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Structural Response - Shepherd 1
Structural Response to Explosions
Joseph E ShepherdCalifornia Institute of Technology
Pasadena, CA USA 91125
Presented at
1st European Summer School on Hydrogen SafetyUniversity of Ulster, August 2007
Structural Response - Shepherd 2
Summary
This lecture will cover the fundamentals of structural response to internal and external loading of structures by explosions of fuel-air mixtures. There will be two parts to the lecture. The first part will review the generation and characterization of pressure waves by deflagrations, detonations, transition from deflagration to detonation inside of vessel, and blast waves from unconfined vapor cloud explosions and detonations of fuel-air clouds. The second part will cover structural response of simple structures with an emphasis on single degree of freedom models and integral characterization of the pressure loading.
Structural Response - Shepherd 3
Outline• Overview• Determining Structural Loads• Determining Structural Response • Examples
Structural Response - Shepherd 4
Overview
• Why carry out structural response analysis? • How do explosions damage structures?• Motivations for considering structural failure in a
safety assessment• Elements of a structural response analysis
Structural Response - Shepherd 5
Why Study Structural Response?
• Before an event as part of a safety assessment activity
– Will structural failure happen?
• After an event as part of an incident investigation
– Why did structural failure happen?
Structural Response - Shepherd 6
How do explosions damage structures?
• Bend, break, or displace load-bearing panels, posts, and beams, possibly causing structural collapse
• Distort and possibly rupture pressure vessels. pipes, valves, and instrumentation, releasing hazardous (toxic or explosive) materials into the environment
• Shock and vibration can break nonstructural components (e.g., glass windows) far from incident.
• Create fragments which can travel long distances, causing facility damage and bodily injury.
• Start fires due to thermal radiation from fireballs and heat transfer from combustion products.
Structural Response - Shepherd 7
Pasadena TX 1989 – C2H4 Flixborough 1974 - cyclohexane
Port Hudson 1974 – C3H8
(20 Kg H2 )
Structural Response - Shepherd 8
Nuclear Blast Wave Damage – 5 psi (34 kPa)
Structural Response - Shepherd 9
Nuclear Blast Wave Effects
1.7 psi (11.7 kPa)5 psi (34 kPa)
Structural Response - Shepherd 10
Motivations for studying structural response
• Immediate life safety consequence – damage to critical structures will lead to injury or death. Examples: Pressure vessels and piping systems containing toxic materials.
• Creating a potential hazard – release of combustible or flammable material could result in fire or explosion that has life safety and secondary hazard generation consequences.
• Economic loss – destruction of high value processing equipment, loss of product, plant downtime, environmental cleanup, compensation of victims, litigation costs.
Structural Response - Shepherd 11
Pasadena TX 1989
Structural Response - Shepherd 12
Elements of Structural Response Analysis
• Define explosion hazard or sequence of events in an actual accident.
– HAZOP or FEMA • Develop a model for the type of explosion that takes
place.– Validate explosion model against existing data or new tests
• Estimate the structural loading• Develop a model for the structure and loading capacity• Estimate response of structure to loading
– Validate structural model against existing data or new tests• Establish pass-fail criteria based on material properties
and maximum deformations or stresses– Use existing databases or carry out material testing
Structural Response - Shepherd 13
Related Subjects• Earthquake engineering
– Strong ground motion excites building motion• Terminal ballistics
– Projectile impact creates stress waves and vibration• Crashworthiness
– Vehicle crash mitigation • Weapons effects
– Conventional (High explosive and FAE)– Nuclear and nuclear simulation testing
TIP – Many recent studies on structural response to blasts have been sponsored to counter terrorism – the results are often restricted to government agencies or official use only.
Structural Response - Shepherd 14
Structural Response - Shepherd 15
Determining structural loads
• Load generally means “applied force” in this context. The primary load is usually thought of as due to pressure differences created by the explosion process. Pressure differences across components of a structure create forces on the structure and internal stresses.
• Three simple cases– External explosion– Blast wave interaction– Internal explosion
Structural Response - Shepherd 16
External Explosion• Explosion due to accidental
vapor cloud release and ignition source starting a combustion wave
• Flame accelerates due to instabilities and turbulence due to flow over facility structures
• Volume displacement of combustion (“source of volume”) compresses gas and creates motion locally and at a distance– Blast wave propagates away
from source Unconfined Vapor Cloud Explosion (UVCE)
Structural Response - Shepherd 17
Blast Wave Interaction
• Blast wave consists of– Leading shock front– Flow behind front
• Pressure loading – Incident and reflected
pressure behind shock– Stagnation pressure from
flow
• Factors in loading– Blast decay time– Diffraction time– Distance from blast origin
Structural Response - Shepherd 18
Internal Explosion
• Can be deflagration or detonation• Deflagration
– Pressure independent of position, slow• Detonation
– Spatial dependence of pressure– Local peak associated with detonation wave
formation and propagation
Structural Response - Shepherd 19
Type of Combustion• The computation of structural loading requires determining the time history of
the pressure applied to the structure. There are two generic situations– Internal explosion– External explosion
• The mode of combustion is important in both situations– Deflagration – slow speed combustion (1-1000 m/s)– Detonation – high speed combustion (1500-3000 m/s)– Deflagration-to-detonation transition (DDT) – accelerating combustion wave with
localized pressure spikes• The mode of combustion depends on many factors
– Composition of mixture: amount of fuel, oxidizer and diluent– Initial temperature and pressure– Type of ignition source– Presence of flame accelerating elements such internal obstructions in tubes, pipe
racks, grates, etc.– Distance of propagation (size of pipe, vessel, or fuel-air cloud)
Structural Response - Shepherd 20
Pressure Generation Mechanism
• Volume expansion due to combustion– Displaces surrounding gas– Confinement due to
• Inertia of gas• Surrounding structure limits motion
• Pressure rise due to – Confinement– Compression of surrounding gas– Generation of blast waves
Structural Response - Shepherd 21
Combustion and Pressure Waves
• Overall Combustion Reaction – major speciesH2 + ½(O2 + 3.76 N2) H2O + 1.88N2
• Combustion results in temperature rise due to conversion of chemical to thermal energy
• Temperature rise creates– Volume expansion (low speed flames)– Pressure rise in constant volume combustion– Pressure rise and flow in detonation and high speed
flames
Structural Response - Shepherd 22
Creation of Pressure Waves by Explosions• Expansion of combustion
products due to conversion of chemical to thermal energy in combustion and creation of gaseous products in high explosives
• Expansion ratio for gaseous explosions depends on thermodynamics
• Expansion rate depends on chemical kinetics and fluid mechanics– Flame speeds – Detonation velocity
Structural Response - Shepherd 23
Creation of flow by Explosions I.• Flames create flow due to expansion of products
pushing against confining surfaces• Consider ignition at the closed-end of a tube
– Expansion ratio
– Flame velocity
– Flow velocity
Burned (u =0) Vf Unburned u > 0
flame
ST
effTfTf SAASV σσ == /
b
uρ
ρσ =
effT
effTf SSVU )1( −=−= σ
Blast wave
u = 0
Structural Response - Shepherd 24
Creation of flow by Explosions II• Detonations and shock waves create flow due to
acceleration by pressure gradients in waves• Consider ignition of detonation at the closed-end of a
tube
Burned (u =0) Burned u >0 Unburned u = 0
Detonation wave
Expansion wave
u
x
Structural Response - Shepherd 25
Loading Histories
• Pressure-time histories can be derived from several
sources
– Experimental measurements
– Analytical models with thermodynamic computation of
parameters
– Detailed numerical simulations using computation fluid
dynamics
– Empirical correlations of data
– Approximate numerical models of blast wave
propagation (Blast-X)
• Characterizing pressure-time histories
– Single peak or multiple peaks
– Rise time
– Peak pressure
– Duration
Slow flame in vessel
High speed flame in vessel
Nonideal vapor cloud explosion
Ideal vapor detonation
Structural Response - Shepherd 26
Pressure Loading Characterization
• Structural response time T vs. loading and unloading time scales τI
• Peak pressure Δ P vs. Capacity of structure• Loading regimes
– Slow (quasi-static), typical of flame inside vessels T << τL or τu
– Sudden, shock or detonation waves τL << T• Short duration – Impulsive τU << T• Long duration - Step load T << τU
τload τunload
Structural Response - Shepherd 27
Preview• Structural response in simplistic terms
– What are structural response times?• Large spectrum for a complex structure• Single value for simple structure
– How do these compare to loading and unloading times of pressure wave?
• Loading time• Unloading time
– Estimate peak deflection and stresses based on these time scale comparisons and peak load
– Compare capacity of structure with expected peak load. Failure can occur to do either
• Excessive stress – plastic deformation or fracture makes structure too weak for service
• Excessive deformation – structure not useable due to leaks in fittings or misfit of components (rotating shafts, etc).
Structural Response - Shepherd 28
Ideal Blast Wave SourcesSimplest form of pressure loading – due concentrated, rapid release of energyHigh explosive or “prompt” gaseous detonation. Main shock wave followed bypressure wave and gas motion, possibly secondary waves.
Structural Response - Shepherd 29
Blast Wave from Hydrogen-Air Detonation
Structural Response - Shepherd 30
Blast and Shock Waves
• Leading shock front pressure jump determined by wave speed – shock Mach number.
• Gas is set into motion by shock then returns to rest
• Wave decays with distance
• Loading determined by– Peak pressure rise– Impulse– Positive and negative
phase durations
Δ P
τ-τ+
Specific impulse!
Structural Response - Shepherd 31
Scaling Ideal Blast Waves I.
• Dimensional analysis (Hopkinson 1915, Sachs 1944, Taylor-Sedov)– Total energy release E = Mq
• M = mass of explosive atmosphere (kg)• q = specific heat of combustion (J/kg)
– Initial state of atmosphere Po or ρo and co
• Limiting cases– Strength of shock wave
• Strong Δ P >> Po
• Weak Δ P << Po
– Distance from source• Near R ~ Rsource
• Far R >> Rsource
Structural Response - Shepherd 32
Scaling Ideal Blast Waves II.• Scale parameters
– Blast length scale Rs = (E/Po)1/3
– Time scale Ts = Rs/co
– Pressure scale• Close to explosion Pexp (usually bounded by PCJ) • Far from explosion Po
• Nondimensional variables– pressure Δ P/Po
– distance R/Rs
– time t/Ts
– Impulse (specific) I/(Po Ts)
Relationships:
Δ P/Po = F(R/Rs)I/(Δ P Ts) = G(R/Rs)
Structural Response - Shepherd 33
Cube Root Scaling in Standard atmosphere
• Simplest expression of scaling (Hopkinson)– At a given scaled range R/M1/3, you will have the same
scaled impulse I/M1/3 and overpressure Δ P– When you increase the charge size by K, overpressure
will remain constant at a distance KR, and the duration and arrival time will increase by K.
Structural Response - Shepherd 34
TNT Equivalent• Ideal blast wave from gaseous explosion equivalent to
that from High Explosive (TNT) when energy of gaseous explosive is correctly chosen
• Universal blast wave curves in far field when expressed in Sachs’ scaled variables
• For ideal gas explosions (detonations) E is some fixed fraction of the heat of combustion (Q = qM)
• For nonideal gas explosions (unconfined vapor clouds), E is quite a bit smaller. Key issues:– How to correctly select energy equivalence?– How to correctly treat near field?
Structural Response - Shepherd 35
Scaling of Blast Pressure – Ideal Detonation
Comparison of fuel-air bag tests to high explosives
Work done at DRES (Suffield, CANADA) in 1980s
Structural Response - Shepherd 36
Scaling of Impulse – Ideal Detonation
Air burst
Surface burst
For the same overpressure or scaled impulse at a given distance, M(surface) = 1/2 M(air)
Structural Response - Shepherd 37
Energy scaling of H2-air blast
Energy Equivalence
100 MJ/kg of H2
or
2.71 MJ/kg of fuel-air mix for stoichiometric.
Structural Response - Shepherd 38
Hydrogen-air Detonation in a Duct
• Blast waves in ducts decay much more slowly than unconfined blasts
Δ P ~ x-1
• Multiple shock waves created by reverberation of transverse waves within duct
• Pressure profile approaches triangular waveshape at large distances.
Structural Response - Shepherd 39
Interaction of Blast Waves with Structures
Blast-wave interactions with multiple structures LHJ Absil, AC van den Berg, J. Weerheijm p. 685 - 290,Shock Waves, Vol. 1, Ed. Sturtevant, Hornung, Shepherd, World Scientific, 1996.
Structural Response - Shepherd 40
Idealized Interactions
Enhancement depends:
Incident wave strength
Angle of incidence
Structural Response - Shepherd 41
Nonideal Explosions
• Blast pressure depends on magnitude of maximum flame speed
• Flame speed is a function of– Mixture composition– Turbulence level– Extent of confinement
• There is no fixed energy equivalent– E varies from 0.1 to 10% of Q
• Impulse and peak pressure depend on flame speed and size of cloud – Sachs’ scaling has to be expanded to include these
Structural Response - Shepherd 42
Pressure Waves from Fast FlamesSachs’ scaling with addition parameter – effective flame Mach number Mf. Numericalsimulations based on ‘porous piston’ model and 1-D gas dynamics.
Tang and Baker 1999
Structural Response - Shepherd 43
What is Effective Flame Speed?
Dorofeev 2006
Consider volume displacementof a wrinkled (turbulent) flame growing ina mean spherical fashion.
Expansionratio
Structural Response - Shepherd 44
Internal Explosion - Deflagration
• Limiting pressure determined by thermodynamic considerations– Adiabatic combustion process– Chemical equilibrium in products– Constant volume
• Pressure-time history determinedby flame speed
fuel-air mixture
Products
Combustion wave
SLVf = Sf + u
Structural Response - Shepherd 45
Burning Velocity
Depends on substance, composition, pressure, temperature
Structural Response - Shepherd 46
Expansion ratio and Flame temperature
• Related to flame temperature through gas law
• E will depend on composition • For fuel-air mixtures, Emax ~7
reactantsreactantsreactants TNTN
VV
E productsproductsproducts ==
NRTPV=
Structural Response - Shepherd 47
Expansion ratio
Structural Response - Shepherd 48
Adiabatic Flame Temperature
• Temperature of products if there are no heat lossesHreactants(Treactants) = Hproducts(Tproducts)
• Simple approximation for lean mixture:Tproduct ~ Treactants + fHc/Cp
Hc = heat of combustion of fuel (42 MJ/kg fuel)Cp = heat capacity of products (including N2, …)
• For stoichiometric HC fuel-air mixtures: Tproducts ~ 2000oC• Decreases for off-stoichiometric, and diluted mixtures, 1100-
1400 oC at flammability limit.• Values are similar for all HC fuels when expressed in terms of
equivalence ratio.
Structural Response - Shepherd 49
Pressure in Closed Vessel ExplosionPeak pressure limited by heat transfer during burn and anyVenting that takes place due to openings or structural failure
Structural Response - Shepherd 50
Adiabatic Explosion Pressure
• Pressure of products if there are no heat losses and complete reaction occurs
• Energy balance at constant volumeEreactants(Treactants) = Eproducts(Tproducts)
Vreactants = Vproducts
Pp = Pr (NpTp/NrTr)• Products in thermodynamic equilibrium• For stoichiometric HC fuel-air mixtures: Pp ~ 8-10 Pr
• Decreases for off-stoichiometric, and diluted mixtures, • Values are similar for all HC fuels when expressed in terms of
equivalence ratio.• Upper bound for peak pressure as long as no significant flame
acceleration occurs
Structural Response - Shepherd 51
Measured Peak Pressure vs Calculated
Structural Response - Shepherd 52
Structural Response - Shepherd 53
Forces, Stresses and Strains
• Loading becomes destructive when forces are sufficient to displace structures that are not anchored or else the forces (or thermal expansion) create stresses that exceed yield strength of the material.
• Important cases– Rigid body motion – fragments and overturning– Deformation due to internal stresses
• Bending, beams and plates• Membrane stresses, pressure vessels
Structural Response - Shepherd 54
Rigid Body Forces due to Explosion
• Pressure varies with position and time over surface – has to be measured or computed
• Local increment of force on surface due to pressure only in high Reynolds’ number flow
Geometry and distribution of pressure will result in moments as well as forces! Be sure to add in contributions from body forces (gravity) to get total force.
Structural Response - Shepherd 55
Consequence of Forces I.
• Rigid body motions– Translation
– Rotation
X’ = X – Xcm distance from center of mass
Structural Response - Shepherd 56
Internal Forces Due to an Explosion
• Force on a surface element dS
• Stress tensor σ
Structural Response - Shepherd 57
Consequence of forces – small strains (<0.2 %)
• Elastic deformation • Elastic strain
• Elastic shear
Youngs’ modulus E, shear modulus E, and Poisson ratio ν are material properties
Structural Response - Shepherd 58
Consequences of forces – large strains
• Onset of yielding for σ ~ σY
• Necking occurs in plastic regime σ > σY
• Plastic instability and rupturefor σ > σu
• Energy absorption by plastic deformation Plot is in terms of engineering stress and strain, apparent
maximum in stress is due to area reduction caused by necking
Structural Response - Shepherd 59
Stress-Strain Relationships
Structural Response - Shepherd 60
Yield and Ultimate Strength
• Yield point σYP determined by uniaxial tension test• Yielding is actually due to stress differences or shear.
Extension of tension test to multi-axial loading:– Maximum shear stress model τmax < σYP/2– Von Mises or octahedral shear stress criterion
• Onset of localized permanent deformation occurs well before complete plastic collapse of structure occurs.
Structural Response - Shepherd 61
Some Typical Material Properties
ρ E G ν σy σu εruptureMaterial (kg/m3) (GPa) (GPa) (MPa) (MPa)Aluminum 6061-T6 2.71 x 103 70 25.9 0.351 241 290 0.05Aluminum 2024-T4 2.77 x 103 73 27.6 0.342 290 441 0.3Steel (mild) 7.85 x 103 200 79 0.266 248 410-550 0.18-0.25Steel stainless 7.6 x 103 190 73 0.31 286-500 760-1280 0.45-0.65Steel (HSLA) 7.6 x 103 200 0.29 1500-1900 1500-2000 0.3-0.6Concrete 7.6 x 103 30-50 20-30 - 0Fiberglass 1.5-1.9 x 103 35-45 - 100-300 -Polycarbonate 1.2-1.3 x 103 2.6 55 60 -PVC 1.3-1.6 x 103 0.2-0.6 45-48 - -Wood 0.4-0.8 x 103 1-10 - 33-55 -Polyethylene (HD) 0.94-0.97 x 103 0.7 20-30 37 -
Structural Response - Shepherd 62
Internal Forces due to Explosions
• Stress waves– Longitudinal or transverse– Short time scale
• Flexural waves– Shock or detonation propagation inside tubes– Vibrations in shells
• tension or compression– Deforms shells
• shearing loads– Bends beams and plates
Structural Response - Shepherd 63
Statics vs. Dynamics
• Static loading T >> τl, τu– Loading and unloading times long compared to
characteristic structural response time– Inertia unimportant– Response determined completely by stiffness,
magnitude of load. • Dynamic loading T · τ
– Loading or unloading time short compared to characteristic structural response time
– Inertia important– Response depends on time history of loading
Structural Response - Shepherd 64
Static Stresses in Spherical Shell
• Balance membrane stresses with internal pressure loading
• Force balance on equator
• Membrane stress
Validate only for thin-wall vessels h < 0.2 R
R
R
Structural Response - Shepherd 65
Static Stresses in Cylindrical Shells
• Biaxial state of stress• Longitudinal stress due to
projected force on end caps.
• Radial (hoop) stress due to projected force on equator
Structural Response - Shepherd 66
Bending of Beams
• Force on beam due to integrated effects of pressure loading
• Pure bending has no net longitudinal stress
• Deflection for uniform loading
Structural Response - Shepherd 67
Stress Wave propagation in Solids
• Dynamic loading by impact or high explosive detonation in contact with structure
• Two main types– Longitudinal (compression, P-waves) – Transverse (shear, S-waves
• Stress-velocity relationship (for bar P-waves) Cl exact for bar
Structural Response - Shepherd 68
Is direct stress wave propagation important?
• Time scale very fast compared to main structural response T ~ L/C
– Average out in microseconds (10-6 s)
• Stress level low compared to yield stress
σ ~ Δ P ~ 10 MPa << σY = 200- 500 MPa
31553205Aluminum32056100Steel
Cs(m/s)Cl (m/s)
Direct stress propagation within the structural elements is usually not relevant for structural response to gaseous explosions.
Structural Response - Shepherd 69
Structural motions
• Element vibrations– Membranes or shells
– Plates or beams
– Modes of flexural motion• Standing waves, frequencies ωi
• Propagating dispersive waves ω(k)
• Coupled motions of entire structure
Structural Response - Shepherd 70
Two Special Situations
• Loading on small objects– Represent forces as drag coefficients dependent on shape and
orientation and function of flow speed.F = ½ ρ V2 CD(Mach No, Reynolds No) x Frontal Area
• Thermal stresses. – Thermal stresses are stresses that are created by differential
thermal expansion caused by time-dependent heat transfer from hot explosion gases. This is distinct from the loss of strength of materials due to bulk heating, which is a very important factor in fires which occur over very much longer durations than explosions.
ε = σ/E + α Δ T
Structural Response - Shepherd 71
Determining structural response
• Issues– Static or dynamic
• depends on time scale of response compared to that of load– impulsive (short loading duration)– sudden (short rise time)– quasi-static (long rise time)
– Elastic or elastic-plastic• depends on magnitude of stresses and deformation
– yield stress limit appropriate for vessels designed to contain explosions
– maximum displacement or deformation limit appropriate for determining or preventing leaks or rupture under accident conditions
Structural Response - Shepherd 72
Simple estimates• Strength of materials approach assuming equivalent static load
– Useful only for very slow combustion (static loads) and negligible thermal load
• Theory of elasticity and analytical solutions– static solutions for many common vessels and components (Roarke’s Handbook)– dynamic solutions available for simple shapes – mode shapes and vibrational periods are tabulated.– Energy methods with assumed mode shapes (Baker et al method)– Analytical models for traveling loads available for shock and detonation waves– Transient thermo-elastic solutions available for simple shapes
• Theory of plasticity – rigid-plastic solutions available for simple shapes and impulsive loads.– Energy methods can provide quick bounds on deformation
• Empirical correlations– Test data available for certain shapes (clamped plates) and impulsive loads– Pressure-impulse damage criteria have been measured for many items and people subjected to blast
loading
• Spring-mass system models– single degree of freedom – multi-degree of freedom– elastic vs plastic spring elements
Structural Response - Shepherd 73
Simple Structural Models
• Ignore elastic wave propagation within structure• Lump mass and stiffness into discrete elements
– Mass matrix M– Stiffness matrix K– Displacements Xi– Applied forces Fi
• Equivalent to modeling structure as coupled “spring-mass” system
• Results in a spectrum of vibrational frequencies ωI corresponding to different vibrational modes– Fundamental (lowest) mode usually most relevant
Structural Response - Shepherd 74
Single Degree of Freedom Models (SDOF)
• Example - radial oscillation of a shell.
• Allow only for radial displacement x of tube surface
• Assumes radial and axial symmetry of load
• Elastic oscillations only• Results in harmonic oscillator
equation (no damping)
P(t)
xh
frequency
R
τp
t
period
Structural Response - Shepherd 75
SODF - Square Pulse
Pulse length τ: 100μsPulse length τ: 10μs
Structural Response - Shepherd 76
SDOF – Static Regime
• Very slow application of load – (quasi-static) no oscillations
T << τu or τL
• Static deflection
FMax
T time
force
displacement
τl
Structural Response - Shepherd 77
SDOF -Impulsive Regime• Sudden load application, short
duration of loading τ << T• Linear scaling between
maximum strain/ displacement and impulse in elastic regime:
• Impulse generates initial velocity
• Energy conservation determines maximum deflection
Structural Response - Shepherd 78
SDOF – Sudden regime
• Quick application of load and long duration τu >> T
• Peak deflection is twice static value for same maximum load
FMax
T time
force
displacement
τ
Structural Response - Shepherd 79
SDOF - Dynamic load factor (DLF)
Structural Response - Shepherd 80
Considerations about material properties
• Simple models: – perfectly plastic, – elastic perfectly plastic
• More realistic models– Strain hardening σY (ε)
– Strain rate effects, σY(dε/dt)
– Temperature effects σY(T)
σ
ε
σ
ε
ε.
.
Structural Response - Shepherd 81
SDOF - Plasticity
• Replace kX with nonlinear relationship based on flow stress curve σ(ε)
• Energy absorbed by plastic work is much higher than elastic work
• Peak deformation for impulsive load scales with impulse squared.
Structural Response - Shepherd 82
SDOF – Pressure- Impulse (P-I)
• Alternative representation of response• For fixed Xmax and pulse shape, unique relation
between peak pressure (P) and impulse (I)
Shock wave with exponential tail
Structural Response - Shepherd 83
Numerical simulation
• Finite element models• static• vibration: mode shape and frequencies• dynamic
– transient response to specified loading– elastic – plastic/fracture
• Numerical integration of simple models with complex loading histories
– spring-mass systems– Elasticity with assumed mode shape
Structural Response - Shepherd 84
Example
• Blast loading of a cantilever beam
– Giordona et al elastic response
– Van Netton and Dewey plastic response
– Baker et al energy method
Structural Response - Shepherd 85
Initial stages of shock diffraction over a cantilever beam
Giordano et al, Shock Waves 14 (1-2), 103-110, 2005.
Structural Response - Shepherd 86
Later stages of diffraction over a cantilever beam
Giordano et al, Shock Waves 14 (1-2), 103-110, 2005.
Structural Response - Shepherd 87
Applied Load and Oscillations of Beam
Giordano et al, Shock Waves 14 (1-2), 103-110, 2005.
Structural Response - Shepherd 88
Plastic Deformation of Blast loaded Cantilever
Van Netten and Dewey, Shock Waves (1997) 7: 175–190
Structural Response - Shepherd 89
Blast Loading
Van Netten and Dewey, Shock Waves (1997) 7: 175–190
Structural Response - Shepherd 90
Shock tube experiments
Van Netten and Dewey, Shock Waves (1997) 7: 175–190
Structural Response - Shepherd 91
Structural Response of Piping to Internal Gaseous Detonation
Structural Response - Shepherd 92
Detonations in Piping
• Accidental explosions• Potential hazard in
– Chemical processing plants– Nuclear facilities
• Waste processing• Fuel and waste storage• Power plants
• Test facilities– Detonation tubes used in laboratory facilities– Field test installations (vapor recovery systems)
Structural Response - Shepherd 93
Hamaoka-1 NPP
Brunsbuettel KBB
Recent Accidental Detonations
Both due to generation of H2+1/2O2 by radiolysis and accumulation in stagnant pipe legs without high-point vents or off-gas systems.
Structural Response - Shepherd 94
Outline• Basic detonation facts• Elastic response of tubes to detonation• Fracture of tubes with detonation loading• Bounding loads
– Deflagration to detonation transition– Reflection of detonation
• Plastic deformation• Interaction with bends and tees• Role of ASME code
Structural Response - Shepherd 95
What is a Detonation Wave?
A supersonic combustion wave characterized by a unique coupling between a shock front and a zone of chemical energy release referred to as the “reaction zone.”
Structural Response - Shepherd 96
Detonation Concepts
• Steadily propagating wave (CJ)• Shock-induced chemical reaction (ZND)• Propagating pressure wave• Induces a flow and pressure variation behind
detonation• Instability of front
Structural Response - Shepherd 97
Chapman-Jouguet (CJ) Model
Thermodynamics and elementary gas dynamicsAdequate to predict ideal wave speed
Combustion wave moves at minimum speed consistent with conservation of mass, momentum and energy across the wave front. Equivalent to productsaway from wave front with a relative velocity equal to the speed of sound “sonicor CJ condition”
Structural Response - Shepherd 98
Structural Response - Shepherd 99
ZND Model
UCJReactantsProductsProducts
•Steady reactive flow behind nonreactive shock•Shock-induced chemical reaction•1D “smooth” flow – no instabilities
Radicals
shoc
k
Structural Response - Shepherd 100
Chemical Length and Time Scales
0.8 1 1.2 1.4
10-2
10-1
100
Normalized velocity, U/UCJ
Indu
ctio
n Zo
ne le
ngth
, cm
0 0.5 10
1000
2000
3000
0
0.01
0.02
0.03
0.04
0.05
OH
T
Distance, cm
Tem
pera
ture
, K
OH
mol
e fra
ctio
n2H2-O2-60%N2
Δ
Structural Response - Shepherd 101
Measured Pressures in Tube
Structural Response - Shepherd 102
Taylor-Zeldovich Expansion Wavecl
osed
end
Lx
particle path
t
0
open
end
2
1 - at rest
3
detonation
expansion fan
Stationary region
Structural Response - Shepherd 103
Propagating Pressure Wave
Structural Response - Shepherd 104
Wave Front Has Structure
End plate soot foil
Structural Response - Shepherd 105
Summary on Detonation Facts
• Detonations have– Characteristic minimum speed (CJ model)– Characteristic peak pressure (CJ model)– Characteristic length scale (ZND model)
• Measure cell width
• Imposes traveling load on tube– Sudden jump in pressure– Decrease in pressure followed by uniform region
Structural Response - Shepherd 106
Detonations Excite Elastic Waves
Structural Response - Shepherd 107
Modeling Structural Response To Detonations
• SDOF model for hoop oscillations• Simplified traveling wave model
– Beam on an elastic foundation• Analytical shell models
– (Tang) with rotary inertia• Numerical simulation
– Shell models (Cirak)– FEM models (LS-Dyna)
Need to add mathematical equations
Structural Response - Shepherd 108
Flexural Waves in Tubes
• Coupled response due to hoop oscillations and bending
• Traveling load can excite resonance when flexural wave group velocity matches wave speed
• Can be treated with analytical and FEM models
Measured strain (hoop)
t (ms)0 2 4 6 8
10-4
Amplification factor
U (m/s)
Structural Response - Shepherd 109
Measuring Elastic Vibration
Structural Response - Shepherd 110
rigid collets
stiff I-Beam
straingages
Precision test rig
Structural Response - Shepherd 111
120o
S1
S2
S3S4 S5 Detonationwave
D=41mm
20mm 20mm
Strain gages:radial spacing: S1, S2, S3axial spacing: S3, S4, S5
vibrometer
S4
S3S5
vibrometer
Gage locations
Structural Response - Shepherd 112
Comparison of shell model with experiment
15o location
Structural Response - Shepherd 113
Fracture
Fracture
External Blast
Structural Response - Shepherd 114
Strain Gage
Locations
Strain Response of Fracturing TubesStrain Response of Fracturing Tubes
Structural Response - Shepherd 115
Structural Response - Shepherd 116
Post-test Al 6061-T6 Specimens (Pcj = 6.2 MPa)
Surface Notch Length = 1.27 cm
Outer diameter: 41.28 mm, Wall thickness: 0.89 mm, Length: 0.914 mSurface notch dimensions: Width: 0.25 mm, Notch depth: 0.56 mm, Lengths: 1.27 cm, 2.54 cm, 5.08
cm, 7.62 cm
Detonation wave direction
Surface Notch Length = 2.54 cm
Surface Notch Length = 5.08 cm
Surface Notch Length = 7.62 cm
Fracture Behavior is a Strong Function of Initial Flaw Length
Structural Response - Shepherd 117
Fracture Threshold Model
Flat Plate Model
analyzed by Newman and Raju (1981)
Actual tube
surface
Fracture Condition:
(ΦΔpR/h)√(πd)/KIc > √(Q)/F
where Q, F = functions of flaw length (2a), flaw depth (d), and wall
thickness (h)
Approximate
Structural Response - Shepherd 118
Note: 1) Parameters on the axes are
non-dimensional2) Threshold is a 3-D surface
ΔP = Pcj - PatmR = Tube mean radiush = Tube wall thicknessd = Surface notch depth2a = Surface notch lengthKIc = Fracture toughnessΦ = Dynamic
Amplification factor
• Tube material: Al6061-T6• Wall thickness: 0.089 to 0.12
cm• d/h: 0.5 to 0.8• Pcj: 2 to 6 MPa• Axial Flaw Length: 1.3 to 7.6
cm• O.D.: 4.13 cm
RuptureNo RuptureThreshold Theory
Fracture Threshold of Flawed Tubes under Detonation Loading
Structural Response - Shepherd 119
Using Prestress to Control Crack Propagation Path
Detonation direction
Structural Response - Shepherd 120
Incipient Crack Kinking
Detonation Direction
Torque Direction(right-hand rule)
Initial Notch
HoopStress
ShearStress
HoopStress
ShearStress
Kinked Incipient Forward and Backward Cracks
Image from Shot 153
Initial Notch
Structural Response - Shepherd 121
Mixed-Mode Fracture
• Experimental data are compared with numerical data by Melin (1994) using a local kII = 0 criteria
Circles: Forward CracksDeltas: Backward Cracks
Stress Intensity Factors
Structural Response - Shepherd 122
Effect of Reflected Shear Wave: Crack Path Direction Reversal
• Cracks initially kinked at angles consistent with principal stresses
• The cracks then reversed directions due to reflected shear waves
• Shear wave travel time: 150 μs
Shot 143
Structural Response - Shepherd 123
Effect of Reflected Shear Wave: Crack Path Direction Reversal
Shear Strain Reversal
Detonation Wave Direction
Rosette 1 (solid) Rosette 2 (dotted)
Structural Response - Shepherd 124
Effect of Reflected Shear Wave: Additional Kinked Crack
Shot 142
Structural Response - Shepherd 125
Application to Pulse Detonation
• Pulse detonation engine use repeated detonations to generate thrust
• In development as primary thrust generator (ramjet-type device) and high pressure combustion chamber for jet engines
Structural Response - Shepherd 126
Testing at WPAFB
Thanks to John Hoke, Royce Bradley and Fred Schauer
Structural Response - Shepherd 127
Crack Opening – deep flaw
After 4700 cycles After 7500 cycles
Structural Response - Shepherd 128
DDT
• Deflagration to detonation transition is a common industrial hazard with gaseous explosions
• Compression of gas by flame increases pressure when detonation finally occurs “pressure piling”.
• Represents upper bound in severity of pressure loading.
Structural Response - Shepherd 129
The path of DDT
Structural Response - Shepherd 130
burned unburned
1. A smooth flame with laminar flow ahead
2. First wrinkling of flame and instability of upstream flow
3. Breakdown into turbulent flow and a corrugated flame
4. Production of pressure waves ahead of turbulent flame
5. Local explosion of vortical structure within the flame
6. Transition to detonation
Structural Response - Shepherd 131
Slow Flame (Deflagration)
Structural Response - Shepherd 132
Fast Flame
Structural Response - Shepherd 133
DDT after Flame Acceleration Period
Structural Response - Shepherd 134
Rapid onset of DDT
Structural Response - Shepherd 135
Structural Response to DDT
Thick walled vessels for elastic responseThin-walled vessels for plastic response and failure
Use bars or tabs as “obstacles” to cause flame acceleration
Structural Response - Shepherd 136
Reflection of near-CJ Detonation
30% H2 in H2-N2O mixture at 1 atm initial pressure
Structural Response - Shepherd 137
DDT near end flange
15% H2 in H2-N2O at 1 atm initial pressure
Structural Response - Shepherd 138
Summary of results for H2-O2 Mixtures
Strains and pressures are a strong function of composition, peak occurs whenDDT is close to the end of the tube.
Structural Response - Shepherd 139
Structural Response - Shepherd 140
Computations of Detonation Reflection
• 3-in Schedule 40 316L pipe 1-m long, 38 mm diam, 4.5 mm wall 240 MPa yield stress
• Reflected CJ detonation. CJ Velocity 2600 m/s, PCJ/Po = 26
• Three initial pressures 3, 6, 9 atm
• LS-DYNA simulation with traveling load model of waves
Structural Response - Shepherd 1413 atm
Structural Response - Shepherd 1426 atm
Structural Response - Shepherd 1439 atm
Structural Response - Shepherd 144
Spatial distribution of Effective Plastic Strain
3 atm
6 atm
9 atm
Structural Response - Shepherd 145
Plastic Deformation
• It is useful to use plastic deformation to accommodate rare events.
• Need to have more data and modeling to determine peak allowable impulses and pressures to avoid rupure.
Structural Response - Shepherd 146
Bends and Tees
• Limited data available• Important for plants and facilities• Some enhancement of hoop load due to wave
reflections• Transverse loads can be quite significant
– Creates bending in tubes– Supporting structures (hangers) can fail– Flange bolts can fail in shear due to transverse loads
Structural Response - Shepherd 147
Detonations and ASME Code Rules
• Not covered under current BPVC VIII or Piping Code B31• Proposed code case for impulsively loaded vessels is under
development by ASME Task Force on Impulsively Loaded Vessels, SWG/HPV, ASME VIII.
• Current impulsive loading code case intended to cover vessels used to contain high explosive detonation.– many common elements associated with dynamic loading– Further work needed to treat gas detonation specific issues
Structural Response - Shepherd 148
Issue for Gaseous Detonation
• Loading is more difficult to define for gases than for HE detonation– More testing is needed to have generic results
• Mixed loading regime, not purely impulsive.• Plastic deformation will require considering entire
loading history.• Traveling load aspects of gaseous detonation
Structural Response - Shepherd 149
Extending the Code
• Ad hoc design practices can be standardized
• Analysis of accidents and DDT harder to standardize
• Designers and analysts might be able to use extended code as a basis for building vessels and
piping to contain gaseous detonation
– Elastically for high frequency or intentional events
– Plastically for rare events or one-time use
• Much work has already been done for impulsively loaded vessels code case development
– Dynamic response of materials
– Stain hardening, strain rate effects
– Fracture safe design
– Plastic instability limits (incomplete)