Plastic Analysis

Noida Institute of Engineering and Technology,
Greater Noida
Plastic Analysis
Aayushi
Assistant Professor
Civil Engg. Department
6/5/2022
1
Unit: 5
Aayushi RCE-502, DOS 1 Unit 5
Subject Name : Design of
Structure I
Course Details : B Tech 5th
Sem

6/5/2022 Aayushi RCE-502, DOS 1 Unit 5 2
Content
 Course Objective
 Course Outcome
 CO-PO & PSO Mapping
 Prerequisite & Recap
 Plastic Analysis
 Topic outcome
 Objective of topic
 Topic outcome & mapping with PO
 Prerequisite & Recap
 Material
 Idealised stress-strain curve for mild steel
 Syllabus of unit 5
 Topic objective

Content
 Idealised Curved in plastic theory
 Process of yielding of a section
 Inverted T section
 Plastic hinge
 Shape factor for various cross section
 Mechanisms of failure
 Beam Mechanism
 Panel / Sway Mechanisms
 Method f Plastic Analysis
 Determination of collapse load
 Numerical of Plastic Analysis
 Plastic Moment
 Factor of safety

Content
 You tube Video Links
 Daily Quiz
 Weekly Assignment
 MCQs
 Old Question Papers
 Expected Questions in University Examination
 Summary
 References

Objective
1
To impart the principles of elastic structural analysis and behavior
of indeterminate structures.
2
To impart knowledge about various methods involved in the
analysis of indeterminate structures...
3
To apply these methods for analyzing the indeterminate structures
to evaluate the response of structures .
4
To enable the student get a feeling of how real-life structures
behave
5
To make the student familiar with latest computational techniques
and software used for structural analysis. .
6/5/2022
5
Course Objective

Students will be able
CO1 To Identify and analyze the moment distribution in beams and frames by Slope
Deflection Method, Moment Distribution Method and Strain Energy Method.
CO2 To provide adequate learning of indeterminate structures with Muller’s
Principle; Apply & Analyze the concept of influence lines for deciding the
critical forces and sections while designing..
CO3 To learn about suspension bridge, two and three hinged stiffening girders and
their influence line diagram external loading and analyze the same.
CO4 To Identify and analyze forces and displacement matrix for various structural.
CO5 To understand the collapse load in the building and plastic moment formed.
CO6 Apply the concepts of forces and displacements to solve indeterminate structure.
Course Outcome

CO PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12
RCE502.1 3 3 1 2 1 - - - 2 - - 3
RCE502.2 3 2 2 2 3 - - 1 2 - 2 2
RCE502.3 3 - 2 2 1 - - - 2 - 2 2
RCE502.4 3 1 2 2 1 - - - 2 - 2 2
RCE502.5 3 2 2 2 1 - - - 2 - 2 2
RCE502.6 3 2 3 2 1 - - - 2 - 2 3
AVG. 3 2 2 2 1 - - - 2 - 2 2
CO-PO and PSO Mapping
COs PSO1 PSO2 PSO3 PSO4
RCE502.1 2 1 2 3
RCE502.2 2 1 2 2
RCE502.3 2 2 2 2
RCE502.4 2 1 2 3
RCE502.5 2 1 2 3
RCE502.6 2 1 2 3
Avg. 2 1 2 3

Prerequisite and Recap
Basics of strength of material
Basics of engineering mechanics
Basic of steel structures

Syllabus of Unit 5
Basics of Plastic Analysis. Applications of Static
and Kinematic theorem for Plastic Analysis of
Beams and Single Storied Frames.

6/5/2022 10
Objective of Unit 5
Plastic analysis method, is utilized to predict the
collapse load factor of Beam and truss structures.

 Plastic analysis method, is utilized to predict the collapse load
factor of Beam and truss structures.
 The obtained collapse load factor is then incorporated into
truss generate truss structures which cannot only maintain
load-carrying capacity under ordinary load conditions
 Collapse under accidental load conditions such as an
extremely large earthquake.
Topic Objective

 A fundamental understanding of the principles of plastic collapse and
energy dissipation of structures.
6/5/2022 12
Topic Outcomes
Once the student has successfully completed this unit, he/she will be able
to:
 To understand the Behavior of structures due to ultimate and accidental
loadings.
 To analysis and design onshore and offshore structures both in
ultimate limit state and accidental limit state under non-linear material
behavior.

6/5/2022 13
Objective of Topic
Topic-1 Name
Plastic Analysis
Objective of Topic-1:
Plastic analysis method, is utilized to predict the
collapse load factor of Beam and truss structures.

6/5/2022 14
Topic Outcome and mapping with PO
Programme Outcomes (POs)
PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12
TO1 3 2 2 2 1 2 2 1
Outcome of Topic-1:
After the successfully competition of this topic students
will be able to predict the collapse load factor of Beam
and truss structures.

Prerequisite and Recap
 Shape factor for various section
 Bending Moment
 Bending Moment Diagram for various
beam with different loads
 Understanding of plastic analysis

Plastic Analysis -Why? What?
Behaviour beyond elastic limit?
• Plastic deformation - collapse load
• Safe load – load factor
• Design based on collapse (ultimate) load – limit design
• Economic - Optimum use of material
Topic: Plastic Analysis

Topic: Materials
Material
Elastic-Perfectly
Plastic
Elastic

Topic: Idealised stress-strain curve of mild steel

Topic: Idealised curve in plastic theory
Idealised Stress - Strain curve in plastic theory

Continuous….
• Elastic analysis
- Material is in the elastic state
- Performance of structures under service loads
- Deformation increases with increasing load
• Plastic analysis
- Material is in the plastic state
- Performance of structures under ultimate/collapse loads
- Deformation/Curvature increases without an increase in load.

Assumptions :
• Plane sections remain plane in plastic condition
• Stress-strain relation is identical both in compression and tension .
Continuous…

• Let M at a cross-section increases gradually.
• Within elastic limit, M = σ.Z
• Z is section modulus, I/y
• Elastic limit - yield stresses reached
My = σy.Z
• When moment is increased, yield spreads into inner fibers.
Remaining portion still elastic
• Finally, the entire cross-section yields
Topic: Process of yielding of a section

Change in stress distribution during yielding
Rectangular cross section
Continuous…

Topic: Inverted T section

• When the section is completely yielded, the section is fully plastic
• A fully plastic section behaves like a hinge - Plastic hinge
Topic: Plastic hinge
Plastic hinge is defined as an yielded zone due to bending in a
structural member, at which large rotations can occur at a
section at constant plastic moment, MP

Mechanical hinge Plastic hinge
Reality Concept
Resists zero
moment
Resists a constant
moment MP
Continuous…
Mechanical Hinge
Plastic Hinge with
MP= 0

• M - Moment corresponding to working load
• My - Moment at which the section yields
• MP - Moment at which entire section is under yield stress
Continuous…

• Moment at which the entire section is under yield stress
• NA divides cross-section into 2 equal parts
Topic: Plastic moment

Continuous…
Couple due to
C=T
= σ y Z
Plastic modulus

Rectangular cross-section:
Section modulus
Plastic modulus
Shape factor
Topic: Shape factor for various cross-sections

Circular section
Continuous…

Triangular section
Continuous…

Topic: I section
Load factor

Topic: Factor of Safety

• A statically determinate beam will collapse if one plastic hinge is
developed .
• Consider a simply supported beam with constant cross section
loaded with a point load P at midspan.
• If P is increased until a plastic hinge is developed at the point of
maximum moment (just underneath P) an unstable structure will be
created.
• Any further increase in load will cause collapse .
• For a statically indeterminate beam to collapse, more than one
plastic hinge should be developed
• The plastic hinge will act as real hinge for further increase of load
(until sufficient plastic hinges are developed for collapse.)
• As the load is increased, there is a redistribution of moment, as the
plastic hinge cannot carry any additional moment.
Topic: Mechanisms of failure

Topic:Beam mechanisms

Continuous…

Topic: Panel mechanism/sway mechanism

• Static method or Equilibrium method
- Lower bound: A load computed on the basis of an assumed
equilibrium BM diagram in which the moments are not greater than MP
is always less than (or at the worst equal to) the true ultimate load.
• Kinematic method or Mechanism method or Virtual
work method
- Work performed by the external loads is equated to the internal work
absorbed by plastic hinges
- Upper bound: A load computed on the basis of an assumed
mechanism is always greater than (or at the best equal to) the true
ultimate load.
Topic: Methods of Plastic Analysis

• Collapse load (Wc): Minimum load at which collapse will occur -
Least value
• Fully plastic moment (MP): Maximum moment capacity for
design - Highest value
Continuous…

1. Simple beam
Topic: Determination of collapse load

2. Fixed beam with UDL
Continuous…

3. Fixed beam with point load 4. Fixed beam with eccentric
Continuous…

Virtual work:
Continuous…

5. Propped cantilever with point load at midspan
Continuous…

6. Propped cantilever with UDL
Continuous…

Problem 1: For the beam, determine the design plastic moment
capacity.
• Degree of Indeterminacy, N = 3 - 2 = 1
• No. of hinges, n = 3
• No. of independent mechanisms ,r = n - N = 2
Topic: Numerical
50 KN 75 kN
1.5 m 1.5 m
7.5 m

Continuous…
Design plastic moment (Highest of the above) = 87.5 kNm

Continuous…
Problem 2: A beam of span 6 m is to be designed for an ultimate
UDL of 25 kN/m. The beam is simply supported at the ends. Design
a suitable I section using plastic theory, assuming σy= 250 MPa.
• Degree of Indeterminacy, N = 2 - 2 = 0
• No. of hinges, n = 1
• No. of independent mechanisms, r = n-N = 1

Continuous…

Problem 3: Find the collapse load for the frame shown.
Continuous…
W • Degree of Indeterminacy, N = 5
- 3 = 2
• No. of hinges, n = 5 (at A, B, C,
E & F)
• No. of independent mechanisms
,r = n - N = 3
• Beam Mechanisms for
members AB & BC
• Panel Mechanism

• Beam Mechanism for AB
6/5/2022
Aayushi RCE-502, DOS 1
Unit 5
52
Continuous…
• Beam Mechanism for BC

Continuous…

 Basic concepts on Plastic
Analysis have been discussed
and the 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.
 Worked examples illustrating
plastic methods of analysis have
been provided.
Summary of Plastic Analysis

Youtube/other Video Links
Youtube Video Links
Topic Links Process
Plastic Analysis https://www.youtube.com/watch?v
=6035iMwKLmY
Click on
the link
Plastic Analysis
Mechanism
https://www.youtube.com/watch?v
=XGxOn7E11AE&list=PLPWQY
202UkqqnmS_dvWwfi9P7gwk-
wKDE
Click on
the link

 Structures designed using elastic analysis may be ______
than those designed using plastic analysis
a) lighter
b) heavier
c) of same weight
d) almost half times the weight
 Both elastic and plastic methods neglect ________
a) live load acting on structure
b) dead load acting on structure
c) deformations due to load
d) influence of stability
Daily Quiz

a) lighter
b) heavier
c) of same weight
Daily Quiz

 What is shape factor ? Determine the shape factor for
rectangular and diamond section .
 Derive the shape factor of rectangular section, triangular
section and circular section.
 Find the shape factor of hollow circular section as shown in
Fig.
Daily Quiz

 What is a plastic hinge?
 What is a mechanism?
 What is difference between plastic hinge and mechanical hinge?
 Define collapse load.
 List out the assumptions made for plastic analysis.
 Define shape factor.
 List out the shape factors for the following sections.
 Mention the section having maximum shape factor.
 Define load factor.
 State upper bound theory.
Weekly Assignment

 Define shape factor and obtain its value for a T-section with the
following dimensions shown in Fig. If the yield stress is 250
N/mm?. Find M,.
 A beam of rectangular cross section b xd is subjected to a bending
moment stress 0.9 M Find out the depth of elastic core.
Weekly Assignment

Which of the following relation about plastic moment is correct?
a) Mp = Zp /fy
b) Mp = Zp + fy
c) Mp = Zpfy
d) Mp = Zp – fy
Which of the following relation is correct about shape factor, v?
a) v = Zp+Ze
b) v = ZpZe
c) v = Zp/Ze
d) v = Ze/Zp
MCQ s

The shape factor does not depend on ___
a) material properties
b) cross sectional shape
c) moment of resistance
d) section modulus
In elastic stage, equilibrium condition is achieved when neutral axis
___________ and in fully plastic stage, it is achieved when neutral
axis ___________
a) is above centroid of the section, divides the section into two parts of
one-third area and two-third area
b) is below centroid of the section, divides the section into two parts of
one-third area and two-third area
c) is above centroid of the section, divides the section into two equal
areas
d) passes through centroid of the section, divides the section into two
equal areas
MCQ s

What is plastic moment of resistance?
a) maximum moment in stress strain curve, the point where the
curvature can increase indefinitely
b) maximum moment in stress strain curve, the point where the
curvature can decrease indefinitely
c) minimum moment in stress strain curve, the point where the
curvature can increase indefinitely
d) minimum moment in stress strain curve, the point where the
curvature can decrease indefinitely
MCQ s

 What is the condition for equilibrium in plastic analysis?
a) bending moment distribution defined by assumed plastic hinges must
not be in static equilibrium with applied loads and reactions
b) shear force distribution defined by assumed plastic hinges must be in
static equilibrium with applied loads and reactions
c) bending moment distribution defined by assumed plastic hinges must
be in static equilibrium with applied loads and reactions
d) shear force distribution defined by assumed plastic hinges must not be
in static equilibrium with applied loads and reactions
 Which of the following is true?
a) ultimate load is reached when a mechanism is formed
b) ultimate load is not reached when a mechanism is formed
c) plastic hinges are not required for beam to form a mechanism
d) frictionless hinges are not required for beam to form a mechanism
MCQ s

Old Question Papers

• Find the shape factor of I section
• Numerical on mechanism of frames
Expected Questions for University Exam

 A two span continuous beam has span lengths AB = 6 m and BC = 6
m and carries a UDL of 30 kN/m on entire length of the beam. A and
C are simply supports. If the load factor is 1.80 and the shape factor
is 1.15 for theI-section. Find the section modulus needed. Assume
yield stress for the material is 250 N/mm2.
 A continuous beam ABC is loaded as shown in Fig. determine the
required plastic moment. If the load factor is 3.2.
Expected Questions for University Exam

 Introduction-Plastic hinge concept-plastic section
modulus-shape factor-redistribution of moments-
collapse mechanism.
 Theorems of plastic analysis - Static/lower bound
theorem; Kinematic/upper bound theorem-Plastic
analysis of beams and portal frames by equilibrium
and mechanism methods.
Summary

 Jain, A. K., “Advanced Structural Analysis “, Nem Chand
& Bros., Roorkee.
 Hibbeler, R.C., “Structural Analysis”, Pearson Prentice
Hall, Sector - 62, Noida-201309
 C. S. Reddy “Structural Analysis”, Tata Mc Graw Hill
Publishing Company Limited,New Delhi.
 Timoshenko, S. P. and D. Young, “ Theory of Structures”
, Tata Mc-Graw Hill BookPublishing Company Ltd., New
Delhi.
 Dayaratnam, P. “ Analysis of Statically Indeterminate
Structures”, Affiliated East-WestPress.
 Wang, C. K. “ Intermediate Structural Analysis”, Mc
Graw-Hill Book PublishingCompany Ltd.
References

References

Plastic Analysis

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