lecture4b-4
Transcript of 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
Plastic collapse of a portal framePlastic collapse of a portal frame
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
Plastic collapse of a portal framePlastic collapse of a portal frame
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
Plastic collapse of a portal framePlastic collapse of a portal frame
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
Plastic collapse of a portal framePlastic collapse of a portal frame
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