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PLASTICITY (inelastic behaviour of materials). R m. R H. . R e. R H. . Permanent plastic strain. arctanE. . Brittle material. Linear elastic material. Elastic materials when unloaded return to initial shape (strains caused by loading are reversible). .

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slide2

Rm

RH

Re

RH

Permanent plastic strain

arctanE

Brittle material

Linear elastic material

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

Plastic strains occurs when loads are high enough

Elastoplastic material

Plastic strains are irreversible

slide3

RH

Re

Linear elasticity

Stiff material with ideal plasticity

Re

Elasticity with ideal plasticity

Re

Re

Stiff material with plastic hardening

Elasto-plastic material with plastic hardening

RH

Typical real material

Different idealisations of tensile diagram for elasto-plastic materials

slide4

z

z

Neutral axis

zmax

for

for

for

Centre of gravity

y

x

M

A

Elasto-plastic bending

Elastic range

Neutral axis

Beam cross-section

Side view

slide5

z’

y’

z

zmax

Centre of gravity

z’

y

A

x’

Elasto-plastic bending

Plastic limit moment

Elastic limit moment

Elastic neutral axis

Plastic neutral axis

slide6

z’

z’

z’

N1

z1’

z2’

x’

y’

y’

N2

CoG ofA2

CoG of A1

Elasto-plastic bending

Plastic limit moment

A1

A

A2

slide7

z’

z1’

z2’

y’

z

y

A

Elasto-plastic bending

Limit elastic moment

Limit plastic moment

A/2

A/2

k1 – shape coefficient

slide8

z

z

2

2

b

h

b

h

(

)

h

h

=

=

=

=

W

W

2

S

A

2

b

spr

pl

yc

1

6

2

4

4

A1

2

é

ù

b

h

æ

ö

(

(

)

)

h

h

h

h

=

-

=

-

-

=

ç

÷

W

S

A

S

A

b

b

ê

ú

pl

yo

1

yo

2

è

ø

h

2

4

2

4

4

ë

û

yc= yo

2

2

W

b

h

b

h

M

pl

=

=

=

=

=

M

R

M

R

k

1

.

5

A2

e

e

6

4

W

M

spr

b

p

3

d

=

W

spr

32

p

2

d

4

d

2

3

(

)

1

d

=

=

=

W

2

S

A

2

pl

yc

1

p

2

4

3

6

W

32

pl

=

=

=

k

1

.

7

p

W

6

spr

A1

d

yc= yo

A2

slide9

5

1

5

2

1

2

6

k = 1.76

2

2

2

1

a

3

Loading plane

15

k = 2.34

k = 1.52

20

9

k = 1.42

8

k = 1.45

10

k = 2.38

7

1

4

5

2

5

5

5

3

4

3

12

6

9

MC riddle:

k=1,5

k=k(a)=?

k=?

slide10

Plastic limit of a cross-section

Elasticity with ideal plasticity

Statically determined

No plastic gain

Homogeneous distribution

Non-homogeneous distribution

Plastic gain

Statically undetermined

slide11

a

a

1

1

P

Limit analysis of structures

Statically determined structures

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

Elastic solution

 Plastic solution

From equilirium:

Stress in bars:

In limit elastic state:

Limit elastic capacity:

Limit plastic capacity:

slide12

a

a

a

P

1

1

2

Limit analysis of structures

Statically undetermined structures

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

Elastic solution

Equilibriuim :

Displacement compatibility:

Elastic limit capacity – plastic limit in bar #2

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

slide13

1,40

1,365

1,30

67,5o

1,20

1,10

a [o]

1,00

90

0

10

20

30

40

50

60

70

80

Capacity of the 3-bar structure due to plastic properties

slide14

z’

x’

Plastic hinge:

Limit analysis of beams

Concept of plastic hinge

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

Beam axis

slide15

Limit analysis of beams

Moment–curvature interdependence

In elastic range:

In plastic range:

k

1

1

slide16

Limit analysis of beams

Plastic hinge

Statically determined structures

Bending moment

Curvature

Plastic zone spreading

slide17

Unstable mechanism!

Limit elastic moment

Limit plastic moment

Shear forces diagram

Statically indetermined structures

slide18

l/2

l/2

Limit analysis by virtual work principle

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

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.