1 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity PLASTICITY (inelastic behaviour of...

1/18M.Chrzanowski: Strength of Materials

SM2-011: Plasticity

PLASTICITY(inelastic behaviour of materials)


SM2-011: Plasticity

RH

Elastic materials when unloaded return to initial shape (strains caused by loading are reversible)

Plastic strains are irreversible

Plastic strains occurs when loads are high enough

RH

Rm

Re

Linear elastic material

Elastoplastic material

Brittle material

Permanent plastic strain

arctanE


SM2-011: Plasticity

RH

Linear elasticity

Re

Elasticity with ideal plasticity

Different idealisations of tensile diagram for elasto-plastic materials

Re

Elasto-plastic material with plastic hardening

Re

Stiff material with plastic hardening

Re

Stiff material with ideal plasticity

RH

Typical real material


SM2-011: Plasticity

Elasto-plastic bending

for

for

for

E

eR

eR

plpl

pl

pl

eH RR

pl

eR

pl

Mx

z

zEz

Elastic range

z

MM MM eRmax

eRmax

y

z

A

Side view

Neutral axis

Beam cross-section

zmax

Neutral axis

Centre of gravity


SM2-011: Plasticity

zEz

MM

eRmaxy

z

A

zmax

Elastic neutral axis

Centre of gravity

Elastic limit moment

eR

eReR

eR

eR

eR

Plastic limit moment

MM MM MMM z’

y’

z’

x’

z


Plastic neutral axis


SM2-011: Plasticity


eR

eR

Plastic limit moment

MM z’

y’

z’

x’A

A1

A2

z1’z2’

CoG of A1

N1

CoG of A2

N2 MM

0N

ee

A

RARANNdAzN 2121'0 2/21 AAA

2122112211 '''''''' SSRzARzARzNzNdAzzM eee

A

2/'''2

'2

'' 212121 zzARzA

zA

RSSRM eee

z’

y’

plee WRAzzRM 2/'' 21


SM2-011: Plasticity

Limit elastic moment

MM

eRmax

MM

sprWM /max

max/ zJW yspr

y

z

A

Limit plastic moment

spre WRM 2/'' 21 AzzWpl

eR

eR

eRmaxz’

y’

z1’z2’

A/2

A/2

ple WRM

k1 – shape coefficient


kWWMM sprpl //


SM2-011: Plasticity

yc= yo

z

A1

A2

d

W S A b h h b hpl yc 2 2

2 4 41

2

Wb h

spr 2

6

W S A S A b h h b h h b h

pl yo yo

1 2

2

2 4 2 4 4

yc= yo

z

b

h

A1

A2 M R

b he

2

6 M R

b he

2

4

kM

M

W

W

pl

spr

1 5.

Wd

spr 3

32

W S Ad d d

pl yc 2 2 12 4

4 2

3 61

2 3

kW

W

pl

spr

32

61 7

.


SM2-011: Plasticity

6

2

2 2 2

k = 1.76

5 5 5

5

5

20k = 1.42

3 4 3

2

7

3

k = 1.52

11

6

4

910k = 2.38

1

8

2

5

9

1k = 1.45

12

15 k = 2.34

k=1,5

k=?

MC riddle:

k=k()=?

Loading plane


SM2-011: Plasticity

Limit elastic capacity

Limit plastic capacity

Ratio of plastic to elastic capacities k

Tension

Bending

Plastic limit of a cross-section

spreWRM

1/ NN

pleWRM

eH RR

pl

eR

pl

ARN e

1/ kMM

ARN e

Elasticity with ideal plasticity

Statically undetermined

Statically determined

No plastic gain

Plastic gain

Hom

oge

neou

s di

strib

utio

n

Non

-ho

mog

eneo

us

dist

ribut

ion


SM2-011: Plasticity

Limit analysis of structures

Length and cross-section area of both bars: l, A

Elastic solution

cos2A

PRe

cos21

PN

cos2 eARP

cos21 A

P

11

P

From equilirium:

Plastic solution

Statically determined structures

Stress in bars:

In limit elastic state:

Limit elastic capacity:

cos2 eARP Limit plastic capacity:

PP


SM2-011: Plasticity

Length and cross-section area of both bars: l, AElastic solution

Displacement compatibility:

Equilibriuim : PNN 21 cos2

cos21 ll cos21

EA

lN

EA

lN cos21 NN

21 cos21

cos

PN

2cos21 eARP

Elastic limit capacity – plastic limit in bar #2

eARN 2

eARNN 21 PARAR ee cos2 )cos21( eARP

22 cos21

PN

Plastic limit capacity – plastic limit in bars #1 and #2

1

2

1

P

1cos21

cos21/

2

PP

Statically undetermined structures

P P

Limit analysis of structures


SM2-011: Plasticity

0 20 5030 40 60 70 80 901,00

1,20

1,3651,40

1,10

10

67,5o1,30

1cos21

cos21/

2

PP

Capacity of the 3-bar structure due to plastic properties

PP /


SM2-011: Plasticity

Limit analysis of beams Concept of plastic hinge

eR

eR

z’

x’

eR

eR

eR

eR

Trace of the cross-section plane according to the Bernoulli hypothesis

x

zz

x

0

1

Beam axis

Plastic hinge: MM


SM2-011: Plasticity

Moment – curvature interdependenceIn elastic range:

1

EJ

MIn plastic range:

MM

/

MM /

1

1

)(?

fM

MM EJ

M

kMM /

k

Limit analysis of beams


SM2-011: Plasticity

Statically determined structures P PP

4/lPM

EJPl 4/ EJlP 4/

4/lPM

kllpl /11

Bending moment

Curvature

Plastic zone spreading

Plastic hinge

4/PlM

Limit analysis of beams

kM

M

lM

lMPP

/4

/4/


SM2-011: Plasticity

PStatically indetermined structures

P P

Limit elastic moment Limit plastic moment

Unstable mechanism!

lMlMlMP /6/2/4

16/3Pl

16/5,2 Pl16/3 lPM Shear forces diagram

lM /4

lM /2lMP 316

PPP PPP

kM

M

M

l

l

MPP 125,1

16

18

16

36/


SM2-011: Plasticity

Limit analysis by virtual work principle

In limit plastic state the moment distribution due to given mechanism is known. Example: P

M

M

l/2l/2

MM

P

M

PMM 22/3 lPM

lMP /6

On this basis limit plastic capacity can be easily found, however, the ratio of plastic to elastic capacity is unavailable.

In a more complex case one has to consider all possible mechanisms. The right one is that which yields the smallest value of limit plastic capacity.


SM2-011: Plasticity

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