Resistance to Accidental
Explosions
General principles
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Outline
Classification of explosion loads
Dynamic response based on SDOF analogy
Dynamic response charts
ISO-damage (pressure-impulse) diagram
Resistance curves for beams, girders and plates
Ductility limitations
Verification of simple design methods
NUS July 12-14, 2005
Analysis and Design for Robustness of Offshore Structures
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Simple (SDOF) vs. advanced
methods
Impulsive asymptote
11
10
9
SDOF methods – Biggs’ (1964)
(Elastic-plastic/rigid plastic methods, component analysis…)
– Early Design
– Screening of scenarios
USFOS
– Codes (NORSOK, IGN(UK)…
Pressure F/R
•
8
7
6
5
4
Iso-damage curve for ymax /yelastic = 10
Elastic-perfectly plastic resistance
3
2
1
Pressure asymptote
0
0
1
2
3
4
5
6
7
8
9
10
11
Impulse I/(RT)
Iso-damage
curve for
Non-linear static and dynamic
analysis
•
Advanced Methods – NLFEA
blast loading
– Large-scale simulations feasible
– Detail Engineering
– Critical Scenarios
– Quality of analysis?
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EXPLOSION
Classification of response
Impulsive domain I  2meq 
wmax
0
R wdw ,
td/T< 0.3
td
I   F  t  dt = impuls
0
Response independent of load magnitude
Dynamic domain Quasi-static domainwmax 
1
wmax
Fmax
0
Fmax  R w wmax
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R wdw
0.3 < td/T < 3
3 < td/T
Rise time small
Rise time large
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EXPLOSION
Impulsive domain - td/T< 0.3
Feq(t)
Y(td)
Feq(t)
meq
keq(y)
y(t)
td
R(y)= keq(y)·y
•
Conservation of momentum
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Conservation of energy
y max
1
1 I2
2
meq y t d  
  R  y dy
0
2
2 meq
y  td   0
1
y  td  
m eq
•
t

td
0
Feq  t  dt 
I
meq
I  2meq 
y max
0
R  y dy
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EXPLOSION
Quasi-static domain
Feq(t)
Feq(t)
meq
keq(y)
Y(td)
- td/T> 3
R(y)= keq(y)·y
Feq(t)
y(t)
trise
y(t)
td
t
Feq,max y max 
External work
NUS July 12-14, 2005

y max
0
R  y  dy
Strain energy
t
(2)
(1)
• Rise time small (1)
td
trise
• Rise time large (2)
Feq,max  R  y max 
Static solution
Analysis and Design for Robustness of Offshore Structures
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Explosion response -1 DOF analogy
m y  k y  f (t )
Dynamic equilibrium
(x)
= displacement shape function
y(t)
= displacement amplitude
m   m  x  dx   M ii
2
2
= generalized mass
i
f (t )   p (t )  x dx   Fii
= generalized load
k   EI, xx  x  dx
= generalized elastic bending stiffness
i
2
k 0
= generalized plastic bending stiffness
(fully developed mechanism)
k   N, x  x  dx
2
= generalized membrane stiffness
(fully plastic: N = NP)
m
= distributed mass
Mi
= concentrated mass
p
= explosion pressure
Fi
= concentrated load (e.g. support reactions)
xi
= position of concentrated mass/load
NUS July 12-14, 2005
Analysis and Design for Robustness of Offshore Structures
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Dynamic equilibrium- alternative formulation
klm My  Ky  F (t )
klm 
km
kl
= load-mass transformation factor
m
M
f
kl 
F
km 
= mass transformation factor
= load transformation factor
M   mdx   M i
= total mass
F   pdx   Fi
= total load
i
i
K
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k
km
= characteristic stiffness
Analysis and Design for Robustness of Offshore Structures
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EXPLOSION
SDOF analogy – Biggs’ method
f(t)
f(t)
t
Feq(t)
y
Load-mass
transformation
factor
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meq
keq(y)
Dynamic equilibrium:
meq y  t   ceq y  t   k eq  y   Feq  t 
y(t)
klm My  Ky  F (t )
Analysis and Design for Robustness of Offshore Structures
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ymax
t
9
Development of explosion response charts
 klm,u Mu  klm,c Mc  y  K  y y  F(t)
F(t)
R(y) Rel
yel
Displacement [m]
0,20
0,15
Fmax
 Explosion load history
t
y
Dynamic equilibrium
Solve dynamic equation
– numerical integration
Determine maximum
deformation ymax
Shell - plate
Shell - stiffener
beam
Rel/Fmax = 0.31
0,10
Perform analysis for
different duration and load
amplitude
0,05
Rel/Fmax = 0.59
0,00
0,000
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0,005
0,010
Time [secs]
0,015
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EXPLOSION
Classification of resistance curves
R
R
K1
K2
K1
R
K2
K3
K1
K1
w
Elastic
K2
R
w
w
Elastic-plastic
(determinate)
Elastic-plastic
(indeterminate)
R
w
Elastic-plastic
with membrane
K3
K2=0
Rel
K1
Wel
or yel
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w
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Explosion response chart
maximum displacement versus load duration
Governing parameters:
Mechanisme resistance
vs. maximum load
Rel/Fmax
Load duration vs.
eigenperiod td/T
Membrane stiffness, if
any
NUS July 12-14, 2005
Analysis and Design for Robustness of Offshore Structures
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EXPLOSION
Dynamic response chart for pressure pulse-[J.M.Biggs]
Triangular load - rise time = 0.3 td
NUS July 12-14, 2005
Analysis and Design for Robustness of Offshore Structures
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Development of ISO-damage curves from dynamic response
charts for a given pressure pulse
Rel/Fmax =0.05
=0.1
= 0.3
100
= 0.5
= 0.6
= 0.7
Example
yallow/yel =10
Rel/Fmax = 0.8
= 0.9
10
ymax/yel
= 1.0
= 1.1
= 1.2
= 1.5
1
k3 = 0
k 3 = 0.1k 1
F
R
Fmax
Rel
k 3 = 0.2k 1
k 3 = 0.5k 1
k 3 = 0.5k 1 =0.2k 1
=0.1k 1
k1
td
yel
y
0.1
0.1
1
10
td/T
NUS July 12-14, 2005
Analysis and Design for Robustness of Offshore Structures
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Development of ISO-damage curves from dynamic
response charts for a given pressure pulse
Rel/Fmax =0.05
=0.1
= 0.3
100
= 0.5
= 0.6
= 0.7
Example
yallow/yel =10
Rel/Fmax = 0.8
= 0.9
10
ymax/yel
= 1.0
= 1.1
= 1.2
= 1.5
1
k3 = 0
k 3 = 0.1k 1
F
R
Fmax
Rel
k 3 = 0.2k 1
k 3 = 0.5k 1
k 3 = 0.5k 1 =0.2k 1
=0.1k 1
k1
td
yel
y
0.1
0.1
1
td/T
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Pressure = Fmax
Impulse =1/2F
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td
15
EXPLOSION
Iso-damage curve for yallow/yelastic =10.
[W.Baker]
Impulsive asymptote
11
10
9
Pressure F/R
8
7
6
5
Inadmissible domain
4
Iso-damage curve for ymax /yelastic = 10
Elastic-perfectly plastic resistance
3
2
1
Admissible domain
Pressure asymptote
0
0
1
2
3
4
5
6
7
8
9
10
11
Impulse I/(RT)
NUS July 12-14, 2005
Analysis and Design for Robustness of Offshore Structures
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EXPLOSION
Resistance curves
Beams and girders
Tabulated values for elastic-plastic behaviour
Resistance curves based on plastic thory
Plates
Elastic and plastic theory
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Analysis and Design for Robustness of Offshore Structures
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Mass factor km
Load case
F=pL
L
Load-mass factor
klm
Uniform Concen- Uniform
mass
trated
mass
mass
Resistance
domain
Load
Factor
kl
Elastic
0.64
0.50
0.78
Plastic
bending
0.50
0.33
0.66
Plastic
membrane
0.50
0.33
0.66
Elastic
1.0
Concentrated
mass
Maximum
resistance
Rel
Characteristic
stiffness
K
8Mp
384 EI
L
5 L3
0.49
1.0
0.49
0
L
4NP
L
F
L/2
L3
L
L/2
Plastic
bending
Plastic
membrane
F/2
4Mp
F/2
Elastic
1.0
1.0
0.87
1.0
1.0
0.76
0.33
0.33
0.52
1.0
1.0
0.87
0.33
4Mp
0.60
Plastic
bending
1.0
Plastic
membrane
1.0
1.0
1.0
0.56
0.56
1.0
1.0
0.56
4NP
L
6M p
56.4 EI
L
L3
0.56
L
0.78 R  0.28 F
0
2 N P ymax
L
0.525 R  0.025 F
0.52 R el  0.02 F
6M p
L/3 L/3 L/3
2 N P ymax
L
0.75 Rel  0.25 F
L
0.33
0.39 R  0.11F
0.38 Rel  0.12 F
8Mp
48 EI
1.0
Dynamic reaction
V
0
6N P
L
3 N P ymax
L
Transformation factors for beams with various boundary
and load conditions
NUS July 12-14, 2005
Analysis and Design for Robustness of Offshore Structures
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Mass factor km
Load case
F=pL
L
Resistance
domain
Load
Factor
kl
Elastic
0.53
Concentrated
mass
Load-mass factor
klm
Uniform Concen- Uniform
mass
trated
mass
mass
0.41
Maximum
resistance
Rel
L
Elastoplastic
bending
0.64
0.50
0.78

8 M ps  M Pm

L
Plastic
bending
0.50
Plastic
membrane
0.50
0.33
0.66
L/2
Elastic


L3
384 EI
5 L3
 307 EI 


 L3 
L
Dynamic reaction
V
0.36 R  0.14 F
0.39 Rel  0.11F
0.38 Rel  0.12 F
0
2 N p ymax
0.33
4NP
L
0.66

4 M ps  M Pm
L/
2
384 EI
12 M ps
0.77
8 M ps  M Pm
F
Characteristic
stiffness
K
1.0
1.0
0.37
1.0
0.37

L
4 M ps  M Pm
Plastic
bending
1.0
Plastic
membrane
1.0
1.0
1.0
0.33
0.33
1.0
1.0
0.33
0.33

192 EI
L3

L
0.71R  0.21F
0.75 Rel  0.25 F
0
L
4NP
L
2 N P ymax
L
Transformation factors for beams with various boundary and
load conditions
NUS July 12-14, 2005
Analysis and Design for Robustness of Offshore Structures
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Mass factor km
Load case
Resistance
Load
domain
Factor
kl
Elastic
080
Load-mass factor
klm
Concentrated
mass
Uniform Concenmass
trated
mass
Uniform
mass
0.64
0.41
0.51
0.80
Maximum
resistance
Characteristic
linear stiffness
Rel
K1
V
9 M ps
L
F/2
6  M ps  M Pm 
F/2
Elastoplastic
L/3 L/3 L/3
0.87
0.76
0.52
0.87
0.60
bending
L
6  M ps  M Pm 
260EI
L3
1.0
0.56
1.0
0.56
0.48 R  0.02 F
56.4EI
L3
0.52 Rel  0.02 F
L
1.0
Dynamic reaction
0
0.52 Rel  0.02 F
Plastic
bending
1.0
1.0
0.56
1.0
0.56
6N P
L
3 N P ymax
L
Plastic
membrane
New Revision II: Transformation factors for clamped
beam with two concentrated loads
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Analysis and Design for Robustness of Offshore Structures
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Mass factor km
Load case
F=pL
V1
L
Resistan
ce
domain
Load
Fact
or
kl
Elastic
0.58
Concen
-trated
mass
Load-mass
factor k lm
Unifor Concen- Unifor
m mass trated
m mass
mass
0.45
Maximum
resistance
Rel
Characterist
ic stiffness
K
185 EI
8 M ps
0.78
L
Elastoplastic
Bending
V
0.64
0.50
0.78
2

4 M
ps
 2 M Pm
L

4 M ps  2 M Pm
Plastic
bending
0.50
0.33
0.66
Plastic
membra
ne
0.50
0.33
0.66


L3
384 EI
3
5L
 160 EI 



L3 
Dynamic
reaction
V
V1  0.26 R  0.12 F
V2  0.43R  0.19 F
0.39 R  0.11F
 M Ps
L
L
0
4NP
L
0.38 R  0.12 F
 M Ps
L
2 N P y max
L
V1  0.25 R  0.07 F
F
V1
L/2
L/2
Elastic
1.0
1.0
0.43
1.0
0.43
Elastoplastic
V
2
Bending
1.0
1.0
0.49
1.0
0.49
Plastic
bending
1.0
Plastic
membra
ne
1.0
1.0
0.33
1.0
0.33
107 EI
16 M Ps
3L

2 M ps  2 M Pm
L3
48 EI

L

2 M ps  2 M Pm

L3
1.0
0.33
1.0
0.78 R  0.28 F
 106 EI 



L3 
0
L
F/2
V1
 M Ps
L
0.75 R  0.25 F
 M Ps
L
4NP
L
0.33
2 N P y max
L
132 EI
F/2
V2  0.54 R  0.14 F
Elastic
0.81
0.67
0.45
0.83
0.55
Elastoplastic
Bending
V
2
0.87
0.76
0.52
0.87
0.60

6 M Ps
L
2 M ps  3 M Pm
L3
L

2 M ps  3 M Pm
L/3 L/3 L/3

L3
56 EI
 122 EI 



L3 
V1  0.17 R  0.17 F
V2  0.33R  0.33F
0.525 R  0.025 F
 M Ps
L

L
Plastic
bending
Plastic
membra
ne
1.0
1.0
0.56
1.0
0.56
0
0.56
6N P
L
0.52 Rel  0.02 F
1.0
1.0
0.56
1.0
 M Ps
L
3 N P y max
L
Transformation factors for beams with various boundary and load conditions
NUS July 12-14, 2005
Analysis and Design for Robustness of Offshore Structures
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Ductility ratios
( Ref: Interim Guidance Notes)
Table A.6-3 Ductility ratios  beams with no axial restraint
Boundary
Load
conditions
Cross-section category
Class 1
Class 2
Class 3
Cantilevered
Concentrated
Distributed
6
7
4
5
2
2
Pinned
Concentrated
Distributed
6
12
4
8
2
3
Fixed
Concentrated
Distributed
6
4
4
3
2
2
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