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Table of Contents
                            Manual Methods of Plastic Analysis
	Introduction
	Theorems of Plasticity
		Kinematic Theorem (Upper Bound Theorem)
		Static Theorem (Lower Bound Theorem)
		Uniqueness Theorem
	Mechanism Method
	Statical Method
	Uniformly Distributed Loads (UDL)
		Method to Calculate External Work for UDL
	Continuous Beams and Frames
		Partial and Complete Collapse
		Application to Continuous Beams
		UDL on End Span of a Continuous Beam
		Application to Portal Frames
	Calculation of Member Forces at Collapse
	Effect of Axial Force on Plastic Collapse Load
	Problems
	Bibliography
                        
Document Text Contents
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CHAPTER 5
Manual Methods of
Plastic Analysis
5.1 Introduction

In contrast to incremental elastoplastic analysis, classical rigid plastic
analysis has been used for plastic design over the past decades, and
textbooks on this topic are abundant.1–3 Rigid plastic analysis makes
use of the assumption that the elastic deformation is so small that it
can be ignored. Therefore, in using this method of analysis, the mate-
rial behaves as if the structure does not deform until it collapses plas-
tically. This behavior is depicted in the stress–strain diagram shown
in Figure 5.1.

Although classical rigid plastic analysis has many restrictions in
its use, its simplicity still has certain merits for the plastic design of
simple beams and frames. However, its use is applicable mainly for
manual calculations as it requires substantial personal judgment to,
for instance, locate the plastic hinges in the structure. This some-
times proves to be difficult for inexperienced users. This chapter
describes the classical theorems of plasticity. The applications of
these theorems to plastic analysis are demonstrated by the use of
mechanism and statical methods, both of which are suitable for man-
ual calculations of simple structures. Emphasis is placed on the use of
the mechanism method in which rigid plastic behavior for steel mate-
rial is assumed.
5.2 Theorems of Plasticity

There are three basic theorems of plasticity from which manual meth-
ods for collapse load calculations can be developed. Although
attempts have been made to generalize these methods by computers,4

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ε0

fy

f

FIGURE 5.1. Rigid plastic behavior.

140 Plastic Analysis and Design of Steel Structures
the calculations based on these methods are still largely performed
manually. The basic theorems of plasticity are kinematic, static, and
uniqueness, which are outlined next.
5.2.1 Kinematic Theorem (Upper Bound Theorem)

This theorem states that the collapse load or load factor obtained for a
structure that satisfies all the conditions of yield and collapse mecha-
nism is either greater than or equal to the true collapse load. The true
collapse load can be found by choosing the smallest value of collapse
loads obtained from all possible cases of collapse mechanisms for the
structure. The method derived from this theorem is based on the bal-
ance of external work and internal work for a particular collapse
mechanism. It is usually referred to as the mechanism method.
5.2.2 Static Theorem (Lower Bound Theorem)

This theorem states that the collapse load obtained for a structure that
satisfies all the conditions of static equilibrium and yield is either less
than or equal to the true collapse load. In other words, the collapse
load, calculated from a collapse mode other than the true one, can
be described as conservative when the structure satisfies these condi-
tions. The true collapse load can be found by choosing the largest
value of the collapse loads obtained from all cases of possible yield
conditions in the structure. The yield conditions assumed in the
structure do not necessarily lead to a collapse mechanism for the
structure. The use of this theorem for calculating the collapse load
of an indeterminate structure usually considers static equilibrium
through a flexibility approach to produce free and reactant bending
moment diagrams. It is usually referred to as the statical method.

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Mp

Mp
Plastic hinge

x

θ
αθ
α

Continuous beam

End span w

w

FIGURE 5.14. Collapse mechanism at end span of a continuous beam.

150 Plastic Analysis and Design of Steel Structures
the load w or maximize the bending moment Mp of the internal plastic
hinge so that the value of x can be found.

The relationship between the angles of plastic rotation y and a is

yx ¼ a L� xð Þ;
therefore a ¼ yx

L� x :

External work ¼ wxð Þ x
2
yþw L� xð Þ L� x

2

� �
a ¼ wLx

2
y:

Internal work ¼ MpaþMp aþ yð Þ ¼ Mp
Lþ x
L� x

� �
y:

External work ¼ Internal work,

therefore w ¼ Mp
2 Lþ xð Þ

L Lx � x2ð Þ
� �

(5.5)

For minimum w,
dw
dx

¼ 0. It can be proved that if w ¼ Mp
f1 xð Þ
f2 xð Þ

, then
dw
dx

¼ 0 will lead to the following equation:

f1 xð Þ
f2 xð Þ

¼ f
0
1 xð Þ
f
0
2 xð Þ

(5.6)

where f
0
xð Þ represents the first derivative of f xð Þ.

From Equations (5.5) and (5.6),

Lþ x
Lx � x2 ¼

1

L� 2x giving x
2 þ 2Lx � L2 ¼ 0;

therefore x ¼ 0:414L:

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Manual Methods of Plastic Analysis 151
Substit uting x into Equa tion (5.5) gives w ¼ 11 :65 Mp
L2

.

This is the stand ard soluti on of the colla pse load for UD L acting
on the en d span of a con tinuous beam.

Example 5.6 What is the maxim um load factor a that the beam sho wn
in Figure 5.15 can sup port if Mp ¼ 93 kNm?
20α kN/m

6m 8m

10α kN/m

FIGURE 5.15. Example 5.6.
Soluti on
Left span

20a ¼ 11: 65Mp
L2

¼ 11: 65 93
62

� �
¼ 30kN =m ;
theref ore a ¼ 1:5 :
Right span

10 a ¼ 11 :65Mp
L2

¼ 11: 65 93
82

� �
¼ 17kN =m;
theref ore a ¼ 1: 7:
Hence, th e maximu m load factor a ¼ 1:5
5.6.4 Application to Portal Frames

A portal frame us ually involves high degrees of indetermi nacy . Ther e-
fore, there are alw ays a large numbe r of partial and comple te collap se
mechani sms (som etimes term ed ba sic mechani sms) that can be com-
bined to form new collapse mechani sms wit h some plast ic hinges
becomi ng elast ic (unloa ding) again. For comple x frames, it requires
substantial judgment and experience in using this method to identify
all possible partial and complete collapse mechanisms.

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Manual Methods of Plastic Analysis 161
5.6. Identify the critical collapse mechanism for the portal frame
with one support pinned and the other fixed shown in
Figure P5.6 and calculate the common factor P at collapse. Plas-
tic moment ¼ Mp.
3L

3P

P

L

L

L

FIGURE P5.6. Problem 5.6.
5.7. Determine the collapse load factor a for the pin-based portal frame
shown in Figure P5.7. For all members, Mp ¼ 200;Np ¼ 700; all in
consistent units. The members are made of I sections with the

yield condition given by m ¼ 1:18ð1�bÞ where m ¼ M
MP

and

b ¼ N
NP

for b > 0:15 otherwise m � 1:
80α

4

80α

4

70α

160α

4

FIGURE P5.7. Pin-based portal frame.

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162 Plastic Analysis and Design of Steel Structures
5.8. Determine the value of P at collapse for the column shown in
Figure P5.8. The plastic moment of the column is Mp.
P

L

2L

P
2L

B

C

A

D

E

L

FIGURE P5.8. Problem 5.8.
Bibliography

1. Neal, B. G. (1977). The plastic methods of structural analysis, London.
Chapman and Hall.

2. Horne, M. R. (1971). Plastic theory of structures, Oxford. MIT Press.
3. Beedle, L. S. (1958). Plastic design of steel frames, New York. Wiley.
4. Olsen, P. C. (1999). Rigid plastic analysis of plastic frame structures. Comp.

Meth. Appl. Mech. Eng., 179, pp. 19–30.

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