##### Document Text Contents

Page 1

f05-24-H8298.eps

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

Page 2

f05-19-H8298.eps

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.

Page 12

f05-15-H8298.eps

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:

Page 13

f05-03-H8298.eps

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.

Page 23

f05-21-H8298.eps

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.

Page 24

f05-25-H8298.eps

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.

f05-24-H8298.eps

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

Page 2

f05-19-H8298.eps

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.

Page 12

f05-15-H8298.eps

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:

Page 13

f05-03-H8298.eps

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.

Page 23

f05-21-H8298.eps

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.

Page 24

f05-25-H8298.eps

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.