Mat Ch35

PLASTIC ANALYSIS

Teaching Resources IIT Madras, SERC Madras, Anna Univ., INSDAG Calcutta

PLASTIC ANALYSIS
IntroductionFundamentals of plastic theoryBending of beams symmetrical about
both axes
General Requirements for utilising
plastic design concepts
Plastic hinges2

PLASTIC ANALYSIS - 2
Fundamental conditions for plastic
analysis
Rigid plastic AnalysisStabilityEffect of axial load and shearPlastic analysis for more than one
condition of loading


INTRODUCTION
Elastic design method limits the usefulness to the
allowable stress of the material, which is well below the

elastic limit
In plastic design method, the ultimate load is regarded
as the design criterion.
Plastic design method is rapid and provides a rational
approach for the analysis.
Plastic design method can be easily applied in the
analysis and design of statically indeterminate framed

structures.


Basis of Plastic Theory

Ductility of Steel

fy

A

B

Stress f

O

Strain

Idealised stress strain curve for steel in tension

Strain hardening commences

Strain hardening


Stress - Strain Curve for perfectly plastic materials

Basis of plastic theory - 2

Perfectly Elastic Material

Stress

Strain

Compression

Tension

fy

fy


Fully Plastic Moment of a Section

Assumptions

Material obeys Hooke's law until the stress reaches the upper yield value; on further straining, the stress drops to the lower yield value and thereafter remains constant.

Homogeneous and isotropic in both the elastic

and plastic states.

Plane transverse sections remain plane and

normal to the longitudinal axis after bending.


Fully Plastic Moment of a Section- 2

The yield stresses and the modulus of elasticity have the same value in compression and tension.

No resultant axial force on the beam.
Cross section of the beam is symmetrical about
an axis through its centroid parallel to plane of

bending.
Every layer of the material is free to expand and
contract longitudinally and laterally under stress.


Fully Plastic Moment of a Section- 3

Total compression , C = Total tension , T

C = fy .A1

T = fy . A2

fy

fy

A2

G2

A1

G1


Elastic stresses in beams

Bending of Beams Symmetrical about both Axes

Neutral

Axis

f

f fy

f


Teaching Resources for Steel Structures

IIT Madras, SERC Madras, Anna Univ., INSDAG Calcutta

Bending of Beams Symmetrical about both Axes - 2

Stresses in partially plastic beams

Plastic Zone (Compression)

Plastic Zone (Tension)

Neutral

Axis

(a)Rectangular

section

(b) I - section

(c) Stress distribution

for (a) or (b)

fy

fy


Bending of Beams Symmetrical about both Axes - 3

b

d

fy

fy

Neutral

Axis

(b) I - section

Stresses in fully plastic beams

(a) Rectangular

section

(c) Stress distribution

for (a) or (b)


General Requirements for Utilising
Plastic Design Concepts

Shape factor varies from 1.15 to 1.8

S= (Plastic Modulus)/(Elastic Modulus)

1.00

0.87

0.67

o

Rotation

a

b

(S 1.00)

(S 1.15)

(S 1.50)

(S 1.80)

Moment rotation curves




Plastic hinge - a yielded zone at which an infinite
rotation can take place at a constant plastic moment.
Theoretically, a plastic hinge is assumed to form
at a point of plastic rotation.
There is a constant moment at the Plastic hinge, with
a value equal to Plastic moment Capacity of the

cross section

Plastic Hinges




Hinged Length of a Simply Supported Beam with

Central Concentrated Load

Plastic Hinges

It has been shown that

My = 2/3Mp and

L/2

L/2

MY

MY

Mp

W

x

b

h




Mechanism condition

The ultimate or collapse load is reached when a mechanism is formed.
Equilibrium condition
Fx = 0, Fy = 0, Mxy = 0
Plastic moment condition
Bending moment should not be more than the plastic moment.

Fundamental Conditions for Plastic Analysis


Mechanism

- Beam Mechanism

- Panel or Sway Mechanism

- Gable Mechanism

- Joint Mechanism

- Combined Mechanism




Mechanism - 2

Beam Mechanism

Simply supported beam

(fails due to formation of one hinge)

Propped cantilever beam

(fails when 2 hinges are formed)




Beam Mechanism

Mechanism - 3

1

3

2

Fixed beam

Three hinges are formed in the following order as shown:

1, 2, 3




Mechanism - 4

Panel, Gable and Joint Mechanisms

(a) Panel Mechanism

(b) Gable Mechanism

(c) Joint Mechanism




Mechanism - 5

Combined Mechanism

(Two hinges on the beam + 2 hinges at the base)

Combined Mechanism

Two hinges developed on the beam




Lower Bound or Static Theorem

A load factor (s ) computed on the basis of an arbitrarily assumed bending moment diagram which is in equilibrium with the applied loads and where the fully plastic moment of resistance is nowhere exceeded will always be less than (or at best equal to) the load factor at rigid plastic collapse, (p).

Load Factor and Theorems of

Plastic Collapse

Plastic analysis is governed by three theorems




Upper Bound or Kinematic Theorem

A load factor (k) computed on the basis of an arbitrarily assumed mechanism will always be greater than, (or at

best equal to) the load factor at rigid plastic collapse (p )

Uniqueness Theorem

If both the above criteria are satisfied, then the resulting load factor corresponds to its value at rigid plastic collapse (p).

Load factor and theorems of plastic collapse - 2




Rigid plastic analysis

W

Plastic zone

Yield zone

Stiff length

2

M

Simply supported beam at plastic hinge stage

L

B. M. D.

Moment rotation curve




Rigid plastic analysis - 2

P=0

P=0

MB

A

B

C

W / unit length

L

Loading

MP

MP

MP

MP

2

Collapse

Encastre Beam

MA




Continuous Beams

Rigid Plastic Analysis - 3

10W

10W

6W

2L

2L

3L

2L

2L

3L




Collapse pattern 3:

Collapse pattern 2:

Collapse pattern 1:


10W

2

10W

6W

10W

6W

10W

2

3

5

10W

10W

6W

2

3

5




Mechanism Method
In this method, various alternative plastic failure
mechanisms are evaluated.
The plastic collapse loads are obtained by equating
the internal work to the external work done by

loads.
For gabled frames and other such frames, the
kinematics of collapse is somewhat complex.





Therefore it is convenient to use the instantaneous centres of rotation of the rigid elements of the frame to evaluate displacements corresponding to different mechanisms.
Rigid Body Rotation

x

y

o

x

F

P

P

r


Rectangular Portal Framework and Interaction Diagrams


H

a

a

a

V

(a)

H

V

H

V

(c)

2

(d)

H

V

2

(b)

Possible Failure Mechanisms





Interaction Diagram

0

2

4

6

2

4

6

Va / Mp

A

B

C

D

Ha / Mp


Frames not of Constant Section Throughout


H

a

a

a

V

(a)

H

V

(b)

H

(c)

V

2

(d)

H

2

V

2





0

2

4

6

2

4

6

Interaction Diagrams

8

8

A

B

C

D

Va / Mp

Ha / Mp


For plastically designed frames three stability criteria to be considered for ensuring the safety of the frame.
General Frame Stability Local Buckling Criterion Restraints
Stability




If a member is subjected to the combined action of bending moment and axial force, the plastic moment capacity will be reduced.

Effect of Axial Load and Shear

Effect of axial force on plastic moment capacity

C

T

fy

fy

C

fy

b

d

C

T

fy

fy

d/2

y1

Total stresses = Bending + Axial

compression




It is necessary to perform separate calculations, one for each loading condition, the section being determined by the solution requiring the largest plastic moment.
This is because the principle of Superposition does
NOT hold for plastic Analysis

Plastic Analysis for more than one condition of loading




- Basic concepts on Plastic Analysis have been

discussed

- methods of computation of ultimate load causing

plastic collapse have been outlined

- Theorems of plastic collapse and alternative

patterns of hinge formation triggering plastic

collapse have been discussed.

Concluding Remarks




THANK YOU




1

y

S

1

2

y

p

y

M

S

1

M

=

L

3

1

x

=

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Mat Ch35