theory of elasticity chapter 9 two-dimensional solution

Theory of Elasticity

Chapter 9

Two-Dimensional Solution

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

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Two-Dimensional Solution in Polar Coordinate

• 9.1 Polar Coordinate Formulation( 极坐标下的求解 )• 9.2 Coordinate Transformation of Stress Compone

nts （应力分量的坐标变换）• 9.3 Axisymmetrial stresses and corresponding displaceme

nts （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）

Chapter Page 9 2



Chapter Page

Practical Engineering need solutions in polar coordinate

Aeroengine and its rotor system

Some symmetry and circular structure

9 3



Chapter Page

Stress Concentration in Practical Engineering

First civil airline Comet: May 2nd,1953 , London-JohnstonAir disaster: Jan.10,1954, and April 8,1954

9 4



Chapter Page

Stress Concentration is the main reason that cause the air disasters

9 5


9.1 Polar Coordinate Formulation

Chapter Page

r

X

Y cosrx sinry

22 yxr x

y1tan

Polar Coordinates

rrxrx

r

x

sincos

rryry

r

y

cossin

9 6



Chapter Page

r

X

YPolar Coordinates

rrrrrrrx

2

22

2

22

2

22

2

2 11cossin2

11sincos

rrrrrrry

2

22

2

22

2

22

2

2 11cossin2

11cossin

rrrrrrryx

2

222

2

2

22

22 11sincos

11cossin

9 7



Chapter Page

Basic Equations in Polar Coordinates （极坐标下的基本方程）

StressesStresses

2

2

2

11

rrrrr 2

2

r

rrr

1

Laplace operatorsLaplace operators 2

2

22

2

2

2

2

22 11

rrrryx

2

2

2

22

24 11

rrrrBiharmonic operatorsBiharmonic operators

011

2

2

2

22

2224

rrrr

9 8



Chapter Page

x

y

O

d

r

dr

P

A

B

C

rr

rk

rk

ddrr )(

rd

drrr

r

drrr

r

drr

d

r

r

1

rr

rr

0 rk

021

krrrrr

Equilibrium Equations( 平衡方程 )

9 9



Chapter Page

Geometrical Equations 几何方程

r

urr

u

rr

ur 1

r

u

r

uu

rr

r

1

9 10



Chapter Page

Physical Equations 物理方程

)(1

rr E

)(1

rE

rrr EG

)1(21

)1

(1 2

rr E

rrr EG

)1(21

)1

(1 2

rE

121

EE

Plain Stress Plain Strain

9 11



Chapter Page

Boundary Conditions( 边界条件 )

l

r

0

00

r

0

0 r

0qlr

9 12


9.2 Coordinate Transformation of Stress Components

Chapter Page

Polar Coordinate Cartesian Coordinate

2sin2cos

22 rrr

x

2sin2cos

22 rrr

y

2cos2sin

2 rr

xy

Cartesian Coordinate Polar Coordinate

2sin2cos22 xy

yxyxr

2sin2cos22 xy

yxyx

2cos2sin2 xy

yxr

9 13


9.3 Axisymmetrial stresses and corresponding displacements( 轴对称应力和位移 )

Chapter Page

axisymmetric

field quantities are independent of the angular coordinate

X √

9 14


9.3 Axisymmetrial stresses and corresponding displacements

2

2

2

11

rrrrr 2

2

r

rrr

1

Chapter Page

2

2

dr

d dr

d

rr

1 0 r

011

2

2

2

22

2224

rrrr

axisymmetric )(r 0

axisymmetric case

axisymmetric case0

12

2

2

dr

d

rdr

d

9 15



Michell solution of the biharmonic equation

Chapter Page

01

2

2

2

dr

d

rdr

d DCrrBrrA 22 lnln

2

2

dr

d dr

d

rr

1 0 r

CrBr

Ar 2)ln21(

2

CrBr

A2)ln23(

2

0 rr

cossin4

KIHrE

Bru

sincos)1(2 KICr BrrBr

r

A

Eur )31()1(ln)1(2)1(

1

H,I,K associated with the rigid-body motion

Plane stress case

9 16



Chapter Page

CrBr

Ar 2)ln21(

2

CrBr

A2)ln23(

2

0 rr

Function of “hole” on distribution

No hole:

A and B vanish.

Otherwise when r=0, stress component become infinite

.constr

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

Chapter Page

plane stress conditions

CrBr

Ar 2)ln21(

2

CrBr

A2)ln23(

2

0 rr

Axisymmetric problem

Boundary Conditions

0arr 0

brr

aarrq

bbrr

q

9 18


9 19


aqCaBa

A 2)ln21(

2

bqCbBb

A 2)ln21(

2

Chapter Page

3 unknowns, 2 Equations ?

for multiply connected regions, the compatibility equations are not sufficient to guarantee single-valued displacements.

Single or multiply connected region?

cossin

4KIHr

E

Bru

sincos)1(2 KICr BrrBr

r

A

Eur )31()1(ln)1(2)1(

1 B = 0



Chapter Page

aqCaBa

A 2)ln21(

2

bqCbBb

A 2)ln21(

2

B = 0

),(22

22

ab qqab

baA

22

22 )(2

ab

bqaqC ba

bar q

bara

q

abrb

2

2

2

2

2

2

2

2

1

1

1

1

ba q

bara

q

abrb

2

2

2

2

2

2

2

2

1

1

1

1

9 20


Chapter Page

bar q

bara

q

abrb

2

2

2

2

2

2

2

2

1

1

1

1

ba q

bara

q

abrb

2

2

2

2

2

2

2

2

1

1

1

1

bqq ab ),0(0

ar qr

a2

2

aqr

a2

2

Demonstration of Saint-Venant’s Principle

9 21

9.5 Stress concentration of the circular hole

Chapter Page

Review:

bar q

bara

q

abrb

2

2

2

2

2

2

2

2

1

1

1

1

ba q

bara

q

abrb

2

2

2

2

2

2

2

2

1

1

1

1

9 22


Chapter Page

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.

maxK

9 23


Chapter Page

Examples:

9 24


Chapter Page

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 qx

r r

A

2sin2cos22 xy

yxyxr

2cos

22

qq

2cos2sin2 xy

yxr

2sin

2

q

9 25


Chapter Page

b

0arr

0arr

2cos22

qqbrr

2sin2

qbrr

r r


9 26


Chapter Page

Problem 1

2

qr

b

a

2cos2

qr

2sin2

qr

b

a

Problem 2


＋＝b

0arr

0arr

2cos22

qqbrr

2sin2

qbrr

r r

9 27


Solution of Problem 1

Chapter Page

B.C.0

arr

0arr

2

qbrr

0brr

2

qr

b

a

21

1

2

2

2

2

q

bara

r

21

1

2

2

2

2

q

bara

0 r

when b>>a

21

2

2 q

r

ar

2

12

2 q

r

a

0 r

9 28


Solution of Problem 2

Chapter Page

2cos2

qr

2sin2

qr

b

a

Problem 2

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。

2

2

2

11

rrrr

rrr

1

Assume ： 2cos)(rf

011

2

2

2

22

2

rrrr

9 29


Chapter Page

011

2

2

2

22

2

rrrr

02cos)(9)(9)(2)(

32

2

23

3

4

4

dr

rdf

rdr

rfd

rdr

rfd

rdr

rfd

0)(9)(9)(2)(

32

2

23

3

4

4

dr

rdf

rdr

rfd

rdr

rfd

rdr

rfd

224 1

)(r

DCBrArrf 2cos)(rf

2cos12

24

rDCBrAr

9 30


Chapter Page

2cos12

24

rDCBrAr

2cos2

qr

2sin2

qr

b

a

2

2

2

11

rrrr 2cos)

642(

42 r

D

r

CB

2

2

r

2cos)

6212(

42

r

DBAr

rrr

1 2sin)62

26(42

2

r

D

r

CBAr B.C.

,0A ,4

qB

,2qaC 4

4qaD

2cos312 4

4

r

aq

2cos)3

1)(1(2 2

2

2

2

r

a

r

aqr

2sin)3

1)(1(2 2

2

2

2

r

a

r

aqrr

9 31


Chapter Page

Superposition of Solution 1and 2

2cos312

12 4

4

2

2

r

aq

r

aq

2cos)3

1)(1(2

)1(2 2

2

2

2

2

2

r

a

r

aq

r

aqr

2sin)3

1)(1(2 2

2

2

2

r

a

r

aqrr

G. Kirsch Solution

9 32


Chapter Page

x

y

q1

q2

q2

q1

x

y

q1q1

x

y

q2

q2

9 33


Chapter Page

Stress concentration of ellipse hole

x

y

qq2a

2b

)2

1()( max b

aq

9 34

Homework

• 4-8

• 4-13

• 4-15

• 4-16

Chapter Page 9 35

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 )

Chapter Page

theory of elasticity chapter 9 two-dimensional solution

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