PLASTIC ANALYSISINTRODUCTION

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.

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2

Strain hardening

fyA B

Stress f

O Strain

Idealised stress – strain curve for steel in tension

Strain hardening commences

Basis of Plastic TheoryDuctility of Steel

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Stress

Strain

Compression

Tensionfy

fy

Stress - Strain Curve for perfectly plastic materials

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.

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• 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.

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Elastic stresses in beams

Neutral Axis

f

f fy

f

Bending of Beams Symmetrical about both Axes

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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

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b

d

fy

fy

Neutral Axis

(b) I - section

Stresses in fully plastic beams

(a) Rectangular section

(c) Stress distribution for (a) or (b)

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Total compression , C = Total tension , T

C = fy .A1

T = fy . A2

fy

fy

1y

A2

G2

A1

G1

2y

•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

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Plastic Hinges

Hinged Length of a Simply Supported Beam with Central Concentrated Load

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L/2 L/2

MY MYMp

W

xb

h

Plastic Hinges

L3

1x

It has been shown that

My = 2/3Mp and

• 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.

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Fundamental Conditions for Plastic Analysis

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Mechanism

- Beam Mechanism

- Panel or Sway Mechanism

- Gable Mechanism

- Joint Mechanism

- Combined Mechanism

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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

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Mechanism - 3

1

3

2

Fixed beam

Three hinges are formed in the following order as shown:

1, 2, 3

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(a) Panel Mechanism (b) Gable Mechanism (c) Joint Mechanism

Mechanism - 4

Panel, Gable and Joint Mechanisms

Mechanism - 5

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

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Combined Mechanism

Two hinges developed on the beam

Lower Bound or Static Theorem

For a given structure and loading, if there exists any distribution of bending moment throughout the structure which is both safe and statically admissible with a set of loads ‘W’, the value of W must be less than or equal to the collapse load Wc

i.e, W <= Wc

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Load Factor and Theorems of Plastic Collapse

Plastic analysis is governed by three theorems

Upper Bound or Kinematic Theorem

For a given structure and loading, the value of ‘W’ found to correspond to any assumed mechanism must be either greater than or equal to the collapse load Wc

i.e, W >= Wc

Uniqueness Theorem

Static and kinematic theorem can be combined to form a theorem which gives unique value for collapse load. This theorem is called uniquenness theorem

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Rigid plastic analysis

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Simply supported beam at plastic hinge stage

L

WPlastic zone Yield zone Stiff length

2

M

B. M. D. Moment – rotation curve

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Rigid plastic analysis - 2

P=0 P=0

MB

A BC

W / unit length

L

Loading

MPMP

MP MP

2

Collapse

MA

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Continuous Beams

10W10W 6W

2L 2L 3L 2L 2L 3L

Rigid Plastic Analysis - 3

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10W

2

10W 6W

10W 6W10W

2 35

10W 10W 6W

235

Collapse pattern 3:

Collapse pattern 2:

Collapse pattern 1:

Rigid Plastic Analysis - 4