This document provides an introduction to plastic theory for the analysis of structures. It discusses key concepts like the idealized stress-strain curve, assumptions of plastic theory, formation of plastic hinges, and the concept of shape factor. Shape factor is the ratio of the plastic moment capacity to the elastic moment capacity of a cross-section. Several examples are provided to demonstrate calculating shape factors for different cross-sections, including rectangular, I-sections, unsymmetrical sections, and diamond-shaped sections.

1. Presented by
Dr. Subhash V. Patankar
Department of Civil Engineering,
Sanjivani College of Engineering, Kopargaon,
Dist:Ahmednagar, Maharashtra, India.
E-mail: patankarsubhashcivil@sanjivani.org.in,
Mobile No. : 8087482971

2. CONTENTS
1) Introduction
2) Concept of Plastic Theory
3) Idealised stress-Strain Curve
4) Shape Factor
5) Uniqueness Theorem
6) Static method of analysis of beams
7) Kinematical method of analysis of beams
8) Examples

3. INTRODUCTION
Theory of elasticity:
 Generally in analysis of structures, we have assumed that
materials behaved elastically in which stress is
proportional to strain.
 It obey the Hookes Law.
 This concept is used in analysis and design of structure
subjected to various types of loading.
 For the design of structures, service loads or working
loads are consider.
 Only up to yield point of material is considered and
hence reserve capacity of material is not utilised.
 So, the section is thick and not economical.

4. Theory of Plasticity:
 From experimental results of actual stress-strain curve
of mild steel, it is observe that the material is neither
fail at yield point (yield stress) nor fail at ultimate
point (Maximum stress).
 But it fails at maximum strain. That means elastic
materials have reserve capacity/ strength which is not
utilised in concept of theory of elasticity.
 If load increases beyond elastic limit, It was observed
that the section did not fail as yield strength
developed only at top & bottom fibres of section.
 Towards NA, development of stress is less than yield
value.

5.  So, to utilised such large reserve strength of materials,
plastic theory was introduced by Kazinczy.
 It is assumed that the structure does not failed/collapse
at yield load but there is a plasticizasion of inner and
inner fibers.
 If further increase in load beyond elastic limit, it is
observed that there is formation of plastic hinge at NA
in such a way that both zone (tension & compression)
rotate at imaginary point.
 This concept is used in deign of steel structures as
well as RCC.

6. Idealised Stress-strain Curve
€
Elastic range
Strain Hardening
Actual & Idealised Stress-Strain Curve for M.S.
σ
Plastic range
Elastic range
Plastic range
σ
€

7. Assumptions used in Plastic Theory
 Plane section before bending remains plane after bending.
 Stress-Strain diagram is a straight line up to yield point and
beyond that, it is horizontal line till the failure occur i.e.
strain hardening effect is neglected.
 Stress strain diagram is same for both tension and
compression.
 Area of cross section above and below NA is same
 The effect of axial load and shear on fully plastic moment
capacity of the section is neglected.
 Sufficient number of plastic hinges are formed to transform
the structure into a collapse mechanism.
 Plastic hinges are imaginary points where both zones are
rotated and it will developed only at the critical sections.
 Plastic bending moment remains constant.

8. Idealised Stress-Strain Curve for M.S. in TOP
Fy
Compression
Tension
Strain, €
Stress, σ
Fy

9. Mp
C
T
C/S of Beam Fb=Fy Fy Fy Mp
BMD
L
Development of Plastic Hinge
M= Mp
W=0 to Wu
Concept of Plastic Hinge

10. Concept of Plastic Hinge
 Consider a simply supported beam of span L is
subjected to a point load ‘W’ at center as shown in
fig.
 If point load is zero, bending moment is zero and
hence bending stress is also zero. (Neglect self weight
of beam)
 If load W increases then bending stress developed in
the section also increases linearly up to elastic limit
i.e. σb σy.

11.  But beyond elastic limit, if the load continuously
increase then inner and inner fibres get yielded but the
sections does not fail.
 Beyond this stage, if load further increases continuously
then stress cannot be increase beyond yield stress at top
and bottom fibre. But their plasticization take place.
This stage is called elasto-plastic stage.
 If load continuously increase beyond this stage then
gradually complete plasticization of both zones above
and below NA take place.

12.  At this stage, both tension and compression zones are
connected to each other with very small point (hinge)
on which they rotate. This imaginary hinge is called as
plastic hinge.
 Plastic Moment (Mp) is the maximum moment
developed at critical section due to formation of plastic
hinge.
 Mathematically Mp = σy.Zp

13. Shape Factor
 It is the ratio of plastic moment (Mp) to the yield or
elastic moment (Me or My) of a given section.
 It is denoted by S
Mathematically, S = =
S =

14. Calculation of Shape Factor
1. Calculate Elastic Moment,
Me =
2. Calculate Plastic Moment,
Mp =
3. Calculate Shape Factor,
K =

15. Mp
C
T
C/S of Beam σb = σy Mp
Plastic Moment:
Apply flexure formula,
σb = M/Z,
Compressive force C and
Tensile force T forms the
couple,
C = σy. Ac, and leaver arm (a) = Yc + Yt
As per assumption, total area of section in compression must be
equal to total area in tension below plastic Neutral axis.
Ac = At = A/2
As the section is symmetrical, we can say Yc Yt = D/4,
Mp = C. (Yc + Yt)
Mp = σy.(A/2). (Yc + Yt) and also Mp = σy.Zp
Mp =σy.Zp

16. Mp
C
T
C/S of Beam σb = σy Mp
(TOP)
C/S of Beam σb = σy (TOE)
Example1: To find the shape
factor of rectangular C/s of a beam
In theory of elasticity,
Me = σb.Ze,
σb=σy & Z = b.D2/6,
Me = σy(b.D2/6)
In theory of Plasticity,
Mp = σy. Zp
= σy.(A/2). (Yc + Yt)
For rectangular section,
A = b.D, Yc = Yt = D/4
Mp = σy. b.D. (D/4 + D/4)/2
= σy.b. D2/4
Shape Factor = Mp/Me
K =Ks = 1.5

17. Example2: Calculate the shape factor ‘s’ for a beam
cross-section shown in Fig.

19. 1. Elastic moment (Me) :
I-section is symmetrical about Y-Y axis. So, Yt + Yb

20. 2. Plastic moment (Mp) :
3. Shape factor (S) :

21. Example 3: Find shape factor for an unsymmetrical I
section as shown in Fig.

22. 1. Calculate Elastic moment

24. 2. Plastic moment (Mp):
As section is unsymmetrical, So from equal area
concept, find Plastic N.A.
Let plastic neutral axis is at a distance ‘h’ from top
flange
h = 32.5 mm from top

25. Yt =Y2 =66.85 mm
3. Shape factor (S):
= 1.688

26. Example4: Calculate the shape factor for the cross-section
shown in Fig. If fy = 250 N/mm2. Calculate the plastic
moment capacity of the section, Mp.

27. Example5: Calculate the shape factor for the cross-section
shown in Fig.

28. Example6: Calculate the shape factor for the cross-section shown
in Fig. If fy = 250 N/mm2. Calculate the plastic moment capacity
of the section, Mp.
h
b

29. Example7: Calculate the shape factor for the diamond shape
cross-section as shown in Fig. If fy = 250 N/mm2. Calculate the
plastic moment capacity of the section, Mp.
d
b

30. Thank You