EMSD Ch 7

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Chapter VII

3-D problems with axial symmetry

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VII-2

3-D problems with axial symmetryPressurized thick-walled cylinder

( )r ru u r e

1

0r

d dr u

dr r dr

Balance equation in terms of displacements:

2 1graddiv curlcurl 0

1 2

Gu G u F

graddiv 0u

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VII-3


Gradient and divergence in cylindrical coordinates

Displacements

1grad

1 1div

r z

zr

e e er r z

v vv r v

r r r z

21

plane strain statein direction z

; 0 ; 0r zcu c r u ur

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VII-4


Strains

r

1 1 1

2 2u 1 1 1

r 2

sym.

r r z r

z

z

u uu u u u

r r r r r z u u u

r z r

u

z

2 21 12 2

; ; 0r z r rz z c c

c cr r

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VII-5


Stresses (Hookes law in plane strain state)

1 0

1 01 1 21 2

0 02

r r

r r

z r

E

1 2 1 2

2 2

1

;1 1 2 1 1 2

; 01 1 2

r

z r rz z

c c c c E E

r r

Ec

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VII-6


Boundary conditions

j ji in T

r

r i

ri

n er a p

T p e

r= a (inner wall)

r= b (outer wall)

r

r e

re

n er b p

T p e

pi

pe

a

b

n=er

n=-er

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VII-7


Constants:

2 2

1 2 2

1 2 1i ea p b p

C E b a

2

2 2

1

1 1

i ep pC

Ea b

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VII-8


Complete results in plane strain state:

2 2

2 22 2

2 2

2 2

2 22 2

2 2

2 2

2 2

2 2 2 2

2 2

2

1 11 2

2

i e i er

i e i e

i ez

r i e i e

a p b p p p

r rb a

a b

a p b p p p

r rb a

a b

a p b p

b a

u a p b p r a b p prG b a

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VII-10


pe =30N/mm2

pi =100N/mm2

a=20 mm

b=30 mm

r

r

r

r = a r = b

-30

+82

-100

+152

r

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VII-11

3-D problems with axial symmetryParticular case: thin-walled cylinder

2

2

i e

ea R

eb R

p p p

pe

pi

abR

e

p

Reif

z

plane stressplane strain

2 2

plane stress plane strain

0 ;

0 or

or 1 1

r

z

r r

pR

e

pRe

pR pRu u

Ee Ee

0r

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VII-12

3-D problems with axial symmetrySpherical coordinates

sin sin

sin cos

cos

x r

y r

z r

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VII-13

3-D problems with axial symmetryPressurized thick-walled sphere

spherical coordinates: r

22

graddiv 0

d 1 d0

d d

r r

r

u u r e

u

r ur r r

3 3

3 33 3

3 3

3 3

3 33 3

3 3

1

2

i e i e

r

i e i e

a p b p p p

r rb a

a b

a p b p p p

r rb a

a b

pi

pe

a

b

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VII-14

3-D problems with axial symmetryParticular case: thin-walled sphere

Thin-wall: ; ;2 2 i e

e ea R b R p p p

1A

01

B1

2 0

r

r

ur

upR re u

2 1 1cos

1 23 4 1

sin cotg2 1 2 sin

ru a b

rc

u a br

a, b, care constant

sinif 0 : except if cotg

sin 1 cos

b bu c b b

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VII-15

3-D problems with axial symmetrySpherical coordinates with symmetry of revolution

...Symmetry of revolution 0 and 0

1; ; cotg

1; 0

r r rr

rr r

u

u uu u u

r r r r r

u u u

r r r

1 21 1 2

Hooke's law:

2 2 21 2 1 2 1 2

0

r r

r r r

ij ij ll ij

r

E

G e G e G e

e G

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VII-16

3-D problems with axial symmetrySolution to Kelvins and Boussinesqs problems

KELVIN BOUSSINESQ

Balance equation in terms of displacements:

Homothety form of the solution is

Equilibrium

2 1graddiv curl curl 0

1 2

Gu G u F

f r g

2 21 1 1

; ;g g u g

r r r

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VII-17

3-D problems with axial symmetrySolution to Kelvins and Boussinesqs problems

2

2

2

2

2 2 1cos 2

1 2

cos 1cos 2

1 cos

1 1cos 2

1 cos

sin 1sin 2

1 cos

and = constant

r

r

u

Ga Gb

r

Ga Gbr

Ga Gbr

Ga Gbr

a b

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VII-18

3-D problems with axial symmetrySolution to Kelvins problem

KELVIN

3 4 sin 1sin

2 1 2 1 cos

2 1 1cos

1 2

3 4 1

sin2 1

if he 0

2

nce

r

bu a

r

u ar

u a

b

r

a ? balance of vertical forces

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VII-19

3-D problems with axial symmetrySolution to Kelvins problem

Vertical balance

0 cos sin 2 sin d 0

1 28 1

r rP r r

PaG

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VII-20

3-D problems with axial symmetrySolution to Boussinesqs problem

BOUSSINESQ

2

sin 1sin 2

0 2 0i

1 co

0 O

s

fK2

r

r Ga Gbr

a b

20 cos sin 2 sin d 01 2 1 2

;2 4

r rP r r

P Pa b

G G

Vertical balance

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VII-21

3-D problems with axial symmetryBoussinesqs problem applied to a distributed load

Principle:

Substitute Pwith q dAthen integrate (generallynumerical integration)

one gets the general solution to the problem

of stress determination under a foundation

Assumptions:

1. the ground is a linear elastic material and follows Hookes law

2. the vertical load distribution q is known

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VII-22

3-D problems with axial symmetryBoussinesqs problem applied to a distributed load

Practically, it can be hardto determine the load

distribution:only Mand Pare known

Flexible foundation slab: linear distribution of pressure

Rigid foundation slab:

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VII-23

3-D problems with axial symmetryParticular case: load distributed over a circle

In plane view

OM r

d d d

load d d d

A s s

p A ps s

Vertical displacement in M (surface):

2

2

sin 1 23 4 with d and

2 4 1 cos

1 1 1 1d d d d d d d d

2 2 2

Pw u P p A r s

Gr

p pw s s s w p s p s

Gs G G E

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VII-24

3-D problems with axial symmetryHertzs problem

2 2 22

2 2 2 2 2

2 2

2 1

2 1

2

1 2 2

1 2

1 2

1 2

1 2

2

idem2 2

MN 2

with2

R R z z r R

r rz z

R R

r R R

z z rR R

R R

R R

2

2

2

2

1 2 1 22

1

1

1

1 d d

d d1

d d

w p sE

w w k k p s

w p sE

2 2

1 2

1 2

1 2

1 1with andk k

E E

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VII-25


1 2 1 2

2

1 2 1 2 1 2

w w z z

z z w w r w w r

Centers of both spheres are getting closer by :

21 2 d dk k p s r

Integral equation forp =p(r)

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VII-26


2 2 2

d d d d

d area of the diagram .

sin2

p s p s

p s k

a r k

Solution: diagram ofp hemisphere the radius of which is a

2

22 2 2 2 2 0

2

d d sin d 2 where2 4

pp s k a r k a r k

a

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VII-27


hence:

1 2

1 2

2210

1 2 11

2

2

2

2

2

1with

4

1

R R

R R

pa k k k

E

kE

2

2 1 2 02 2 20

1 2 2

01 2

22

4

4

ak k p

pk k a r r r

a pk k

a

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VII-28


One can expressp0 as a function of the compression force P

and the solution is:

30Contact area

0 2

2d volume of hemisphere

3

3

2

pP p A k a

a

Pp a

1 23

2223

1 2

3is the radius of the contact surface

4is the reduction of the distance9

between both spheres16

P k ka

P k k

diagram ofp is known

everything is determined (BOUSSINESQ)

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VII-29


1st case: 2 identical balls (same E,,R) = 0.3

2nd case: Ball on a planar surface

2 2

3 3302 2

2 41.109 1.23 0.388

2

P R P PE a p

E E R R

1

3

2

32

2

30 2

1.109

1.23

0.388

R

PRa

E

P

E R

PEp

R

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VII-30


2nd case: evolution of the stresses along axis z

max1

2r z

the largest max is located atz= a / 2

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EMSD Ch 7