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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