##### Document Text Contents

Page 2

Analysis and Design of Steel

and Composite Structures

Page 225

204 Analysis and design of steel and composite structures

7.3.3 Plastic analysis using the mechanism method

The plastic design of beams is to determine all possible collapse mechanisms and the corre-

sponding values of full plastic moments and then design the beams based on the mechanism

which provides the largest full plastic moment. In the plastic analysis using the mechanism

method, the following considerations should be taken into account:

• All possible mechanisms of collapse should be investigated.

• Plastic hinges tend to form at the ends of members, at positions of concentrated loads

and at the point of maximum bending moment.

• The mechanism and Mp of each span in a continuous beam should be investigated

individually.

• At each support of a continuous beam, the plastic hinge forms in the weaker member

with a smaller value of Mp.

The propped cantilever beam shown in Figure 7.10a is used to illustrate the mechanism

method (Baker and Heyman 1969; Horne and Morris 1981). The propped cantilever of span

L is subjected to slowly increase uniformly distributed load w. The collapse mechanism is

given in Figure 7.10b, which is composed of two rigid rinks. The central hinge is located

some distance x from the right-hand support. The angle of rotation at the left-hand end of

the beam is assumed to be θ1. Other rotations can be determined from the geometry in terms

of θ1. This gives θ2 = (L−x)θ1/x and θ3 = Lθ1/x. The resultant force acting on each rigid rink is

shown in Figure 7.10b. Under the resultant force on each rigid rink, the rigid rink undergoes

a mean displacement of δ/2, where δ is the displacement at the point of central plastic hinge.

The work equation can be written as

[ ( ) ]

( )

( )w L x wx

L x

M M

L

x

p p− + ×

−

= +

θ

θ θ1 1 1

2

(7.15)

(b)

δ

L–x x

wxw(L–x)

(a)

w

L

θ2θ1

θ3

Figure 7.10 (a) Propped cantilever and (b) collapse mechanism.

Page 226

Plastic analysis of steel beams and frames 205

The full plastic moment can be obtained from the preceding equation as

M

wLx L x

L x

p =

−

+

2

(7.16)

The maximum full plastic moment is Mp = wL2/11.66 when x = 0.414L (Horne and

Morris 1981).

Example 7.2: Largest plastic moment of two-span continuous beam

A two-span continuous steel beam with different uniform cross sections under factored

concentrated loads is schematically depicted in Figure 7.11a. Determine the largest full

plastic moment of the continuous beam.

1. Mechanism 1

Mechanism 1 is shown in Figure 7.11b. At the support, the plastic hinge is correctly

located in the weaker member. Since no mechanism has been assumed in the second span,

(a)

(b)

(c)

(d)

2θ

3θ

3θ

2θ

2θ

θ

θ

θ

δ

δ

δ

20 kN

20 kN

20 kN

20 kN 20 kN 20 kN

20 kN

20 kN

20 kN

20 kN

2.5 2.5

5 m 6 m

2 2 2

Mp 2Mp

20 kN 20 kN

θ

Figure 7.11 Mechanisms of two-span continuous beam: (a) continuous beam, (b) mechanism 1, (c) mecha-

nism 2 and (d) mechanism 3.

Page 450

Notations 429

τy Shear yield stress

ν Poisson’s ratio or shape factor

ξ Factor that is a function of combined slenderness and imperfection

parameter

ξm Moment redistribution parameter

ψ Degree of shear connection at the cross section with γ = 1.0 and complete

shear connection

ψa Reduction factor used to reduce the uniformly distributed live loads

ψc, ψs, ψl Combination, short-term and long-term factors, respectively

ω, ω1, ω2, ω3 Variable and initial values of the variables

ζ Ratio of structural damping to critical damping of a structure or

coefficient

Analysis and Design of Steel

and Composite Structures

Page 225

204 Analysis and design of steel and composite structures

7.3.3 Plastic analysis using the mechanism method

The plastic design of beams is to determine all possible collapse mechanisms and the corre-

sponding values of full plastic moments and then design the beams based on the mechanism

which provides the largest full plastic moment. In the plastic analysis using the mechanism

method, the following considerations should be taken into account:

• All possible mechanisms of collapse should be investigated.

• Plastic hinges tend to form at the ends of members, at positions of concentrated loads

and at the point of maximum bending moment.

• The mechanism and Mp of each span in a continuous beam should be investigated

individually.

• At each support of a continuous beam, the plastic hinge forms in the weaker member

with a smaller value of Mp.

The propped cantilever beam shown in Figure 7.10a is used to illustrate the mechanism

method (Baker and Heyman 1969; Horne and Morris 1981). The propped cantilever of span

L is subjected to slowly increase uniformly distributed load w. The collapse mechanism is

given in Figure 7.10b, which is composed of two rigid rinks. The central hinge is located

some distance x from the right-hand support. The angle of rotation at the left-hand end of

the beam is assumed to be θ1. Other rotations can be determined from the geometry in terms

of θ1. This gives θ2 = (L−x)θ1/x and θ3 = Lθ1/x. The resultant force acting on each rigid rink is

shown in Figure 7.10b. Under the resultant force on each rigid rink, the rigid rink undergoes

a mean displacement of δ/2, where δ is the displacement at the point of central plastic hinge.

The work equation can be written as

[ ( ) ]

( )

( )w L x wx

L x

M M

L

x

p p− + ×

−

= +

θ

θ θ1 1 1

2

(7.15)

(b)

δ

L–x x

wxw(L–x)

(a)

w

L

θ2θ1

θ3

Figure 7.10 (a) Propped cantilever and (b) collapse mechanism.

Page 226

Plastic analysis of steel beams and frames 205

The full plastic moment can be obtained from the preceding equation as

M

wLx L x

L x

p =

−

+

2

(7.16)

The maximum full plastic moment is Mp = wL2/11.66 when x = 0.414L (Horne and

Morris 1981).

Example 7.2: Largest plastic moment of two-span continuous beam

A two-span continuous steel beam with different uniform cross sections under factored

concentrated loads is schematically depicted in Figure 7.11a. Determine the largest full

plastic moment of the continuous beam.

1. Mechanism 1

Mechanism 1 is shown in Figure 7.11b. At the support, the plastic hinge is correctly

located in the weaker member. Since no mechanism has been assumed in the second span,

(a)

(b)

(c)

(d)

2θ

3θ

3θ

2θ

2θ

θ

θ

θ

δ

δ

δ

20 kN

20 kN

20 kN

20 kN 20 kN 20 kN

20 kN

20 kN

20 kN

20 kN

2.5 2.5

5 m 6 m

2 2 2

Mp 2Mp

20 kN 20 kN

θ

Figure 7.11 Mechanisms of two-span continuous beam: (a) continuous beam, (b) mechanism 1, (c) mecha-

nism 2 and (d) mechanism 3.

Page 450

Notations 429

τy Shear yield stress

ν Poisson’s ratio or shape factor

ξ Factor that is a function of combined slenderness and imperfection

parameter

ξm Moment redistribution parameter

ψ Degree of shear connection at the cross section with γ = 1.0 and complete

shear connection

ψa Reduction factor used to reduce the uniformly distributed live loads

ψc, ψs, ψl Combination, short-term and long-term factors, respectively

ω, ω1, ω2, ω3 Variable and initial values of the variables

ζ Ratio of structural damping to critical damping of a structure or

coefficient