Transcript Title

SM2-011: Plasticity
PLASTICITY
(inelastic behaviour of materials)
M.Chrzanowski: Strength of Materials
1/18
SM2-011: Plasticity
Rm

RH
Elastic materials when unloaded return to initial
shape (strains caused by loading are reversible)
Brittle material
Linear elastic
material

Plastic strains occurs when loads are
high enough

Re
RH
Elastoplastic
material
arctanE
Plastic strains are irreversible
M.Chrzanowski: Strength of Materials

Permanent plastic
strain
2/18
SM2-011: Plasticity
Different idealisations of tensile diagram for elasto-plastic materials


RH

Re
Re
Linear elasticity
Stiff material with
ideal plasticity
Elasticity with ideal
plasticity





RH

Re
Re
Typical real material

M.Chrzanowski: Strength of Materials
Elasto-plastic material with
plastic hardening

Stiff material with
plastic hardening

3/18
SM2-011: Plasticity

Elasto-plastic bending
RH  Re
  pl
  E

 pl
 Re
Centre of
gravity
Neutral axis
  pl     pl
  Re
for
   pl
   Re
for
   pl
M M
 max  Re
Elastic range
z
for
z
M M
 max  Re
zmax
y
Neutral axis
x
M
A
Beam cross-section
Side view
M.Chrzanowski: Strength of Materials
 z 
 z   E   z 
4/18
SM2-011: Plasticity
Elasto-plastic bending
Plastic limit moment
Elastic limit moment
Elastic neutral axis
z
z’
Centre of
gravity
A
M M
M M
  Re
zmax
M M  M
  Re
M z’ M
  Re
 max  Re
y
x’
y’
 z 
M.Chrzanowski: Strength of Materials
 z   E   z 
  Re
  Re
  Re
Plastic neutral axis
5/18
SM2-011: Plasticity
Elasto-plastic bending
Plastic limit moment
M z’ M
z’
z’
  Re
CoG of A1
A1
N1
y’
N2
A
x’
N 0
z1’
z2’ y’
A2
  Re
M M
CoG of A2
N  0    z ' dA  N1  N 2  A1 Re  A2 Re
A1  A2  A / 2
A
M    z ' z ' dA  N1  z '1  N 2  z '2  Re A1  z '1  Re A2  z '2  Re S ' 1  S '2 
A






A
A

M  Re S '1  S '2  Re  z'1  z'2   ARe  z '1  z'2 / 2 M  Re  z ' 1  z '2  A / 2   ReWpl
2
2

M.Chrzanowski: Strength of Materials
6/18
SM2-011: Plasticity
Elasto-plastic bending
Limit elastic moment
Limit plastic moment
M M
 max  M /Wspr
M  Re  Wspr
M M
 max  Re
Wspr  J y / zmax
z
M  Re  W pl


W pl  z ' 1  z '2  A / 2 
 max  Re
z’
  Re
y
z1 ’
z2 ’
A
A/2
y’
A/2
  Re
M / M  Wpl / Wspr  k
M.Chrzanowski: Strength of Materials
k1 – shape coefficient
7/18
SM2-011: Plasticity
z
b h2
Wpl  2 S yc A 1   2 b h h 
2 4
4
b h2
Wspr 
6
A1


   b h2
Wpl  S yo A 1   S yo A 2   b h h  b h   h ÷
2 4  2
4  
4
h
yc= yo
A2
M  Re
b h2
6
b
M  Re
k 
M

M
W pl
b h2
4
 1.5
W spr
z
Wspr 
A1
d
A2
yc= yo
M.Chrzanowski: Strength of Materials
p d3
32
p d2 4 d 2 d3

Wpl  2 S yc A 1   2 1
2 4
3p
6
k 
W pl
W spr

32
6 p
 1.7
8/18
SM2-011: Plasticity
5
2
2
3
20
6
k = 1.42
k = 1.76
k = 1.52
7
5
2
2
5 5 5
2
1
1
3
4
3
5
1
9
10
8
k = 2.38
k = 1.45
15
k = 2.34
1
4
2
6
12
9
MC riddle:
a
Loading plane
k=1,5
M.Chrzanowski: Strength of Materials
k=k(a)=?
k=?
9/18
SM2-011: Plasticity

Plastic limit of a cross-section
RH  Re
Elasticity with ideal
plasticity
  pl
 Re
Nonhomogeneous
distribution
Homogeneous
distribution
Statically determined
Limit elastic
capacity
No
plastic gain
Tension
Bending
Limit plastic
capacity
N  Re A N  Re A
M  ReWspr

 pl
M  ReWpl
Ratio of plastic to
elastic capacities k
N / N 1
M / M  k 1
Plastic gain
Statically undetermined
M.Chrzanowski: Strength of Materials
10/18
SM2-011: Plasticity
Limit analysis of structures
Statically determined structures
Length and cross-section area of both bars: l, A
Elastic solution
1
a a
1
 Plastic solution
P
2 cos a
From equilirium:
N1 
Stress in bars:
1 
P
2 A cos a
In limit elastic state:
Re 
P
2 A cosa
Limit elastic capacity:
P  2 ARe cosa
Limit plastic capacity:
P  2 ARe cos a
PP
P
M.Chrzanowski: Strength of Materials
11/18
SM2-011: Plasticity
Limit analysis of structures
Statically undetermined structures Length and cross-section area of both bars: l, A
2
Elastic solution
Equilibriuim :
2 N1 cosa  N 2  P
1
a a
1
Displacement compatibility:
l1  l2 cosa
P PP
N1 
P cos a
1  2 cos 2 a
N1l N 2l

cos a
EA EA

N2 
Elastic limit capacity – plastic limit in bar #2
N2  ARe
N1  N 2 cosa
P
1  2 cos 2 a
P  ARe 1 2 cos2 a 
Plastic limit capacity – plastic limit in bars #1 and #2
N1  N2  ARe
ARe  2 ARe cos a  P
1  2 cos a
P/P 
1
2
1  2 cos a
M.Chrzanowski: Strength of Materials
P  ARe (1  2 cos a )
12/18
SM2-011: Plasticity
P/P
1  2 cos a
P/P 
1
2
1  2 cos a
1,40
1,365
1,30
67,5o
1,20
1,10
1,00
a [o ]
0
10
20
30
40
50
60
70
80
Capacity of the 3-bar structure due to plastic
properties
M.Chrzanowski: Strength of Materials
90
13/18
SM2-011: Plasticity
Limit analysis of beams
Concept of plastic hinge
z’
  Re
  Re
  Re
   
 
x’
  Re
Trace of the cross-section
plane according to the
Bernoulli hypothesis
x 
z

1

 z
M.Chrzanowski: Strength of Materials
Beam axis

x     

Plastic hinge:
  Re
  Re
  0
M M
 
14/18
SM2-011: Plasticity
Limit analysis of beams
Moment – curvature interdependence
In elastic range:
In plastic range:
M
1


EJ

?
M  f ( )
M
  
EJ
M M
M M
M /M k
 
M /M
k
1
 /
1
M.Chrzanowski: Strength of Materials
15/18
SM2-011: Plasticity
Limit analysis of beams
Statically
P determined structures
P
Plastic hinge
P
Bending moment
M  Pl / 4
Curvature
M  Pl / 4
4M / l M
P/P 

k
  Pl / 4 EJ
  Pl / 4 EJ
4M / l M
M  Pl / 4
 
Plastic zone spreading
l pl  l 1 1 / k 
M.Chrzanowski: Strength of Materials
16/18
SM2-011: Plasticity
PPP
P indetermined
Statically
structures
Unstable mechanism!
P
P
Limit elastic moment
Limit plastic moment
2,5Pl / 16
3Pl / 16
Shear forces diagram
M  3Pl / 16
P  16M 3l
P  4M / l  2M / l  6M / l
4M / l
2M / l
P/P 
M.Chrzanowski: Strength of Materials
6M 3l
18 M

 1,125k
l 16M 16 M
17/18
SM2-011: Plasticity
Limit analysis by virtual work principle
In limit plastic state the moment distribution due
to given mechanism is known. Example:
P
On this basis limit plastic capacity can be easily
found, however, the ratio of plastic to elastic
capacity is unavailable.
l/2
l/2
M
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.
3M    P    l / 2
M



M    2M    P  
P
P  6M / l
M.Chrzanowski: Strength of Materials
M M
 M
18/18
SM2-011: Plasticity
stop
M.Chrzanowski: Strength of Materials
19/18