1. Plastic analysis is based on the assumptions of small displacements and idealized material properties. It involves determining the collapse load and development of plastic hinges. 2. The uniqueness theorem states that if equilibrium, yield criteria, and a collapse mechanism are satisfied, the computed collapse load will be equal to or greater than the true collapse load. Incremental analysis can be used to track plastic hinge formation. 3. Determining deformations at collapse involves computing angle discontinuities at plastic hinge locations using virtual work principles. Inelastic analysis considers loads between initial yield and collapse.

1. PART - III
1. Plastic Analysis
2. Deformations at Collapse
3. Inelastic analysis
4. Artificial stiffening

2. • Nonlinearity: Material Nonlinearity as well as Geometric Nonlinearity
(Large Displacements) are defined in Part I, slide 7. Combined effect of both
the nonlinearities is a research domain, and not included herein.
• Superposition and sequence of Loading: It is only in
the case of structures satisfying, (a) Materials that are linearly elastic, and
(b) Geometric Linearity ( Small Displacements), superposition of loading cases
without regard to sequence of loading is acceptable.
• Plastic Analysis: For the idealized material property. It is based on
‘small displacements’.
• Inextensibility: The assumption of ‘Inextensibility’ while reducing the
number of unknown displacements, assumes ‘small displacements’. Obviously
‘superposition’ is allowed.
Basics relevant to Plastic Analysis

3. Approach to Plastic analysis
• It is based on Force Method of Analysis
• Determinate structures collapse with development of a Plastic
hinge. No warning
• The Moment curvature Relation ship assumes “Elastic and
Plastic” material property.

4. Plastic Analysis
(Virtual Work technique )
The Collapse load is computed by Satisfying Equilibrium using
Virtual Work technique. The same principle (Complementary
Virtual Work) is used for computing Angle Discontinuities at the
location of plastic hinges .

5. Incremental Load Approach for the
Plastic Analysis
• Commence with elastic Analysis,
• Find the Load P1 at the development of First Plastic Hinge,
• Track the development of the subsequent Plastic hinges by
superposing Incremental ‘ΔP proportional loading’, and
corresponding P2, …., Pi, …….PC are determined sequentially.
• If the Given load P is close to P1 or P2. …‘i.e early plastic hinges,
the ‘ΔP approach’ is recommended
• Find angle discontinuities using virtual Work Principles (Technique).

6. Approach to Plastic analysis
• It is based on Force Method of Analysis
• Determinate structures collapse with development of a Plastic
hinge. No warning
• The Moment curvature Relation ship assumes “Elastic and
Plastic” material property.

7. PLASTIC ANALYSIS
UNIQUENESS THEOREM:
1. EQUILIBRIUM,
2. YIELD CONDITION M < = MP , and
3. COLLAPSE MECHANISM.
If all three conditions are satisfied, the collapse load is Unique.
Upper Bound and lower Bound methods are based on
satisfying only two conditions. If the Collapse loads based on
UB and LB are same, the load is the Pc .
The Plastic analysis also assumes small deformations

8. CONCEPT
Uniqueness Theorem:
1. Assume Mechanism,
2. Satisfy Equilibrium, and
3. Satisfy Yield Criteria, maximum moments Mi < or = Mp.
Computed Collapse Load is larger than or equal to True Collapse Load
Alternatively
1. Satisfy yield criteria,
2. Satisfy Equilibrium, and
3. Check for Mechanism.
Computed Collapse Load is smaller than or equal to True Collapse Load

9. Collapse Mechanism
Quiz (Frame)
The aspect of sketching collapse mechanism is an essential concept of Uniqueness Theorem
Consider the following as a Quiz F
c
D E
1. Natural Hinges at the supports A and B
2. Plastic Hinges at the joints E and F. It is a good practice to show
plastic hinges at the end of the members. Member Mpi same
3. λ Pi loading as shown
4. Draw collapse mechanism for the chosen plastic hinges
5. After your attempt with Quiz, we will have ‘online’ interaction.
λ P3
λP4
A B
λ P1
λ P2
L
h/2
ha b
d

10. Failure Collapse
Quiz (Beam)
1. After your attempt with the Quiz, we will have ‘online’ interaction.
Hinge P tons
A P/4 B C
2. Natural hinge in left span. The plastic moment of resistance, Mp is same
for all the members
3. All loads are vertical
4. Sketch possible failure mechanisms
5. Find the collapse load , Pc
• Analyze the problem, if support B is a spring support of stiffness K.
Assume K = 1 t/cm ,and L =3 m.
Discuss the behavior with some typical numerical values of K
L L L 3L

11. Collapse Mechanism (contd.)
Quiz
Derived Geometry Collapse Mechanism
• Quiz for students
• Sketch Collapse Mechanism
• Identify centers of rotation
• For the assumed degree of freedom δ
at joint E, find the angles of rotation
at the joints A, E, and F
• Obtain horizontal deflections at D, E,
and F. Similarly obtain vertical
deflection at F.

12. Full Load Analysis
• Commence with the Plastic analysis, and find Pc or Pn
• Find the location of the Last Plastic Hinge, n
• Track backwards sequentially for the earlier Pn-1 …
• It is obvious that the choice of a method depends on
a given structure. If n is a large value Full Load
analysis is the obvious choice.

13. Identification of Development of the
Last Plastic Hinge at Collapse
We can than track back
the plastic hinges in the
reverse sequence till the
given load is reached.
(Refer slide 15 for the Incremental loads)

14. The Maximum Angle Discontinuity
• Determination of maximum angle
discontinuity is a design requirement
• It should be ensured that the maximum angle
discontinuity is within the acceptable values
• The maximum angle discontinuities take place
at the location of the first few plastic hinges.

15. DEFORMATIONS AT COLLOAPSE
• Angle discontinuities are computed in two
steps:
1. Find the last Plastic hinge at Collapse,
2. At all other plastic hinges, the maximum
moments Mpi are known but at the plastic
hinges, the computation of angle
discontinuities is a ductibility based design
requirement.

16. INELASTIC ANALYSIS
P1< P < PC
The analysis requires:
1. Incremental Analysis, P1 P2 …. Pi and P
2. Full Load Analysis: Collapse load Pc = Pn
collapse at nth plastic hinge. Track Pn-1
Pn-2
.
.
…. Pi and then P .

17. Artificial Stiffening
Artificial stiffening for drawing
For statical diagram by augmentation
• Accept partial collapse
Augmentation is used for reaching
complete collapse
• The structural capacity is
enhanced by stiffening the
impending plastic hinges.
This is at the cost of the
reduced Ductility.

18. OTHERWISE ADOPT APPROPRIATE STRENGTH THEORIES AND RIGOROUS
INELASTIC ANALYSIS, BASED ON MOMENT -CURVATURE RELATION SHIP. THIS
ASPECT IS ELABORATED IN PART IV.
PLASTIC ANALYSIS AS PRESENTED ASSUMES:
Maximum stress theory applied to steel or Under Reinforced Concrete
Structures,