lecture4b-4

Module 4 : Plastic Analysis (2)Module 4 : Plastic Analysis (2)

Dr Yan ZhugeDr Yan Zhuge

CIVCIVE3011E3011 Structural Structural Analysis and Computer Analysis and Computer

ApplicationsApplications

Dr Yan ZhugeDr Yan Zhuge

CIVCIVE3011E3011 Structural Structural Analysis and Computer Analysis and Computer

ApplicationsApplications

Plastic collapse of a portal framePlastic collapse of a portal frame

• Frame is more complex than the simple beam structures

• There are various possibilities for the failure mechanism

• Total number of hinges =

degree of redundancies + 1

Plastic collapse of a portal framePlastic collapse of a portal frameMp = 100kNm

EI = 100kNm2

P1 = 10 kN

P2 = 10 kN

P2 kNP1 kN

5m

10m5m The whole structure is in elastic range

11.4

7.8

21.2

20.8

25.7

= 1

Max bending moment


The bending moment at E reaches Mp, a plastic hinge is formed

Plastic hinge5m

10m5m

= 3.9 44.4

30.4

82.7

80.9

100

P1 kNP2 kN

A

B

CD

E


64.2

31.4

100

97.3

100

5m

10m5m

Plastic hinge

= 4.60

The bending moment at C reaches Mp, now there are two plastic hinges.

A

B

C

D

E


46.7 kN46.7 kN

5m

10m5m

Plastic hinge

= 4.6766.8

33.4

100

100

100

The bending moment at D reaches Mp, now there are three plastic hinges.

E

D

C

B

A


50 kN

5m

10m5m

Plastic hinge

c = 5.0

50 kN

100

50

100

100

100

Four plastic hinges are formed, the structure is changed into a mechanism and the corresponding load is called the collapse load.

A

BC

D

E

Portal frame with pinned support Portal frame with pinned support

VH

L

h

The value of Mp is constant throughout

Two hinges will be Two hinges will be required to form a required to form a

mechanismmechanism

A beam Mechanism pM

pM

pM

pM pM

A Sway Mechanism

cH

cV

cH

cV8

42

LVM

ML

V

MW

cp

pc

p

2

2

hHM

MhH

MW

cp

pc

p

Collapse is caused by the vertical force alone

Collapse is caused by the horizontal force alone

“Combined Mechanism”“Combined Mechanism”There is a third possibility in which the two independent mechanisms are combined to produce the “Combined Mechanism”

pM

pM

cHcV

It is a combination of the beam and side sway mechanism with a “cancelling out” of the joint rotations at B such that B remains a rigid, without the formation of a plastic hinge.

B

Virtual work equationVirtual work equation

hHL

VMMMM

WM

ccpppp

p

224

external work, beam mechanism

external work, beam mechanism

external work, sway mechanism

external work, sway mechanism

internal work, beam mechanism

internal work, beam mechanism

internal work, sway mechanism

internal work, sway mechanism

internal work at hinge which

disappears

internal work at hinge which

disappears

48

24

hHLVM

hHL

VM

ccp

ccp

Which is the “most likely” mechanism?

Which is the “most likely” mechanism?

This is a difficult question to answer, because the actual collapse mechanism depends on the relative values of the forces H and V, see the graph below.

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5Hh/Mp

VL

/Mp

sway mechanism

beam mechanism

collapse

Combined mechanismpermissible region

Interaction diagram (ID)

(2,4)

Notes on interaction diagram (ID)Notes on interaction diagram (ID)• The horizontal line states beam collapse when

VL/Mp =8

• Similar arguments can be used for the other mechanisms, and the arrows in the ID indicate safety.

• The shaded area indicates combinations of V and H that are safe against collapse by any of the possible mechanisms.

• Point (2,4) represents over-collapse because the sway and combined mechanism will form simultaneously.

A more complicated caseA more complicated case

3m

5m

4m 6m

H

V

Plastic moments:

Beam = 400kNm

Columns = 200kNm

Collapse mechanisms

Beam mechanism

Sway mechanism

Combined mechanism

1

2

3

4

5

Beam mechanismBeam mechanism

V = 200 + 400( +

substituting for and

V x 4

V = 250kN

Beam mechanism

= 4= 6

= (2/3)5m

4m 6m

3m

V

At a connection between two members, the plastic hinge forms at a BM equal to the plastic moment of the weaker member

MP = 400 kNm

MP = 200 kNm

Sway mechanismSway mechanism

H = 2 x 200 + 2 x 200

substituting for and

H x 5

H = 213.3kN

Sway mechanism

= 5= 3

= (5/3)5m

4m 6m

3m

H

Plastic hinges form at the top and bottom of each column. The tops of the columns move sideways by the same amount, so the rotations in each column are different

Combined mechanismCombined mechanism

V + H =

4V + 5H

Combined mechanism

= (2/3)

= (5/3)5m

4m 6m

3m

H

V

Beam mech Sway mech

Interaction diagram (ID)Interaction diagram (ID)

Notes on Interaction Diagram (ID)Notes on Interaction Diagram (ID)• The collapse mechanism depends on the relative

magnitudes of H and V• The ID shows that the collapse is under combined

mechanism with the H=166.6 kN and V=208.3 kN (assume V=1.25H).

• The corresponding bending moment diagram:

How to draw the BMD?How to draw the BMD?

The Free Body Diagram (FBD) of 4-5The Free Body Diagram (FBD) of 4-5

kNH

kNmM

MH

MH

p

p

p

3.133

200 and

32

23

5

5

5

5H

kNmM p 200

m3

kNmM p 200

Contd.Contd.

kNmM p 200

2M

kNH 3.331

m5

kNH 6.166The Free Body Diagram (FBD) of 1-2The Free Body Diagram (FBD) of 1-2

kNmM

M

HMM p

5.33

200-533.3

05

2

2

12

Contd.Contd.

2M

1VkNmM p 400

m4

The Free Body Diagram (FBD) of 2-3The Free Body Diagram (FBD) of 2-3

kNV

kNV

V

VMM p

1003.1085.208

3.108

40033.5 4

04

5

1

1

12

Collapse Mode & Load FactorCollapse Mode & Load FactorFor a frame of given Mp and L, any values of V and H will give us a point on the Interaction Diagram. If this point lies outside the boundary then the values of V and H will be inadmissible as the frame will have already collapsed. If the point lies within the boundary then a line drawn from the origin through the point gives information regarding:

i.The Mode of Collapse

ii.The Load Factor

For that particular Case

For a frame of given Mp and L, any values of V and H will give us a point on the Interaction Diagram. If this point lies outside the boundary then the values of V and H will be inadmissible as the frame will have already collapsed. If the point lies within the boundary then a line drawn from the origin through the point gives information regarding:

i.The Mode of Collapse

ii.The Load Factor

For that particular Case

Pitched portal framePitched portal frame

Pitched portal mechanism

Sway mechanism

Combined mechanismBeam mechanism can not develop in the sloping rafters

H

V

AB

C

Sloping members Sloping members

l

h

v

L/2

kh

h = (l sin= lsin = kh

v = (l cos= lcos = (L/2)

Horizontal deflection = vertical projection x plastic rotation

Vertical deflection = horizontal projection x plastic rotation = deflection of a beam with same span

l vv

h

A

B

Symmetric pitched portal frame

Symmetric pitched portal frame

2/L

h

kh1H

V

2/L

3H

2H

l

pMpM

pM3

The analysis is more complicated than the rectangular portal frame, only required for assignment not for exam.

Possibilities to form a beam mechanism

Possibilities to form a beam mechanism

The internal work is the same in each case

Virtual work (pattern (b))Virtual work (pattern (b))Internal work = Mp + 3Mp x 2 + Mp( +) + Mp

= 4Mp(2+k)

External work: Case (a), (b) = VL/2 (same as beam mechanism)

Case (c) = VL/2 + H2 2kh

Case (d) = VL/2 + H3 kh

The horizontal forces determine which pattern will occur

Example – pitched portal frameExample – pitched portal frame

Pitched-portal mechanism Pitched-portal mechanism

)(225.02 ppp MMMLHLV

pMLHV 5.4)5.0(

L

MHV p5.45.0

5.0

25.02

2

LL

Lh

LLAbh 25.0sinsin

LLABV coscos

Vertical deflection is the horizontal projection of AB multiplied by the plastic rotation.Horizontal deflection is the vertical projection of AB multiplied by the plastic rotation.

Sway mechanism Sway mechanism

ppp MMMLH

pMLH 3

L

MH p3

L

Combined mechanism (pitched-portal + sway)Combined mechanism (pitched-portal + sway)

pppp MMMMLHLHV 35.4)5.0(

pMLHV 5.5)5.1(

L

MHV p5.55.1

The interaction diagram The interaction diagram

The collapse load is and (V=5H). L

MV p1.4

L

MH p8.0

The collapse load is and (V=H)

L

MV p2.2

L

MH p2.2

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