Boundary Conditions(边界条件) l r 9 12

14. 9.2 Coordinate Transformation of Stress Components
Theory of Elasticity
Chapter
Page
9.2 Coordinate Transformation of Stress Components
Polar Coordinate  Cartesian Coordinate
Cartesian Coordinate  Polar Coordinate 9 13

15. 9.3 Axisymmetrial stresses and corresponding displacements(轴对称应力和位移)
Theory of Elasticity
Chapter
Page
9.3 Axisymmetrial stresses and corresponding displacements(轴对称应力和位移)
axisymmetric
field quantities are independent of the angular coordinate √ X 9 14

16. 9.3 Axisymmetrial stresses and corresponding displacements
Theory of Elasticity
Chapter
Page
9.3 Axisymmetrial stresses and corresponding displacements
axisymmetric
axisymmetric case
axisymmetric case 9 15

17. 9.3 Axisymmetrial stresses and corresponding displacements
Theory of Elasticity
Chapter
Page
9.3 Axisymmetrial stresses and corresponding displacements
Michell solution of the biharmonic equation
Plane stress case
H,I,K associated with the rigid-body motion 9 16

18. 9.3 Axisymmetrial stresses and corresponding displacements
Theory of Elasticity
Chapter
Page
9.3 Axisymmetrial stresses and corresponding displacements
Function of “hole” on distribution
No hole:
A and B vanish.
Otherwise when r=0, stress component become infinite
A plate without a hole with no body forces (axissymmetrical)
If there is hole at the origin, we will investigate it next 9 17

19. 9.4 Hollow cylinder subjected to uniform pressures
Theory of Elasticity
Chapter
Page
9.4 Hollow cylinder subjected to uniform pressures
plane stress conditions
Axisymmetric problem
Boundary Conditions 9 18

20. 9.4 Hollow cylinder subjected to uniform pressures
Theory of Elasticity
Chapter
Page
3 unknowns, 2 Equations ?
Single or multiply connected region?
for multiply connected regions, the compatibility equations are not sufficient to guarantee single-valued displacements.
B = 0 9 19

21. 9.4 Hollow cylinder subjected to uniform pressures
Theory of Elasticity
Chapter
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9.4 Hollow cylinder subjected to uniform pressures
B = 0 9 20

22. 9.4 Hollow cylinder subjected to uniform pressures
Theory of Elasticity
Chapter
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Demonstration of Saint-Venant’s Principle 9 21

23. 9.5 Stress concentration of the circular hole
Theory of Elasticity
Chapter
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9.5 Stress concentration of the circular hole
Review: 9 22

24. 9.5 Stress concentration of the circular hole
Theory of Elasticity
Chapter
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9.5 Stress concentration of the circular hole
What is stress concentration?
The stress concentration near a hole is a critical issue concerning the strength of a solid structure. The stress concentration can be measured by the stress concentration coefficients that are the ratios between the most severe stress at the critical point (or termed hot spot) and the remote stress. 9 23

25. 9.5 Stress concentration of the circular hole
Theory of Elasticity
Chapter
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9.5 Stress concentration of the circular hole
Examples: 9 24

26. 9.5 Stress concentration of the circular hole
Theory of Elasticity
Chapter
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9.5 Stress concentration of the circular hole
Solution:
Selection of coordinate
To analyse stress concentration near the hole, it is convenient to use polar coordinate.
Problem in polar coordinate
Boundary conditions in polar coordinate: A A 9 25

27. 9.5 Stress concentration of the circular hole
Theory of Elasticity
Chapter
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9.5 Stress concentration of the circular hole b
Problem in polar coordinate 9 26

28. 9.5 Stress concentration of the circular hole
Theory of Elasticity
Chapter
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9.5 Stress concentration of the circular hole b a
Problem 1
Problem in polar coordinate b ＝
Problem 2 ＋ b a 9 27

29. 9.5 Stress concentration of the circular hole
Theory of Elasticity
Chapter
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9.5 Stress concentration of the circular hole
Solution of Problem 1 b a
B.C.
when b>>a 9 28

30. 9.5 Stress concentration of the circular hole
Theory of Elasticity
Chapter
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9.5 Stress concentration of the circular hole
Problem 2
Solution of Problem 2 b a
由边界条件可假设： σr为 r 的某一函数乘以cos2θ ；τr θ 为r 的某一函数乘sin2θ。
Assume： 9 29

31. 9.5 Stress concentration of the circular hole
Theory of Elasticity
Chapter
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9.5 Stress concentration of the circular hole 9 30

32. 9.5 Stress concentration of the circular hole
Theory of Elasticity
Chapter
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9.5 Stress concentration of the circular hole b a
B.C. 9 31

33. 9.5 Stress concentration of the circular hole
Theory of Elasticity
Chapter
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9.5 Stress concentration of the circular hole
Superposition of Solution 1and 2
G. Kirsch Solution 9 32

34. 9.5 Stress concentration of the circular hole
Theory of Elasticity
Chapter
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9.5 Stress concentration of the circular hole x y q1 x y q1 q2 x y q2 9 33

35. 9.5 Stress concentration of the circular hole
Theory of Elasticity
Chapter
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9.5 Stress concentration of the circular hole
Stress concentration of ellipse hole x y q 2a 2b 9 34

36. Theory of Elasticity
Chapter
Page
Homework
4-8
4-13
4-15
4-16 9 35

37. 期中考试 Theory of Elasticity 2005年11月23日，下午6：00－8：00
Chapter
Page
期中考试
2005年11月23日，下午6：00－8：00
地点：（三）218 （一班、二班、三班前15号）
（三）202 (三班16号以后，四班，七班)