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

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

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

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

Two-Dimensional Solution in Polar Coordinate Theory of Elasticity Chapter Page 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

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

9.1 Polar Coordinate Formulation Theory of Elasticity Chapter Page 9.1 Polar Coordinate Formulation Polar Coordinates 9 6

9.1 Polar Coordinate Formulation Theory of Elasticity Chapter Page 9.1 Polar Coordinate Formulation Polar Coordinates 9 7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

9.5 Stress concentration of the circular hole Theory of Elasticity Chapter Page 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

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

9.5 Stress concentration of the circular hole Theory of Elasticity Chapter Page 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

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

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

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

9.5 Stress concentration of the circular hole Theory of Elasticity Chapter Page 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

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

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

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

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

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

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

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)