NUS July 12-14, 2005 Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course 1 Resistance to Accidental Explosions General

NUS July 12-14, 2005 Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course

1

Resistance to Accidental Explosions

General principles


2

Outline

Classification of explosion loadsDynamic response based on SDOF analogyDynamic response chartsISO-damage (pressure-impulse) diagramResistance curves for beams, girders and platesDuctility limitationsVerification of simple design methods


3

Simple (SDOF) vs. advanced methods

• SDOF methods – Biggs’ (1964)(Elastic-plastic/rigid plastic methods, component analysis…)

– Early Design

– Screening of scenarios

– Codes (NORSOK, IGN(UK)…

• Advanced Methods – NLFEA– Large-scale simulations feasible

– Detail Engineering

– Critical Scenarios

– Quality of analysis?

Iso-damage curve for blast loading

USFOSNon-linear static and dynamic analysis

Blast loading Transient dynamic analyses


4

Impulsive domain - td/T< 0.3

Response independent of load magnitude

Dynamic domain - 0.3 < td/T < 3

Quasi-static domain - 3 < td/T

EXPLOSIONClassification of response

w R wdwmax

wmax 1

0Fmax

Rise time small

maxmax wwRF Rise time large

I mRwdweq

wmax20 , I Ftdt

td0 = impuls


5

• Conservation of momentum

EXPLOSIONImpulsive domain - td/T< 0.3

Feq(t)

meq

keq(y)Y(td)

t

Feq(t)

y(t)td

d

d

t

d eq0

eq eq

y t 0

1 Iy t F t dt

m m

R(y)= keq(y)·y

• Conservation of energy

max

max

0

0

22

2

2

1

2

1

y

eq

y

eqdeq

dyyRmI

dyyRm

Itym


6

• Rise time small (1)

External work Strain energy

EXPLOSIONQuasi-static domain - td/T> 3

Feq(t)

meq

keq(y)Y(td)

t

Feq(t)

y(t)td

maxy

eq,max max0

F y R y dy

R(y)= keq(y)·y

• Rise time large (2)

Static solution

eq,max maxF R y

t

Feq(t)

y(t)

td

(1) (2)trise trise


7

Explosion response -1 DOF analogy)( tfykym D y n a m i c e q u i l i b r i u m

( x ) = d i s p l a c e m e n t s h a p e f u n c t i o n

y ( t ) = d i s p l a c e m e n t a m p l i t u d e

22i

iiMdxxmm = g e n e r a l i z e d m a s s

i

iiFdxxtptf )()( = g e n e r a l i z e d l o a d

dxxEIk xx2

, = g e n e r a l i z e d e l a s t i c b e n d i n g s t i f f n e s s

k 0 = g e n e r a l i z e d p l a s t i c b e n d i n g s t i f f n e s s

( f u l l y d e v e l o p e d m e c h a n i s m )

dxxNk x2

, = g e n e r a l i z e d m e m b r a n e s t i f f n e s s

( f u l l y p l a s t i c : N = N P )m = d i s t r i b u t e d m a s s

M i = c o n c e n t r a t e d m a s s

p = e x p l o s i o n p r e s s u r e

F i = c o n c e n t r a t e d l o a d ( e . g . s u p p o r t r e a c t i o n s )

x i = p o s i t i o n o f c o n c e n t r a t e d m a s s / l o a d


8

Dynamic equilibrium- alternative formulation

)(tFKyyMklm

l

mlm k

kk = load-mass transformation factor

M

mkm = mass transformation factor

F

fkl = load transformation factor

i

iMmdxM = total mass

i

iFpdxF = total load

mk

kK = characteristic stiffness


9

EXPLOSIONSDOF analogy – Biggs’ method

f(t)

t

f(t)

Feq(t)

meq

keq(y)y

eq eq eq eqm y t c y t k y F t

Dynamic equilibrium:y(t)

t

ymax

)(tFKyyMklm Load-mass transformation factor


10

Development of explosion response charts

m,u u m,c ck M k M y K y y F(t) l lDynamic equilibrium

Explosion load history

Solve dynamic equation – numerical integration

Determine maximum deformation ymax

Perform analysis for different duration and load amplitude

F(t)

tt

0,00

0,05

0,10

0,15

0,20

0,000 0,005 0,010 0,015Time [secs]

Dis

plac

emen

t [m

]

Shell - plate Shell - stiffenerbeam

Rel/Fmax = 0.31

Rel/Fmax = 0.59

FmaxR(y)

y

Rel

yel


11

EXPLOSIONClassification of resistance curves

RRRR

w w w w

K1K1K1K1

K3

K2

K2K2

Elastic Elastic-plastic(determinate)

Elastic-plastic(indeterminate)

Elastic-plasticwith membrane

K2=0

K1

RK3

w

Rel

Wel or yel


12

Explosion response chartmaximum displacement versus load duration

Governing parameters:

Mechanisme resistance vs. maximum load

Rel/Fmax

Load duration vs. eigenperiod td/T

Membrane stiffness, if any


13

EXPLOSIONDynamic response chart for pressure pulse-[J.M.Biggs]

Triangular load - rise time = 0.3 td

0,1

1

10

100

0,1 1 10

td/T

=0.1 = 0.7= 0.6= 0.5Rel/Fmax=0.05 = 0.3

= 1.1= 1.0

=

= 0.9

Rel/Fmax= 0.8

= 1.2= 1.5

yel y

R

Rel

F

Fmax

td0.30td

k1

k3 = 0.5k1 =0.2k1 =0.1k1

k3 = 0

k3 = 0.1k1

k3 = 0.2k1

k3 = 0.5k1


14

Development of ISO-damage curves from dynamic response charts for a given pressure pulse

Example yallow/yel =10

0.1

1

10

100

0.1 1 10

td/T

y max

/yel

=0.1 = 0.7= 0.6= 0.5Rel/Fmax=0.05 = 0.3

= 1.1

= 1.0

= 0.9

Rel/Fmax= 0.8

= 1.2

= 1.5

yel y

R

Rel

F

Fmax

td

k1

k3 = 0.5k1 =0.2k1 =0.1k1k3 = 0

k3 = 0.1k1

k3 = 0.2k1

k3 = 0.5k1


15

Example yallow/yel =10

0.1

1

10

100

0.1 1 10

td/T

y max

/yel

=0.1 = 0.7= 0.6= 0.5Rel/Fmax=0.05 = 0.3

= 1.1

= 1.0

= 0.9

Rel/Fmax= 0.8

= 1.2

= 1.5

yel y

R

Rel

F

Fmax

td

k1

k3 = 0.5k1 =0.2k1 =0.1k1k3 = 0

k3 = 0.1k1

k3 = 0.2k1

k3 = 0.5k1

Pressure = Fmax

Impulse =1/2Fmaxttd

Development of ISO-damage curves from dynamic response charts for a given pressure pulse


16

EXPLOSIONIso-damage curve for yallow/yelastic =10. [W.Baker]

0

1

2

3

4

5

6

7

8

9

10

11

0 1 2 3 4 5 6 7 8 9 10 11

Impulse I/(RT)

Pre

ssu

re

F/R

Pressure asymptote

Impu

lsiv

e as

ympt

ote

Iso-damage curve for ymax/yelastic = 10

Elastic-perfectly plastic resistance

Inadmissible domain

Admissible domain


17

EXPLOSIONResistance curves

Beams and girders

Tabulated values for elastic-plastic behaviour

Resistance curves based on plastic thory

Plates

Elastic and plastic theory


18

M a ss fa c to r k m L oa d -m a s s fa c to rk lmL oa d ca s e R es is ta n ce

d om a inL oa d

F a cto rk l

C on c en -tra tedm a s s

U n ifo rmm a s s

C on c en -tra tedm a s s

U n ifo rmm a s s

M a xim u mres is ta n c e

R e l

C h a ra c ter is tics tiffn es s

K

D yn a m ic r ea c tionV

E la s tic

P la s ticb en d in g

P la s ticm em b ra n e

0 .6 4

0 .5 0

0 .5 0

0 .5 0

0 .3 3

0 .3 3

0 .7 8

0 .6 6

0 .6 6

8 M

Lp

8 M

Lp

3 8 4

5 3

E I

L

0

4 N

LP

0 3 9 0 1 1. .R F

0 3 8 0 1 2. .R Fe l

L

yN maxP2

E la s tic



1 .0

1 .0

1 .0

1 .0

1 .0

1 .0

0 .4 9

0 .3 3

0 .3 3

1 .0

1 .0

1 .0

0 .4 9

0 .3 3

0 .3 3

4 M

Lp

4 M

Lp

4 83

E I

L

0

4 N

LP

0 7 8 0 2 8. .R F

0 7 5 0 2 5. .R Fe l

L

yN maxP2

E la s tic



0 .8 7

1 .0

1 .0

0 .7 6

1 .0

1 .0

0 .5 2

0 .5 6

0 .5 6

0 .8 7

1 .0

1 .0

0 .6 0

0 .5 6

0 .5 6

6 M

Lp

6 M

Lp

5 6 43

. E I

L

0

6 N

LP

0 5 2 5 0 0 2 5. .R F

0 5 2 0 0 2. .R Fe l

L

yN maxP3

F = p L

L

L /2

F

L /2

L /3 L /3 L /3

F /2 F /2

Transformation factors for beams with various boundary and load conditions


19


Mass factor km Load-mass factorklmLoad case Resistance

domainLoad

Factorkl

Concen-tratedmass

Uniformmass

Concen-tratedmass

Uniformmass

Maximumresistance

Rel

Characteristicstiffness

K

Dynamic reactionV

Elastic

Elasto-plastic

bending

Plasticbending

Plasticmembrane

0.53

0.64

0.50

0.50

0.41

0.50

0.33

0.33

0.77

0.78

0.66

0.66

12 M

Lps

8 M M

L

ps Pm

8 M M

L

ps Pm

3843

EI

L

384

5 3

EI

L

3073

EI

L

0

4 N

LP

0 36 0 14. .R F

0 39 0 11. .R Fel

0 38 0 12. .R Fel

L

yN maxp2

Elastic

Plasticbending

Plasticmembrane

1.0

1.0

1.0

1.0

1.0

1.0

0.37

0.33

0.33

1.0

1.0

1.0

0.37

0.33

0.33

4 M M

L

ps Pm

4 M M

L

ps Pm

1923

EI

L

0

4 N

LP

0 71 0 21. .R F

0 75 0 25. .R Fel

L

yN maxP2

F=pL

L

F

L/2L/2


20

New Revision II: Transformation factors for clamped beam with two concentrated loads

Mass factor km

Load-mass factor klm

Load case

Resistance

domain

Load

Factor

kl

Concen-trated mass

Uniform mass

Concen-trated mass

Uniform mass

Maximum resistance

Rel

Characteristic linear stiffness

K1

Dynamic reaction

V

L/3 L/3 L/3

F/2 F/2

Elastic

Elasto-plastic

bending

Plastic bending

Plastic membrane

080

0.87

1.0

1.0

0.64

0.76

1.0

1.0

0.41

0.52

0.56

0.56

0.80

0.87

1.0

1.0

0.51

0.60

0.56

0.56

9 psM

L

6 ps PmM M

L

6 ps PmM M

L

3

260EI

L

3

56.4EI

L

0

6 PN

L

0.48 0.02R F

0.52 0.02elR F

0.52 0.02elR F

L

yN maxP3


21


M a s s f a c t o r k m L o a d - m a s sf a c t o r k l mL o a d c a s e R e s i s t a n

c ed o m a i n

L o a dF a c t

o rk l

C o n c e n- t r a t e d

m a s s

U n i f o rm m a s s

C o n c e n -t r a t e dm a s s

U n i f o rm m a s s

M a x i m u mr e s i s t a n c e

R e l

C h a r a c t e r i s ti c s t i f f n e s s

K

D y n a m i cr e a c t i o n

V

E l a s t i c

E l a s t o -p l a s t i c

B e n d i n g

P l a s t i cb e n d i n g

P l a s t i cm e m b r a

n e

0 . 5 8

0 . 6 4

0 . 5 0

0 . 5 0

0 . 4 5

0 . 5 0

0 . 3 3

0 . 3 3

0 . 7 8

0 . 7 8

0 . 6 6

0 . 6 6

8 M

Lp s

4 2M M

L

p s P m

4 2M M

L

p s P m

1 8 53

E I

L

3 8 4

5 3

E I

L

1 6 03

E I

L

0

4 N

LP

V R F1 0 2 6 0 1 2 . .

V R F2 0 4 3 0 1 9 . .

0 3 9 0 1 1. .R F

M LP s

0 3 8 0 1 2. .R F

M LP s

L

yN maxP2

E l a s t i c


B e n d i n g


P l a s t i cm e m b r an e

1 . 0

1 . 0

1 . 0

1 . 0

1 . 0

1 . 0

1 . 0

1 . 0

0 . 4 3

0 . 4 9

0 . 3 3

0 . 3 3

1 . 0

1 . 0

1 . 0

1 . 0

0 . 4 3

0 . 4 9

0 . 3 3

0 . 3 3

1 6

3

M

LP s

2 2M M

L

p s P m

2 2M M

L

p s P m

1 0 73

E I

L

4 83

E I

L

1 0 63

E I

L

0

4 N

LP

V R F1 0 2 5 0 0 7 . .

V R F2 0 5 4 0 1 4 . .

0 7 8 0 2 8. .R F

M LP s

0 7 5 0 2 5. .R F

M LP s

L

yN maxP2

E l a s t i c


B e n d i n g


P l a s t i cm e m b r an e

0 . 8 1

0 . 8 7

1 . 0

1 . 0

0 . 6 7

0 . 7 6

1 . 0

1 . 0

0 . 4 5

0 . 5 2

0 . 5 6

0 . 5 6

0 . 8 3

0 . 8 7

1 . 0

1 . 0

0 . 5 5

0 . 6 0

0 . 5 6

0 . 5 6

6 M

LP s

2 3M M

L

p s P m

2 3M M

L

p s P m

1 3 23

E I

L

5 63

E I

L

1 2 23

E I

L

0

6 N

LP

V R F1 0 1 7 0 1 7 . .

V R F2 0 3 3 0 3 3 . .

0 5 2 5 0 0 2 5. .R F

M LP s

0 5 2 0 0 2. .R F

M Le l

P s

L

yN maxP3

V 2V 1

F = p L

L

V 1

L / 2 L / 2

F

V 2

V 1

L / 3 L / 3 L / 3

F / 2 F / 2

V 2


22

Ductility ratios( Ref: Interim Guidance Notes)

Table A.6-3 Ductility ratios beams with no axial restraint

Cross-section categoryBoundary

conditions

Load

Class 1 Class 2 Class 3

Cantilevered ConcentratedDistributed

67

45

22

Pinned ConcentratedDistributed

612

48

23

Fixed ConcentratedDistributed

64

43

22

Documents

NUS July 12-14, 2005 Analysis and Design for Robustness of Offshore Structures NUS – Keppel Short Course 1 Resistance to Accidental Explosions General