Module 4 : Plastic Analysis 2

1
Module 4 Plastic Analysis (2)

• Dr Yan Zhuge
• CIVE3011 Structural Analysis and Computer
  Applications

2
Plastic 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

3
Plastic collapse of a portal frame
?P1 kN
Mp 100kNm EI 100kNm2 P1 10 kN P2 10 kN
?P2 kN
5m
10m
5m
The whole structure is in elastic range
20.8
7.8
21.2
? 1
25.7
11.4
Max bending moment
4
Plastic collapse of a portal frame
?P1 kN
?P2 kN
B
D
C
Plastic hinge
5m
A
E
5m
10m
30.4
80.9
The bending moment at E reaches Mp, a plastic
hinge is formed
82.7
l 3.9
100
44.4
5
Plastic collapse of a portal frame
B
D
C
5m
Plastic hinge
A
E
5m
10m
97.3
31.4
The bending moment at C reaches Mp, now there are
two plastic hinges.
100
l 4.60
100
64.2
6
Plastic collapse of a portal frame
46.7 kN
46.7 kN
B
D
C
Plastic hinge
5m
A
E
10m
5m
100
33.4
The bending moment at D reaches Mp, now there are
three plastic hinges.
100
l 4.67
100
66.8
7
Plastic collapse of a portal frame
50 kN
50 kN
B
D
C
Plastic hinge
5m
A
E
10m
5m
100
Four plastic hinges are formed, the structure is
changed into a mechanism and the corresponding
load is called the collapse load.
50
100
lc 5.0
100
100
8
Portal frame with pinned support
The value of Mp is constant throughout
Two hinges will be required to form a mechanism
9

• A beam Mechanism

Collapse is caused by the vertical force alone
A Sway Mechanism
Collapse is caused by the horizontal force alone
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Combined Mechanism

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

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.
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Virtual work equation
internal work, beam mechanism
external work, sway mechanism
internal work, sway mechanism
internal work at hinge which disappears
external work, beam mechanism
12
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.
Interaction diagram (ID)
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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.

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A more complicated case
15
Beam mechanism
MP 400 kNm
Beam mechanism d 4q 6a a (2/3)q
MP 200 kNm
Vd 200q 400(q a) 200a substituting for d
and a V x 4q 600 (1 (2/3))q V 250kN
At a connection between two members, the plastic
hinge forms at a BM equal to the plastic moment
of the weaker member
16
Sway mechanism
Sway mechanism D 5q 3b b (5/3)q
HD 2 x 200q 2 x 200b substituting for D and
b H x 5q 400 (1 (5/3))q H 213.3kN
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
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Combined mechanism
Combined mechanism a (2/3)q b (5/3)q
Vd HD 600 (1 (2/3))q 400 (1 (5/3))q -
200q - 200q 4V 5H 1666.7
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Interaction diagram (ID)
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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 H166.6 kN and V208.3 kN
  (assume V1.25H).
• The corresponding bending moment diagram

20
How to draw the BMD?
The Free Body Diagram (FBD) of 4-5
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Contd.
The Free Body Diagram (FBD) of 1-2
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Contd.
The Free Body Diagram (FBD) of 2-3
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Collapse Mode Load Factor

• 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
• The Mode of Collapse
• The Load Factor
• For that particular Case

24
Pitched portal frame
V
H
B
A
C
Beam mechanism can not develop in the sloping
rafters
25
Sloping members
dh
B
dh
dv
dv
dv
l
lq
a
q
kh
a
Horizontal deflection vertical projection x
plastic rotation Vertical deflection horizontal
projection x plastic rotation deflection of a
beam with same span
A
L/2
dh (lq) sina lsina q khq dv (lq) cosa
lcosa q (L/2)q
26
Symmetric pitched portal frame
The analysis is more complicated than the
rectangular portal frame, only required for
assignment not for exam.
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Possibilities to form a beam mechanism
The internal work is the same in each case
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Virtual work (pattern (b))
Internal work Mp? 3Mp x 2? Mp(? ?) Mp?
4Mp(2k)?
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
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Example pitched portal frame
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Pitched-portal mechanism
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.
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Sway mechanism
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Combined mechanism (pitched-portal sway)
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The interaction diagram
The collapse load is and
(V5H).
The collapse load is and
(VH)