plastic collapse – beam in bending
TRANSCRIPT
8/12/2019 Plastic Collapse – Beam in Bending
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Plastic Collapse – Beam in
Bending
M=Px/2V=P/2
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Elastic Stress Distributions
Bending Shear
t
b
sx=My/Izz txy=Vq/bIzz
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Post-yield Behaviour – Elastic
Perfectly Plastic Material
M
so
so
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Curvature b
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Strain Distribution
-c
c
c
M
b
Stress Distribution
so
so
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Ramberg-Osgood
(Elastic-plastic Material)
Stress-Strain Relationship M Stress Distribution
n
K E
1
s s
Strain Distribution
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Fully-Plastic Moment
• Equilibrium of x-section
d=h/2
so
so
Mo
F=so bh/2
Mo=(so bh/2)(h/2)
= so bh2/4
b
h
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Maximum Elastic Moment
Fc
Ft
2h/3
F c = F t =( so/2)bh/2
M= F t d = ( so bh/4)(2h/3)= so bh2/6
h/2
so
so
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Section and Plastic Modulus
b
h
z zSzz = bh
2
/6
Zzz = bh2/4
= 1.5 Szz Rectangular x-section
Szz Zzz
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Progression of Yield Yone
Leading to Fully Plastic Hinge and Collapse
• Stresses reach Yield Magnitude at extreme fibres
• Yield Zones spreads towards
Neutral axis
• Yield Zones join, are now
spread through entire x-section
• Plastic Hinge causes structural
collapse
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Elastic-Fully Plastic Moment
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Irregular X-section
to maintain equilibrium, net force is zero
dytycedisarea stressdM s )tan)()((
1
2
c
c
dyty M s
1
2
0c
cdyt P s
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Plastic Torsion: Elastic-plastic material
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Stress-strain behaviour in torsion
(a) Assume strains are small, plane section remain planeduring deformation.
(b) Assume shafts have circular x-section: cylinder or tube
(derivations not valid for non-circular x-section)
(c) Stress-strain curve in torsion can then be approximated
from uniaxial stress-strain curve
t
c
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n xy xy
xy H G
1
t t
2/1
3
n
K H
n
K E
1
s s
Uniaxial Tension Pure Shear
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Residual Stresses
(a) Zero Load - Intial Condition
(b) Maximum Load, Mo>M’>My
(c) Zero Load - Unloaded, Plastically Deformed
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Residual Stress Distribution
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Residual Stress development
Elastic-Perfectly Plastic Material
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Special Case: Un-loading after Fully Plastic Loading
Stress distributions through rectangular x-section
(a) Stress at fully plastic load
(b) Stress change during unloading
(c) Net residual stress in un-loaded condition
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Summary
• The collapse load of a structure is reached
when the x-section forms a number of fully plastic hinges sufficent to create a
mechanism.
• Unloding from a plastically deformed stateleaves residual stresses in the material
• The residual stresses are tensile where the
yield was compressive and vice versa.• The x-section remains in equilibrium; i.e.,
the product of residual stresses and area
over which they act is zero