1 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity PLASTICITY (inelastic behaviour of materials)
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1/18M.Chrzanowski: Strength of Materials
SM2-011: Plasticity
PLASTICITY(inelastic behaviour of materials)
2/18M.Chrzanowski: Strength 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
3/18M.Chrzanowski: Strength of Materials
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
4/18M.Chrzanowski: Strength of Materials
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
5/18M.Chrzanowski: Strength of Materials
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
Elasto-plastic bending
Plastic neutral axis
6/18M.Chrzanowski: Strength of Materials
SM2-011: Plasticity
Elasto-plastic bending
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
7/18M.Chrzanowski: Strength of Materials
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
Elasto-plastic bending
kWWMM sprpl //
8/18M.Chrzanowski: Strength of Materials
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
.
9/18M.Chrzanowski: Strength of Materials
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
10/18M.Chrzanowski: Strength of Materials
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
11/18M.Chrzanowski: Strength of Materials
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
12/18M.Chrzanowski: Strength of Materials
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
13/18M.Chrzanowski: Strength of Materials
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 /
14/18M.Chrzanowski: Strength of Materials
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
15/18M.Chrzanowski: Strength of Materials
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
16/18M.Chrzanowski: Strength of Materials
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/
17/18M.Chrzanowski: Strength of Materials
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/
18/18M.Chrzanowski: Strength of Materials
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.
19/18M.Chrzanowski: Strength of Materials
SM2-011: Plasticity
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